the relative impact of the dual-rotor wind turbine and the ... · i hereby certify that this...

The Relative Impact of the Dual-rotor Wind Turbine

and the Conventional Single-rotor Wind Turbine on

the Transient Stability of the Power System

Ehsan Mostery Farahani

Thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Faculty of Engineering and Industrial Sciences

Swinburne University of Technology

2014

i

Abstract

Comparison of the aerodynamic efficiency of the single-rotor and dual-rotor wind turbine

(SRWT and DWRT, respectively) demonstrates the superiority of the latter; at the same

wind speed, the DRWT is almost 9% relatively more efficient than the SRWT. Based on

this reported advantage, the DRWT should possess a good potential to be manufactured and

installed in new wind farms. However, a comparison of the transient responses of these two

types of wind turbine technology is required before concluding that the performance of the

DRWT is superior to that of the SRWT.

In this dissertation, a comparative assessment of the transient responses of the DRWT and

the SRWT is carried out with respect to four categories of power system transient stability:

transient angle stability; transient voltage stability; transient frequency stability; and sub-

synchronous resonance. As a first step, using the multi-objective drive train method, an

accurate dynamic model is developed for the prime mover of the DRWT, including shafts,

corresponding gearboxes, hubs and blades.

The impact of the DRWT on the grid fault-ride-through capability is evaluated and

compared with the SRWT through mathematical modelling, eigenvalue analysis and

numerical simulations. Both quantitative and numerical approaches show that DRWT-

based wind farms improve the transient angle stability margin of the network in comparison

to SRWT-based wind farms. The main reason for this is the lower rate of speed change in

the DRWT.

Further, the transient frequency control capability of the DRWT is compared with that of

the SRWT for three different de-loading methods: a pitching control method; a sub-optimal

method; and a combination of both methods. It is shown that, by appropriate tunning of a

droop controller integrated into the pitch control system, the DRWT is more successful in

arresting the transient frequency deviation in the first and third controlling modes; in

comparison, the SRWT is more advantageous in suboptimal mode. It is also recognized that,

in addition to the released kinetic energy (KE), the transient variation of the aerodynamic

energy is also influential in the inertial response of the wind turbine, whereas, in the

literature prior to this work the only factor considered for the study of the inertial response

was the impact of the released KE.

ii

In the third category, the relative impact of the DRWT and SRWT on the transient

voltage stability margin is also assessed. The current popular evaluation criterion – the

distance between the stable and unstable operating points of the induction generator – is not

followed in this study due to some identified drawbacks. The proposed criterion is the

comparative peak of transient apparent power delivered during the transient period by the

generating units. The validity of the proposed criterion is tested in the maximum power

point tracking (MPPT) mode. For fixed-speed induction generator (FSIG) technology, the

transient margin of the SRWT is found to be higher. In the presence of double-fed

induction generator (DFIG), no significant difference is observed. However, in the case of

DFIG reactive power saturation, the margin of the SRWT is superior to that of the DRWT.

Finally, in the fourth category, it is revealed that the risk of torsional interaction (TI) and

torsional amplification (TA), as subcategories of sub-synchronous resonance (SSR), is

higher in the DRWT than the SRWT. To avoid interactions between the torsional

frequencies and the grid natural frequency, a genetic algorithm (GA) is designed to

optimize the mechanical parameters of the DRWT in order to remove the torsional

frequencies off the high-risk area (22Hz < f < 42Hz). This method dramatically reduces the

risk of TI-SSR and TA-SSR in the DRWT, while also meeting industrial limitations.

iii

Acknowledgements

I am deeply indebted to my Principal Coordinating Supervisor, Dr. Mehran Ektesabi, for

his constant support. I thank him for supervising my research work incessantly and

providing me with valuable advice, and above all for his technical, financial and emotional

support.

I owe my deepest gratitude to my former Principal Coordinating Supervisor (first two

years), Dr. Nasser Hosseinzadeh, whose support and guidance enabled me to accomplish

the work.

I would like to thank my External Supervisor, Dr. Chandra Kumble, for his valuable

technical advice and numerous creative suggestions.

I would also like to thank my parents for their encouragement and love. Thank you for all

the sacrifices you have made to give me a better chance in life.

iv

Declaration

I hereby certify that this thesis, entitled “Relative Impact of Dual-rotor Wind Turbine

on Transient Stability of Power System versus Conventional Single-rotor Wind Turbine” is

my own work, except where due reference is made in the text and that, to the best of my

knowledge, it has not been submitted to this university or to any other university or

institution for the purposes of obtaining a degree.

Signed

Ehsan Mostery Farahani

v

List of Figures

Fig. 2.1: A 30kW dual-rotor wind turbine system ............................................................... 12

Fig. 2.2: Turbine airflow on upstream and downstream [4] ................................................ 14

Fig. 2.3: Effect of auxiliary rotor length on the increase of power [2] ................................ 15

Fig. 2.4: Geometry and dimensions of main and auxiliary rotors [2] .................................. 16

Fig. 2.5: Impact of interval on the power generation growth [2] ......................................... 17

Fig. 2.6: Power of the DRWT versus the SRWT ................................................................. 17

Fig. 2.7 Categories of power system stability regarding the severity of disturbance .......... 19

Fig. 2.8 Classification of power system [11] ....................................................................... 27

Fig. 2.9. DFIG Equivalent circuit under sub-synchronous frequency ................................. 28

Fig. 2.10. Network resonance mode at various wind speed and level of compensation [30]

.............................................................................................................................................. 28

Fig. 2.11 Fixed speed wind turbine with induction generator ............................................. 31

Fig. 2.12 Fully-rated converter wind turbines ...................................................................... 31

Fig. 2.13. DFIG wind turbine ............................................................................................... 32

Fig. 2.14. DFIG capability curve [72] .................................................................................. 38

Fig. 2.15. Diagram of linear NGH damper .......................................................................... 40

Fig. 2.16. Three-phase transformer with blocking filters..................................................... 41

Fig. 2.17. SSR damping controller implemented in SVC [28] ............................................ 41

Fig. 3.1 General form of an N-mass drive train ................................................................... 44

Fig. 3.2. Mechanical elements of a two-mass system .......................................................... 45

Fig. 3.3. Element arrangement of SRWT and DRWT ......................................................... 47

Fig. 3.4. Employed gears in SRWT and DRWT .................................................................. 48

Fig. 3.5. Dynamic model of one stage spur gear box ........................................................... 48

Fig. 3.6. Dynamic model of the 2 stage bevel gear with 3 shafts ........................................ 49

Fig. 3.7. The profile of the stiffness ..................................................................................... 51

Fig. 3.8 Two-mass model of the blades ............................................................................... 55

Fig. 3.9. General mechanical block diagram of variable speed wind turbine ...................... 56

Fig. 3.10. Induction generator circuit model ........................................................................ 59

Fig. 3.11. Torque speed characteristic curve........................................................................ 60

Fig. 3.12. The post-fault excursion of the operating point ................................................... 62

vi

Fig. 3.13. Generator constant speed pitch control system ................................................... 64

Fig. 3.14. Stream tube effect of the auxiliary turbine on the aerodynamic performance of

the main turbine.................................................................................................................... 68

Fig. 3.15. Two rotors are aerodynamically independent ...................................................... 68

Fig. 3.16. Simple power grid connected to either single-rotor or dual-rotor wind turbine . 69

Fig. 3.17. Dynamic response of the wind turbine to a three phase grid short circuit when

system stays stable ............................................................................................................... 71

Fig. 3.18. Instability of single-rotor wind turbine when pitch angle control is disabled .... 73

Fig. 4.1 Impact of generator speed variation on the transient aerodynamic generation ...... 81

Fig. 4.2. A set up for evaluating the impact of momentum of inertia on kinetic energy

generation ............................................................................................................................. 87

Fig. 4.3. Responses of induction generator to sudden rise of energy demand when it is

driven by diesel engine......................................................................................................... 88

Fig. 4.4. Transient excursion of operating points of SRWT and DRWT for high wind

speeds ................................................................................................................................... 91

Fig. 4.5. Sub-optimal control mode for low speed winds .................................................... 94

Fig. 4.6. Combination of pitch control mode and over-speed mode for medium speed winds

.............................................................................................................................................. 97

Fig. 4.7. Control diagram for the SRWT and DRWT coupled to the grid through FRC ..... 98

Fig. 4.8. Employed power system for the tests .................................................................. 102

Fig. 4.9. Responses of generating units of SRWT and DRWT to the load switching in pitch

control mode....................................................................................................................... 103

Fig. 4.10. Responses of generating units of SRWT and DRWT to the load change for sub-

optimal mode ...................................................................................................................... 106

Fig. 4.11. Responses of DRWT-FRC and SRWT-FRC with droop controller in action ... 108

Fig. 5.1. Overlay of torque-speed characteristics of generator and mechanical drive ....... 113

Fig. 5.2 An abnormal condition addressed as voltage unstable by [70] ............................ 114

Fig. 5.3 Voltage collapse due to the voltage instability after a big load switching ........... 115

Fig. 5.4. Excursion of operating point on the aerodynamic curve ..................................... 120

Fig. 5.5. A simple pitch angle controller for regulating the mechanical torque ................ 121

vii

Fig. 5.6. Rate of speed change in the DRWT and SRWT as a response to electromagnetic

step up (the damping factor is neglected) .......................................................................... 123

Fig. 5.7. Slowdown of generators as a response to sudden growth in load ....................... 123

Fig. 5.8. Comparing the excursion of operating points of SRWT and DRWT on the

aerodynamic curve ............................................................................................................. 124

Fig. 5.9. Transient excursion of operating points of SRWT and DRWT while the effect of

both ∆PPI and ∆PEX are involved ........................................................................................ 126

Fig. 5.10. Post-disturbance responses of active power of DRWT and SRWT to the growth

of load................................................................................................................................. 126

Fig. 5.11. Coefficient of active power (QFACT) as a function of slip in generating mode .. 128

Fig. 5.12. Post-disturbance response of active power of the DRWT and the SRWT to the

growth of load .................................................................................................................... 129

Fig. 5.13. Control block diagram of DFIG wind turbine ................................................... 130

Fig. 5.14. Impact of DFIG on steady state voltage stability margin .................................. 131

Fig. 5.15. Capability curve of DFIGs ................................................................................. 132

Fig. 5.16. Physical layout of a typical wind farm .............................................................. 133

Fig. 5.17. Capability curve of grid connection point ......................................................... 134

Fig. 5.18. Employed power system for the tests ................................................................ 137

Fig. 5.19. Response of DRWT and SRWT FSIG-based wind turbine to the large

disturbance ......................................................................................................................... 138

Fig. 5.20. Response of the DRWT and the SRWT DFIG-based wind turbine to disturbance

............................................................................................................................................ 140

Fig. 5.21. Response of DRWT and SRWT DFIG-based wind turbine to disturbance when

DFIG can’t match the reactive power demand .................................................................. 143

Fig. 6.1 Structure of five-mass shaft system of SRWT...................................................... 148

Fig. 6.2. Structure of eight-mass shaft system of DRWT .................................................. 148

Fig. 6.3. Response of the system with identical eigenvalues versus normal system ......... 165

Fig. 6.4. Average and best fitness of each population ....................................................... 167

Fig. 6.5. The optimization flowchart for pushing away the torsional frequencies of the high-

risk range ............................................................................................................................ 168

Fig. 6.6 Test set up to check the damping of the system ................................................... 170

viii

Fig. 6.7. Dynamic response of DRWT after updating the parameters in speed control mode

............................................................................................................................................ 170

Fig. 6.8. Dynamic response of DRWT after updating the parameters in torque control mode

............................................................................................................................................ 171

Fig. 7.1. Gearless DRWT [114] ......................................................................................... 178

Fig. 7.2. Configuration of the employed generator in the DRWT [114] ........................... 178

ix

List of Abbreviations

AGC Automatic Generation Controller

BFG Bottom Frequency Group

DFIG Double-fed Induction Generator

DRWT Dual-rotor Wind Turbine

FACTS Flexible Transmission ac Systems

FRC Fully-rated Converter

FSIG Fixed-speed Induction Generator

GA Genetic Algorithm

GSC Grid Side Converter

HVDC High Voltage Direct Current

IGE Induction Generator Excitation

KE Kinetic Energy

kW Kilo Watt

MW Mega Watt

MPPT Maximum Power Point Tracking

NGH N.G. Hingorani

PI Proportional Integral

RSC Rotor Side Converter

SRWT Single-rotor Wind Turbine

SSR Sub-synchronous Resonance

STATCOM Static Compensator

SVC Static Var Compensator

TA Torsional Amplification

TCSC Thyristor Control Series Capacitor

TI Torsional Interaction

TICU Torsional Interaction between Closely Coupled Units

TSO Transmission System Operator

UFG Upper Frequency Group

ULTC Under Load Tap Changer

x

VSWTs Variable Speed Wind Turbines

WTGS Wind Turbine Generator System

xi

Table of Contents

Abstract ................................................................................................................................. ii

Acknowledgements ............................................................................................................... iv

Declaration ............................................................................................................................. v

List of Figures ....................................................................................................................... vi

List of Abbreviations.............................................................................................................. x

Chapter 1 Introduction ........................................................................................................... 1

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

1.2 Contributions ................................................................................................................ 2

1.3 Thesis Overview ........................................................................................................... 6

1.4 Publications .................................................................................................................. 9

Chapter 2 : Literature Review .............................................................................................. 11

2.1 Introduction ................................................................................................................ 11

2.2 Dual-rotor Wind Turbine ............................................................................................ 12

2.3 Power System Stability Classification ....................................................................... 17

2.3.1 Stability Classification Regarding the Severity of Disturbance .......................... 18

2.3.1.1 Steady State Stability ................................................................................... 18

2.3.1.2 Dynamic Stability ........................................................................................ 18

2.3.1.3 Transient Stability ........................................................................................ 18

2.3.2 Stability Classification Regarding the Time of Interest....................................... 19

2.3.2.1 Short-term Stability ...................................................................................... 19

2.3.2.2 Mid-term and Long-term Stabilities ............................................................ 19

2.3.3 Stability Classification Regarding the quantity sources the instability ............... 20

2.3.3.1 Angle Stability ............................................................................................. 20

xii

2.3.3.2 Frequency Stability ...................................................................................... 22

2.3.3.3 Voltage Stability .......................................................................................... 24

2.3.3.4 Sub-synchronous Resonance ........................................................................ 25

2.3.3.5 Sub-synchronous Resonance Definition and Types ..................................... 27

2.3.3.5.1 Induction Generator Effect .................................................................... 27

2.3.3.5.2 Torsional Interactions (TI) .................................................................... 29

2.3.3.5.3 Torsional Amplifications (TA) ............................................................. 29

2.3.3.5.4 Torsional Interaction between Closely Coupled Units ......................... 30

2.3.3.6 Methods for Analysing the SSR ................................................................... 30

2.3.3.7 Risk of SSR for Different Wind Turbine Technologies............................... 30

2.3.3.7.1 SSR Risk in Fixed Speed Wind Turbines ............................................. 30

2.3.3.7.2 SSR Risk in Fully-rated Converter Wind Turbines (FRC) ................... 31

2.3.3.7.3 SSR Risk in Double-fed Induction Generators ..................................... 31

2.4 Method to Assess the Impact of DRWT on the Transient Stability ........................... 32

2.4.1 Method to Assess the Impact of Wind Farms on Transient Angle Stability ....... 32

2.4.2 Method to Assess the Impact of Wind Farms on Transient Frequency Stability 34

2.4.3 Method to Assess the Impact of Wind Farms on Transient Voltage Stability .... 36

2.4.4 Method to Assess the Impact of Wind Farms on Transient Sub-synchronous

Resonance ..................................................................................................................... 39

Chapter 3 : Impact of DRWT on Transient Angle Stability of Network ............................. 43

3.1 Introduction ................................................................................................................ 43

3.2 Drive Train Modeling through Multi-objective Method ............................................ 44

3.3 Dynamic Model of DRWT and SRWT Components ................................................. 47

3.3.1 Gear Box .............................................................................................................. 47

3.3.2 Blade Bending Model .......................................................................................... 54

xiii

3.3.3 Shaft and Rotor System ....................................................................................... 55

3.4 SRWT and DRWT State Space Model ...................................................................... 57

3.4.1 Induction Generator Model .................................................................................. 57

3.4.2 Turbine Model ..................................................................................................... 57

3.5 Critical Rotor Speed of Induction Generator.............................................................. 58

3.6 Damping Effect of Droop Control System ................................................................. 64

3.7 Aerodynamic Model for DRWT ................................................................................ 66

3.8 Simulation Results ...................................................................................................... 69

3.9 Conclusions ................................................................................................................ 74

Chapter 4 Investigating the Impact of Dual-Rotor Wind Turbine on the Transient Network

Frequency ............................................................................................................................. 76

4.1 Introduction ................................................................................................................ 76

4.2 Impact of Aerodynamic Energy versus Kinetic Energy ............................................. 77

4.2.1 Kinetic Energy ..................................................................................................... 78

4.2.2 Aerodynamic Transient Response According to Area of Operation ................... 79

4.3 Kinetic Energy Released by SRWT and DRWT ........................................................ 84

4.3.1 Impact of Momentum on KE versus Speed Variation ......................................... 85

4.3.2 KE Ratio between SRWT and DRWT ................................................................ 86

4.4 Frequency Control Capability of SRWT and DRWT Based on Control Mode ......... 89

4.4.1 SRWT and DRWT in Pitch Control Mode .......................................................... 90

4.4.2 SRWT and DRWT in Sub-optimal Mode ........................................................... 93

4.4.3 SRWT and DRWT in Combination Mode .......................................................... 96

4.5 Simulation Results .................................................................................................... 100

4.6 Conclusion ................................................................................................................ 109

Chapter 5 The Impact of DRWT on the Short-term Voltage Stability of the Power System

............................................................................................................................................ 111

xiv

5.1 Introduction .............................................................................................................. 111

5.2 Critical Rotor Speed as the Current Approach ......................................................... 112

5.3 Analyzing the Validity of the Critical Speed Method for Voltage Stability

Assessment ..................................................................................................................... 113

5.3.1 Voltage Collapse due to the Voltage Instability or Angle Instability ................ 113

5.3.2 Generator Speed Reaction to the Short-term Voltage Instability ...................... 115

5.3.3 Influential Factors on Voltage Stability and Angle Stability ............................. 116

5.3.4 Disturbances which Lead to Generator Deceleration ........................................ 116

5.4 Proposed Method for Assessing the Impact of DRWT on Large Disturbance Voltage

Stability .......................................................................................................................... 117

5.4.1 Transient Response of Active Power ................................................................. 118

5.4.2 Transient Response of Reactive Power ............................................................. 126

5.5 Performance of DRWT and SRWT at the Presence of DFIG .................................. 130


5.7 Conclusion ................................................................................................................ 144

Chapter 6 Impact of DRWT-based Wind farms on SSR Risk ........................................... 146

6.1 Introduction .............................................................................................................. 146

6.2 Modeling the DRWT and the SRWT for Torsional Studies .................................... 147

6.2.1 State-space model of the SRWT ........................................................................ 148

6.2.2 State Space Model of the DRWT ...................................................................... 152

6.3 Proposed Method to Reduce the Risk of SSR .......................................................... 158

6.4 Genetic Algorithm Settings ...................................................................................... 159

6.4.1 Problem Formulation ......................................................................................... 159

6.4.2 Initialization ....................................................................................................... 160

6.4.3 Chromosome Fitness ......................................................................................... 161

6.4.4 Selection............................................................................................................. 162

xv

6.4.5 Crossover ........................................................................................................... 162

6.4.6 Mutation ............................................................................................................. 162

6.4.7 Crossover Fraction ............................................................................................. 163

6.4.8 Population Size .................................................................................................. 163

6.4.9 Stopping Criteria ................................................................................................ 163

6.5 The Constraints of the Proposed Method ................................................................. 163

6.5.1 No Torsional Frequency in High Risk Range .................................................... 163

6.5.2 Torsional Frequency Combination .................................................................... 164


6.7 Conclusion ................................................................................................................ 171

Chapter 7 Conclusion and Future Works ........................................................................... 173

7.1 Future Works ............................................................................................................ 177

7.1.1 Gearless DRWT ................................................................................................. 177

7.1.2 Impact of DRWT on the Transient Voltage Stability Margin in Sub-optimal

mode............................................................................................................................ 179

7.1.3 Impact of the DRWT on the Risk of the IGE-SSR ............................................ 179

7.1.4 Using GA to Reduce the Risk of SSR in SRWT ............................................... 180

7.1.5 Down Time Evaluation of DRWT in Comparison to SRWT ............................ 180

Bibliography ....................................................................................................................... 181

xvi

Introduction Chapter 1

1.1 Background

In recent times, the replacement of conventional generating units by renewable energy

sources has increased significantly. The main reasons for this development lie in the

limited capacity of fossil fuels like oil and coal, and global warming due to the high rate

of pollution; conversely, renewable generating units are pollution free and don’t require

expensive fuel. However, the technologies of renewable energy are still expensive and

their availability is not as high as the conventional thermal plants. The main types of

energy generated by natural resources are wind, solar, geothermal, ocean wave, and

tidal energy. Among these energy sources, wind energy is one of the fastest-growing

technologies. There are two main categories of wind energy units – horizontal-axis and

vertical-axis; in the latter category, a special form of wind turbine is a drag-type device

called savonius. The former category has the largest share in the market. One of the

challenges in this area is finding a way to increase the aerodynamic efficiency of the

wind turbines as much as possible. The tip portions of the blades in the conventional

horizontal single-rotor wind turbines (SRWT) are not able to capture energy and

introduce a low aerodynamic resistance pathway to the upcoming wind. Consequently,

the wind can escape through these tip portions without delivering any energy, thereby

causing a reduction in efficiency. To overcome this problem, the dual-rotor wind turbine

(DRWT) has been introduced to enhance aerodynamic efficiency by placing an

auxiliary turbine on the route of the escaped wind. Previous reports have shown that, at

the same wind speed, the DRWT is relatively almost 9% more efficient than the SRWT

when the length of the auxiliary turbine is one-half of that of the main turbine and when

the turbines are placed apart at a distance of one-half of the auxiliary blade length.

It has been demonstrated that the steady state performance of the DRWT with respect

to aerodynamic efficiency is better than the corresponding the SRWT at the same

condition. To assess the potential of a technology to successfully gain enough shares in

the market, it should be seen as competitive, from a number of different aspects,

alongside other products or technologies. Therefore, before making any decision about

the commercialization of the DRWT, it should be compared with the SRWT in terms of

the initial investment versus the long-term profit, energy efficiency, the amount of down

time and the impacts of this technology on the characteristics of the local power system. 1

The current research aims to be a starting point in the investigation of the impact of

DRWT-based wind farms on the electrical characteristics of the local grid. Before

connecting any new generating unit, regardless of its technology, some feasibility

studies should be performed to approve the grid connection. Among the many

characteristics requiring investigation, the main focus of this study is on the effect of the

DRWT on the transient stability of the grid. Specifically, the impact of the DRWT on

four subcategories of the transient stability margin is investigated here. These

subcategories are: transient angle stability; transient frequency stability; transient

voltage stability; and transient sub-synchronous resonance. These categories of stability

have been chosen for investigation based on the connection impact studies scheduled by

ABB Company1 – a pioneer in the electrical industry.

Although other configurations of the DRWT are introduced to enhance the

performance of this technology, the focus of this dissertation is on the T gearbox type of

the DRWT. For example, one promising configuration is created if two rotors are

directly coupled to an asynchronous electrical machine; one rotor is connected to the

stator of the induction generator and the other rotor is coupled to the rotor of the

induction generator. Due the lack of a gearbox, this version of the DRWT is termed a

‘gearless DRWT’ here and has different transient characteristic in comparison with the

T gearbox DRWT.

1.2 Contributions

The contributions of this investigation are fourfold. Firstly, it provides a starting point

for analysing, comparatively, the relative impact of the DRWT-based wind farm and the

SRWT on the transient angle stability margin of the power system. In this regard the

following findings are mentioned here:

- The dynamic model of the mechanical drive of the DRWT is formed using a multi-

objective approach.

- The dynamic model of the three-shaft bevel gear is developed here as an interface

between the generator and the auxiliary and main turbines.

- The natural damping characteristic of the DRWT and the SRWT is compared

through eigenvalue analysis. It is found that, after adding the dynamic equations of

the auxiliary turbine to the state space model of the SRWT to create the model of the

1 http://www.abb.com/industries/db0003db004333/c12573e7003305cbc12570130037e75d.aspx?tabKey=2 2

DRWT, the real part of the eigenvalues are shifted more to the left. It means the

DRWT presents more natural damping than the SRWT.

- It is found that the acceleration and deceleration rates of the DRWT are less than

that of the SRWT under the same conditions. This is due to the extra momentum

inertia added by the auxiliary turbine.

- Through employing the most popular method for analysing the large-disturbance

stability of induction generators – called as ‘critical rotor speed’ method – it is

recognized that the transient angle stability margin of the DRWT is higher than that

of the SRWT. The main reason is the lower acceleration rate of the DRWT. During

the fault, the generator accelerates and the stable operating point approaches the

unstable operating point (critical speed). The criterion of the critical rotor speed

method is the minimum distance that the stable and unstable (critical) operating

points reach during the transient period. The higher the minimum distance reached,

the more the transient margin is achievable.

- The degree of the damping effect of the droop loop integrated into the pitch angle

control system is assessed for both the DRWT and the SRWT. It is shown that the

degree of damping of the DRWT is higher than that of the SRWT.

- The stream tube effect of the auxiliary turbine is included in the calculation of the

average wind speed on the main turbine. Thus, the aerodynamic model of the

DRWT in this study is more accurate in comparison with the introduced

aerodynamic model of the DRWT; this was ignored by the only report previously

published in the electrical area.

The second contribution of this study lies in its initiation of an evaluative analysis of

the relative effects of the DRWT-based wind farm and the conventional SRWT on the

transient frequency stability margin of the local power system. The study is performed

for three different de-loading modes, including: pitch control de-loading mode; sub-

optimal curve de-loading mode; and a combination of both modes – termed here the

‘combination mode’. The finding in this area is as follows:

- Through using sensitivity analysis, it is shown that the inertial response

characteristic of the wind turbines is also influenced by the transient excursion of

the operating point along the aerodynamic curve, in addition to the kinetic energy

(KE) released by the body-mass of the turbine. While in previous reports wind

turbines were treated as synchronous generators and the KE was introduced as the

3

only factor dominating the inertial response of the wind turbines. In cases where the

excursion occurs in the under-speed area, it has a weakening effect on the inertial

response, while the transient displacement of the operating point in the over-speed

area has a boosting effect on the inertial response.

- The KE discharging potential of the DRWT and SRWT is compared and it is

revealed that the SRWT releases slightly more KE than the DRWT.

- It is recognized that in pitch de-loading mode, the transient frequency support

capability of the DRWT is better than that of the SRWT. The main reason is the

higher weakening effect of the operating point excursion on the inertial response of

the SRWT. This is due to the greater speed fall in the under-speed area of the

aerodynamic curve in comparison with the speed drop of the DRWT, under the

same conditions.

- It is found that in the sub-optimal de-loading mode, the SRWT is more effective in

limiting the network transient frequency deviation, in comparison with the DRWT.

This is because of the higher boosting effect of the operating point transient

displacement on the inertial response of the SRWT versus that of the DRWT. The

higher boosting effect is due to the relatively greater speed reduction of the SRWT

in the over-speed area compared to that of the DRWT.

- It is shown that in the combination mode, by appropriate tunning of a droop

controller integrated into the pitch control system, the DRWT is more successful in

arresting the transient frequency deviation. The term ‘appropriate tunning’ refers to

a suitable selection of the droop factor for the auxiliary turbine of the DRWT.

The third contribution consists of a comparative evaluation of the effects of the DRWT-

based wind farm and the conventional SRWT-based wind farm on the transient voltage

stability margin of the local power system. A new method is proposed as the assessment

tool, and the validity of the prediction is tested in the maximum power point tracking

(MPPT) mode for three energy conversion scenarios, including: FSIG, DFIG under

nominal conditions and DFIG when its supplied reactive power reaches the limit of its

capacity. A big load switching is chosen as the type of disturbance which leads to

voltage instability. The associated findings are as follows:

- The accuracy of the current most popular approach for assessing the transient

voltage stability margin of the IG-based generating units, called the ‘critical rotor

speed’ method is examined here and it is concluded that it cannot cover all aspects

4

of the transient voltage stability. In other words, this method was originally

introduced for calculating the transient angle stability margin of the induction

generators, and the predictive ability of this method in terms of the margin of the

transient voltage stability does not include all the influential factors.

- A method is proposed for assessing the impact of the DRWT on the voltage stability

margin of the network. The criterion of this method is the peak of the transient

apparent power delivered by the wind turbines during the transient period. The

higher the apparent power generated by the generating unit during the transient

period, the less the transient voltage stability margin can be predicted for the local

network.

- For the first scenario, with FSIG as the energy conversion system and big load

switching as the disturbance, it is found that the maximum transient apparent power

reached by the DRWT is higher than that of the SRWT in MPPT mode.

Consequently, for this scenario the transient voltage stability margin of the SRWT is

more than that of the DRWT, which confirms the prediction of the proposed

method.

- For the second scenario the energy conversion system is changed to DFIG. In reality

there is significant internal reactive power loss between the individual wind turbines

and the grid due to interface electrical components like cables, transformers and

transmission lines. Internal reactive losses in the wind farm have a significant

decreasing effect on the capacitive area of the capability curve, especially when the

wind farm is operating close to it nominal rating. So, to obtain the capability curve

of the wind farm, instead of an algebraic summation of the capability curves of the

individual wind turbines suggested by the current reports, it is recommended here to

calculate the capability characteristic curve of the grid connection bus of the wind

farm in order to make the study more practical. It is seen that, as long as the required

capacitive reactive power is matched by the wind farm, there is no significant

difference between the transient voltage support performances of both types of wind

turbine. However, when the reactive power supplied by the wind farm reaches the

reactive power capability curve of the grid connection bus during the transient

period, the SRWT maintains its advantage over the DRWT for the same reason as in

the first scenario. This signifies that the turbine with higher transient apparent power

generation capability introduces less voltage stability margin when the DFIG is

5

saturated, which confirms the validity of the method proposed for assessing the

transient voltage stability margin.

The fourth contribution marks a beginning in the risk assessment of sub-synchronous

resonance (SSR) in the DRWT-based wind farms as against that of the SRWT. The state

matrix for both the SRWT and DRWT is formed and the torsional frequencies are

calculated. It is found that the number of torsional frequencies is eight and five for the

DRWT and SRWT, respectively. It signifies the higher risk for the DRWT regarding the

torsional interaction SSR (TI-SSR) and torsional amplification SSR (TA-SSR) as

subcategories of the SSR. In this thesis, the main focus is on SSR risk reduction at the

design stage of the prime mover of the DRWT, rather than subsequently designing

control systems to dampen the SSR oscillations. A solution is proposed to reduce the

risk of neighbouring of the complementary of the grid natural frequency and the

torsional frequencies in the DRWT system. At first, a frequency range is identified as

the high-risk range that the complementary of the natural frequency of the grid normally

falls in. The high- risk area is recognized as falling between 22Hz and 42Hz. Then, a

GA is designed to delimit the torsional frequencies of the shaft system of the DRWT

from the high-risk range through optimizing the mechanical parameters of the DRWT.

It this way, it is possible to reduce the risk of the adjacent of the torsional frequencies

and the grid natural frequency. To do so, 12Hz and 52Hz are defined as the two target

frequencies outside of the high-risk range, and the GA is put in charge of making the

torsional frequencies, which are already inside the high-risk area, as close as possible to

the target frequencies. Whenever a chromosome meets the stopping criterion defined by

the fitness function, it will be checked by two constraints. A constraint is assigned to

supervise the margin between the torsional frequencies. If two torsional frequencies

approach each other, then the damping characteristics of the dynamic response will be

degraded. Another constraint should be assigned to check if there is any torsional

frequency in the high-risk area. If any torsional frequency falls in the area, then the

chromosome is rejected and the GA continues searching the solution space. The

designed GA is run and it should succeed in removing the torsional frequencies from

the high-risk area without harming the dynamic response of the DRWT.

1.3 Thesis Overview

The dual-rotor wind turbine was originally introduced to increase the aerodynamic

efficiency of the wind energy generating units and to enable the lifting of the energy 6

production of the conventional SRWT, through employing an auxiliary turbine. To

make this technology commercially viable, it should be able to compete with other

technologies in this area. This thesis deals with the impact of the DRWT on the power

system transient stability and the main purpose of the study is to open the way to further

research in this field in the future. To fulfil this target, four subcategories of transient

stability are covered here to meet the technical requirements of power system operators.

This research contains five main chapters, as follows:

Chapter 2 presents the review of the literature. Firstly, previous reports concerning the

DRWT and the main reasons for the initial development of this technology are

discussed. Then, the stability of the power system is classified in terms of the severity of

disturbance, that is, the stability vis-à-vis large and small disturbances; the time of

interest, such as short-term (transient) and long-term stabilities; and the quantity that

originates the instability, including voltage, angle and frequency. The factors that

dominate each type of instability are described here briefly. Finally, the pros and cons of

the previous approaches are discussed to inform the analysis of the impact of wind

farms on the angle transient stability, the frequency transient stability, the voltage

transient stability and the transient sub-synchronous resonance; this discussion is

located in the last section of this chapter.

In Chapter 3, the impact of the DRWT on the fault-ride through capability of the local

network is compared with that of the SRWT, through eigenvalue analysis, the critical

rotor speed method and the numerical simulation results. Initially, dynamic models of

the components used in the prime mover of both the SRWT and the DRWT are

developed. The dynamic models of individual components are linked to each other by

adopting a multi-objective approach to form dynamic models for the associated prime

movers of the DRWT and SRWT. Then, the state space models of the SRWT and the

DWRT are obtained as the material for eigenvalue analysis. Next, the transient angle

stability margins of the DRWT and the SRWT are compared analytically using the

critical rotor speed method. After that, the relative degree of the damping effect of the

integrated droop loop in the two types of wind turbine is studied quantitatively.

Subsequently, the aerodynamic model of the DRWT is modified through the inclusion

of the stream-tube effect of the auxiliary turbine in the model. Finally, numerical

simulations are carried out to verify the results obtained by analytical study and

eigenvalue analysis.

7

In Chapter 4, the impact of the DRWT on the transient frequency stability margin of

the local network is compared with that of the SRWT. At first the transient variation of

aerodynamic energy is recognized as another influential factor affecting the energy

generation capability of the wind turbine during the transient period, in addition to the

released kinetic energy (KE). Then the potential of the SRWT and the DRWT is

explored regarding the KE discharging throughout the transient period. Next, the

transient frequency control characteristic of the DRWT and the SRWT is assessed and

compared with regard to three different de-loading modes, such as the pitch angle de-

loading mode, the sub-optimal de-loading mode and a combination of these modes.

After that, the degree of impact of the integrated droop controller on the frequency

support characteristic of the SRWT and the DRWT is studied. Finally, numerical

simulations are performed to verify the validity of the analytical claims discussed in the

previous sections.

In Chapter 5, the impact of the DRWT on the transient voltage stability margin of the

local network is compared with that of the SRWT. At first the current popular method

termed as the ‘critical rotor speed’ method, has been already introduced for analysing

the short-term voltage stability margin of the induction generator-based wind farms, is

described briefly. Then, it is explained why this approach is not appropriate for

evaluating the impact of the DRWT on the transient voltage stability margin. Next, a

new criterion is proposed for evaluating the relative transient voltage supporting

characteristic of the DRWT and SRWT. To test the validity of the proposed method,

three scenarios are designed which all of them operate in maximum power point

tracking mode (MPPT). For the first scenario FSIG technology is used as the energy

conversion system. For the other two scenarios, the voltage regulation performances of

the DRWT and the SRWT are compared when the technology of the energy conversion

is DFIG. The relative transient voltage control capability is then assessed in two cases

for both the DFIG-based DRWT and the DFIG-based SRWT. In the second scenario,

the reactive power limit is not hit by the generated reactive power, while, for the third

scenario, the DFIG is saturated with respect to reactive power limits. In this study, the

capability curve of the connection bus of the wind farm, instead of the ideal capability

curves of the individual DFIGs, is considered as the limiter to the supplied reactive

power. Finally, the theoretical claims made in previous sections are checked through

simulation results.

8

In Chapter 6, the sub-synchronous resonance (SSR) risk of a series-compensated

DRWT-based wind farm is compared with that of the SRWT-based wind farm, and a

method is proposed to alleviate the risk of SSR for the DRWT. At first, the state space

model of the DRWT and the SRWT suitable for torsional frequency calculation is

developed. Then, it is proposed to optimize the mechanical parameters of the DRWT by

a genetic algorithm (GA) to avoid any neighbouring between the complementary of the

grid natural frequency and torsional frequencies of the DRWT. Next, the GA is

configured and its associated fitness function is given. After that to make the method

more realistic and practical, two constraints are introduced. Finally, the performance of

the proposed method is evaluated with regard to SSR risk reduction. The dynamic

response of the DRWT is also tested through numerical simulation to check if there is

any negative impact on the dynamic damping of the system due to the application of the

method.

1.4 Publications

1- E. M. Farahani, N. Hosseinzadeh, M. M. Ektesabi, “Comparison of Dynamic Responses of Dual and Single Rotor Wind Turbines under Transient Conditions”, IEEE ICSET 2010, Kandy, Sri Lanka.

2- E.M. Farahani, N. Hosseinzadeh, M.M. Ektesabi, “SSR Risk Alleviation in Dual-rotor Wind Turbine by Employing Genetic Solutions”, IEEE, AUPEC, Brisbane, Australia, 2011.

3- E.M. Farahani, N. Hosseinzadeh, M. Ektesabi, “Comparison of fault-ride-

through capability of dual and single-rotor wind turbines”, Science Direct, Renewable Energy, Vol. 48, December 2013, Pages 473–481.

4- E.M. Farahani, N. Hosseinzadeh, M.M. Ektesabi, C. Kumble, “Investigating the Impact of Dual-Rotor Wind Turbine on the Transient Network Frequency Deviation”, submitted to IEEE Transaction on Power Systems.

5- E.M. Farahani, M.M. Ektesabi, C. Kumble, N. Hosseinzadeh, “Investigating the Impact of Dual-Rotor Wind Turbine on the Short-term Voltage Stability Margin of the Power System”, submitted to Renewable Energy.

9

http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Farahani,%20E.M..QT.&searchWithin=p_Author_Ids:38232094400&newsearch=true

http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Hosseinzadeh,%20N..QT.&searchWithin=p_Author_Ids:37409311600&newsearch=true

http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Ektesabi,%20M.M..QT.&searchWithin=p_Author_Ids:37860378400&newsearch=true

http://www.sciencedirect.com/science/article/pii/S0960148112003606



http://www.sciencedirect.com/science/journal/09601481

http://www.sciencedirect.com/science/journal/09601481/48/supp/C







: Literature Review Chapter 2

2.1 Introduction

The majority of wind turbines currently in operation have the conventional concept

design. That is, a single-rotor wind turbine (SRWT) connected through a spur gearbox

to a generator. Recently, a dual-rotor wind turbine (DRWT) has been introduced to the

market. It has been proven that the steady-state performance of the DRWT system for

extracting energy is better than for the SRWT. However, analysing the relative impact

of the DRWT on the transient stability margin of the power system requires further

research. The margin of the transient stability can be explored from different aspects

such as margin of the angle stability, margin of the frequency stability, and margin of

the voltage stability. Based on some references sub-synchronous resonance (SSR) can

also be categorized as a part of the power system transient.

The main contribution of this dissertation lies in its initiation of an evaluative analysis

of the relative effects of the DRWT-based wind farm and the conventional SRWT on

the different aspects of the stability margin of a local power system. In this chapter a

review is done regarding the approaches that have been introduced to assess the impact

of the wind farms on the transient characteristics of the power system. This chapter is organized as follows: in section 2.2, the history of the initiation of the

DRWT and its impact on the aerodynamic efficiency is mentioned; in section 2.3 the

transient stability is classified with regard to different criteria such as severity of the

disturbance including small-disturbance and large-disturbance stabilities, time of

interest including short-term and long-term stabilities, and the quantity which its margin

is of interest including angle, frequency and voltage; The methods for assessing the

impact of the wind farms on the margins of the transient angle stability, transient

frequency stability, transient voltage stability are given in 2.4. The risk of SSR, as a part

of power system transient, is also evaluated at the presence of wind energy system in

this section.

11

2.2 Dual-rotor Wind Turbine

The dual-rotor wind turbine (DRWT) was introduced for the first time by [1]. In this

development, an auxiliary turbine was added to the conventional horizontal wind

turbine. The prototype of this technology was installed and tested successfully under the

rating of 30kW. The prototype of this technology, which is also called a ‘counter-

rotating wind turbine system’, is presented in Fig. 2.1. As depicted in the latter figure,

the DRWT has two sets of blade; one of them is the main rotor and the other one is the

auxiliary rotor. The auxiliary turbine rotates clockwise and is placed in an upwind

location, while the main turbine rotates counter-clockwise and is located downwind.

The term ‘upwind location’ means that the turbine positioned in this spot is hit by the

wind first. The idea of the auxiliary turbine was developed due to the inefficient

aerodynamic performance of the main turbine. Some portion of the blades in the main

turbine – the inner 30% portion – plays a less effective role in producing aerodynamic

torque because of the little sweeping velocity of this portion. This portion is termed a

‘dead zone’ and is unfortunately not so small as to be ignored [2].

Fig. 2.1: A 30kW dual-rotor wind turbine system

The auxiliary rotor is employed to make up for the dead zone in the main turbine,

through producing extra aerodynamic torque of its own accord. The length of the

auxiliary blade is almost one-half of the main turbine. The basic information regarding

the 30kW DRWT is given in Table 2.1.

12

Main rotor Auxiliary rotor

No. of blades 3 3

Rotor diameter 11 m 5.5 m

Rotor position Downwind Upwind

Air foil NACA 0012 NACA 4415

Built-in twist (o) -2 0

Rotor RPM 150 300

Blade materials Glass/epoxy Glass/epoxy

Rotation Counter-clockwise Clockwise

Pitch control Variable Variable

Table 2.1: Characteristics of the 30kW prototype dual-rotor wind turbine [2]

From a structural aspect, the DRWT introduces some disadvantages. This is due to the

presence of the additional rotor, in comparison to the conventional SRWT. However,

the DRWT shows some stoning features over conventional wind systems, especially

with regard to the aerodynamic characteristics. The beneficial sides can be as follows: i)

higher aerodynamic efficiency just by adding a small turbine; ii) a free yaw

characteristic is feasible due to the locating of the generator in the tower, which is a

non-rotating area. The weight of the nacelle is thereby reduced significantly; iii) in

cases where there is the same power rating, the lower gear ratio is required due to the

higher tip speeds achieved by the smaller blade length [2].

Analysis of the aerodynamic characteristics of the DRWT is more complicated than for

the SRWT, due to the wake effect of the auxiliary turbine. Here, it is worthwhile

explaining the wake effect briefly for clarity. Wind turbines capture energy from winds

to generate electricity; thus, according to the law of energy, the wind downstream

should have a lower energy in comparison to the wind upstream [3]. As a result, the

wind leaving the turbine has a decreased velocity and is disturbed. This impact of the

turbine on the wind is called, variously, the ‘wake effect’ or ‘stream tube effect’ of the

turbine. As the wind leaves the blades, the wake starts to spread and return, little by

little, to a free stream condition. If another turbine is located in the wake of the

13

upstream turbine, then it is termed as be ‘shadowed’ by the turbine making the wake.

Fig. 2.2 exemplifies the above discussion regarding the wake effect. The size of the

arrows shows the velocity of the wind in different areas around the turbine [4].

Fig. 2.2: Turbine airflow on upstream and downstream [4]

The wake has two main impacts: i) reduction of energy production due to the decrease

of wind velocity; ii) higher dynamic mechanical loading on the turbines downstream

due to the generated turbulence. It is essential to include the wake effects in the design

process of wind farms to increase the aerodynamic energy efficiency and to reduce the

mechanical tension of the wind turbines [5].

Since, in the DRWT, the wind first moves through the auxiliary wind turbine, the wind

that then approaches the main turbine is partially disturbed. The aerodynamic

characteristic of the disturbed wind is more complex than that of a normal wind. In

addition to quantitative analyses, some other experimental tests have been carried out to

investigate the wake behaviour of the blades [6], [7]. Neff and Meroney [6] conducted

some tests on wind tunnels to evaluate the three dimensional wind features close to a

rotating turbine for different amounts of flow. The results contain the radial, axial and

rotational wind speeds measured at close to 60 spots upwind, downwind and on the

sides of the turbine. Magnusson [7] explored the flow downstream of wind turbines

using the blade element and the momentum theory, and compared the simulation

outcomes with experimental data achieved from a wind farm located in Sweden. Since

the auxiliary turbine is smaller than the main turbine, then it is placed upwind of the

main turbine; otherwise, the auxiliary turbine will be shadowed by the main turbine.

Thus, the wake effect of the auxiliary turbine must be included in obtaining the

aerodynamic characteristics of the DRWT. For the best performance of DRWT, the

proper distance between the two turbines, as well as the relative size of the turbines,

should be investigated.

14

A study has been done by [2] to investigate the impact of both the distance between

the turbines and the blade size of the auxiliary turbine on the amount of increase of

aerodynamic efficiency of the DRWT. To reach this stage, parametric analysis was done

by the latter. The change of power increase is given as a percentage: ((P-P0/P0)*100,

where P is for the case when the wake effect is included and P is the power when the

wake effect is excluded. In Fig. 2.3, the increase in power is sketched as a function of

the size of the auxiliary blades. The horizontal axis is the ratio between the sizes of the

auxiliary (DAR) and the main (DMR) turbines. The length of the auxiliary turbine is

variable from zero to the length of the main turbine.

Fig. 2.3: Effect of auxiliary rotor length on the increase of power [2]

As demonstrated in Fig. 2.3, as the length of the auxiliary turbine is increased, the

captured energy from wind increases until the length approaches the 5/8 of the main

rotor size. The amount of rise in the aerodynamic power is contributed to by two main

factors: one is the power rise due to the energy captured by the auxiliary rotor, and the

other one is the change in aerodynamic performance of the main turbine due to the wake

effect of the auxiliary turbine. The interpretation of the former factor is quite

straightforward – the bigger the auxiliary blade size, the higher the energy that can be

captured from the wind. The latter factor can be explained as follows: when the

auxiliary turbine is fairly small in comparison to the main blade, then the area of the

main blade, which is covered by the stream tube from the auxiliary rotor, is only a small

portion of the main turbine disk area. In this area of the main rotor, the wind speed is

reduced and consequently a lower power is achieved. However, in the outer area of the

15

stream tube, which includes the larger portion of the main disk, the wind speeds up (due

to the wake effect) and this assists in the capture of more kinetic energy from the

accelerated wind by the outer region of the blades, which are not covered by the stream

tube. But, when the size of the auxiliary blade exceeds a certain level (5/8 by Fig. 2.3),

the portion covered by the stream tube becomes more than the outer area, leading to a

reduction of the aerodynamic efficiency even less than in the case where there is no

auxiliary turbine. The curve on which the triangles stand contains the effect of both

factors, while the curve with circles includes only the energy captured by the main blade

in the presence of the wake effect (the power produced by the auxiliary turbine is

ignored). As indicated in Fig. 2.3, when the size of the auxiliary blades is one-half of the

main blades, the power is increased up to 20%, which is the maximum improvement in

aerodynamic efficiency achieved by adjusting the size of the auxiliary turbine. The

geometry and dimensions of both the auxiliary and the main turbine are presented in

Fig. 2.4. For the main rotor, an extension bar is placed between the hub and the blade to

dodge the manufacturing expenses of making complex geometry blades without

significantly degrading the aerodynamic effectiveness.

Fig. 2.5 shows the impact of the placement of the main and auxiliary rotors on the

power generation of the DRWT. The length of the auxiliary turbine is set to one-half of

the main turbine. The horizontal axis is the interval distance normalized by the auxiliary

turbine diameter. It can be seen that, as the interval becomes bigger, the power

generation is enhanced. Around 9% growth of power can be obtained when the distance

between the rotors is about one-half of the diameter of the auxiliary turbine. There is no

improvement when the rotors are nearby.

Fig. 2.4: Geometry and dimensions of main and auxiliary rotors [2]

16

Fig. 2.5: Impact of interval on the power generation growth [2]

In Fig. 2.6, the relative power production of the 30 kW DRWT is plotted in comparison

with the corresponding SRWT, for different wind speeds. It can be seen that the power

curve of the DRWT is located above the associated curve of the SRWT. For instance, at

a wind speed of 10.6 m/s, the energy generation of the DRWT is, remarkably, 21%

more than that of the SRWT.

Fig. 2.6: Power of the DRWT versus the SRWT

2.3 Power System Stability Classification

Power system stability was addressed for the first time in 1920 as a problem for power

systems [8], [9]. The consequence of instability can lead to a whole power system

blackout, which signifies the importance of taking care of the stability of the system

[10]. Traditionally, stability was known as ‘transient stability’. However, as power

17

systems developed and had to be interconnected, and new technologies of control

systems were introduced, then some other forms of instability came to the fore, such as

frequency, voltage and small signal instabilities. To design a system with a high safety

margin with respect to stability, it is essential to have a good understanding of the

nature of each type of stability and the associated influential factors they possess. The

classification of power system stability offered by [11] has been carried according to

certain criteria, as follows:

• The severity of the disturbance, which can be, variously, very small and gradual,

small and sudden, and large and sudden.

• The quantity that is the main source of instability, such as angle, voltage or

frequency.

• The time span of interest that must be considered in order to evaluate the

stability, whether it is short-term or long-term.

2.3.1 Stability Classification Regarding the Severity of Disturbance

In the study of electric power systems, several different types of stability descriptors are

encountered. Stability can be categorized into three classes in terms of the severity of

the disturbance, as follows:

2.3.1.1 Steady State Stability

This term refers to the stability of a power system subjected to gradual and small

variations of demand; the system should restore the stability in normal conditions when

there is no oscillation in quantity.

2.3.1.2 Dynamic Stability

This term refers to the stability of a power system subjected to a fairly small and

unexpected disturbance. The system can be linearized without affecting the accuracy of

the studies and stabilized by a linear and continuous supplementary stability control.

This sort of study is quite promising in terms of assessing the stability margin of the

system for the initial operating points.

2.3.1.3 Transient Stability

This term refers to the strength of stability of a power system subjected to a sudden and

severe perturbation. It may be outside the ability of the linear complementary stability

18

control to take proper action to supress the abnormal condition. The system instability

may occur at the first swing unless a more effective countermeasure is taken, usually of

the discrete type, such as dynamic resistance braking or fast valving for the electric

energy surplus area, or load shedding for the electric energy deficient area. For transient

stability analysis and control design, the power system must be described by nonlinear

differential equations. Fig. 2.7 illustrates the above mentioned statements graphically:

Fig. 2.7 Categories of power system stability regarding the severity of disturbance

2.3.2 Stability Classification Regarding the Time of Interest

With regard to the time frame of interest needed to establish whether or not the power

system will remain stable, after any disturbance, the stability is categorized into three

types, including short-, mid- and long-term. They will be described in more detail

below.

2.3.2.1 Short-term Stability

After a very severe disturbance, the system may become unstable in less than 10

seconds. In this case, it is not possible to stabilize quantities such as voltage, frequency

or active power, and they may begin to oscillate with incremental amplitude or

sustained rise or fall of the quantity. Only the fast acting control systems, like

governors, should be included in the study.

2.3.2.2 Mid-term and Long-term Stabilities

In the case where post-fault oscillations are damped, the system is called ‘short-term

stable’. However, the system still cannot be described as a stable system. A power

system is called stable when all its quantities meet acceptable standard boundaries.

Normally, after stabilizing the quantities, following a very severe disturbance, the

configuration of the power system changes due to the disconnection of transmission

lines, generators or putting of new generating units into operation. Consequently, the

voltage, frequency and power flow in the transmission lines have been deviated from 19

their initial spots and they settle down in new operating points. Now it is time for slower

control systems to take necessary actions and bring back the aforementioned quantities

to the designed value. With respect to the time span, this response of the power system

can be divided into two stages: in the first stage, some control systems try to adjust the

bus voltages and power flows to restore them to their permitted limits; for example, the

static voltage controller (SVC) or automatic generation control (AGC) have

comparatively shorter time-constant responses when measured against other energy

suppliers. This stage is called ‘mid-term stability’. The time span for the mid-term

stability is between 10 second and a few minutes. When the mid-term stability period

has elapsed and the quantities are not yet resumed, then the big energy suppliers, like

thermal boilers, which have high time constants, would react to the deviations in order

to restore the normal situation. This stage, which is fairly slow and is considered as the

last response of the power system to the disturbance, is termed ‘long-term stability’. The

time frame of interest is from a few minutes to 10’s of minutes. If, after this period, the

quantities are not placed in accepted limits, then the power system is said to be ‘long-

term unstable’. Usually, the mid-term and long-term instability issues are due to

inadequate reactive/active power reserves, poor coordination of protection and control,

or insufficient equipment reactions.

2.3.3 Stability Classification Regarding the quantity sources the instability

The instability of power systems is sourced mainly from three quantities of power

system such as relative angle difference between the rotors of the generators, voltage of

the buses, and frequency of the power system. Each of these quantities will be discussed

below.

2.3.3.1 Angle Stability

The capability of generating units in a power system to restore their synchronism with

the grid after a severe disturbance is called ‘rotor angle stability’. The angle stability

depends on the strength of the generators to resume/maintain equilibrium between

mechanical and electromagnetic torques for each individual generating unit. Normally,

angle instability is in the form of growing or un-damped angle oscillations that result in

a loss of synchronism with the grid or another group of generators. In steady state

situations, there is a balance between the output electromagnetic torque and the

mechanical input torque of all generators and, consequently, the speed oscillations are 20

quite close to zero. At the occurrence of any disturbance, the equilibrium is not valid

anymore and may lead to deceleration or acceleration of the rotors. If one of the

generators has higher relative acceleration in comparison to the adjoining units, then its

rotor position will be ahead of the corresponding slower machine. So, some portions of

the power of the slower machine will be transferred to the faster one, according to the

angle-power relationship. This reaction of the active power slows down the accelerated

generating unit and helps the generators to maintain the synchronism. If the rotor angle

exceeds a certain level, then any further rotor angle extension causes active power

reduction, which leads to further angular separation between the generator and the other

units or networks. This interaction may become progressive and end up in instability. In

any condition, the margin of the angle stability is strongly influenced by the amount of

the restoring torque [11]. Loss of synchronism can happen between one generator and

the grid, or between the groups of generators, while the synchronism is retained within

each group after separation of the groups from each other. Angle stability can be

categorized into the following subcategories:

Small-disturbance Rotor Angle Stability: This type of disturbance is related to the

capability of the power grid to maintain the synchronism under small perturbations. If

the perturbations are quite small, the linearization of the associated equations does not

affect the accuracy of the studies [11], [12] and [13]. Small-disturbance stability is

strongly influenced by the initial condition of the state variables of the power system.

This type of disturbance can appear in two forms: i) aperiodic or non-oscillatory growth

of rotor angle, due to insufficient synchronizing torque, or ii) rotor oscillations with

accumulative amplitude because of lack of damping torque. The former type of small-

disturbance stability is almost removed from the network through continuous acting of

the generator automatic voltage regulators. However this problem may occur if the

excitation systems hit the limits and remain constant during the transients. With respect

to the largeness of the small-disturbance stability, it might be global or local. The global

problems are due to the interactions among the large group generating stations and

introduce extensive impacts on the network. In this scenario, the rotor angle of a group

of generators in one region swing against a set of generating units in another area. The

characteristics of this phenomenon are very complicated and different in nature from

local oscillations. These oscillations are called ‘inter-area mode oscillations’, and are

influenced by the load characteristics. Local plant mode oscillations involve a small 21

portion of the network and normally are due to the rotor angle swinging of a single

machine against the rest of the grid. This mode of oscillation is influenced by the

automatic voltage control performance of the generator, the transmission system as

viewed by that generating unit, and the amount of generation of the plant. The

instability may happen between 10 and 20 seconds after a disturbance.

Large-disturbance Rotor Angle Stability or Transient Stability: This type of stability

consists of the capability of the power network to restore synchronism after being

subjected to a severe disturbance, such as loss of generating units or short circuits. In

this circumstance the rotor angle experiences a large journey from its initial operating

point. The transient stability margin is most affected by both the severity of the

disturbance and the initial condition. This sort of instability normally appears in the

form of sustained angular separation due to the lack of enough synchronizing torque for

the first swing following the disturbance. Sometimes, large disturbance instability is due

to the superposition of a local mode with a slow inter-area swing mode; this makes a

large deviation for the rotor angle. So the rotor angle may exceed the critical point on

the power-angle curve and become transiently unstable while there is no disturbance on

the network [11]. This type of instability happens between 3 and 5 seconds after a

disturbance, which may be extended to 10 and 20 seconds for very bulky systems with

large inter-area swings. Both types of angle instability are classified as short-term

phenomena in power systems.

2.3.3.2 Frequency Stability

In the case of an imbalance between energy demand and power delivery, the

frequency deviates from its nominal value. The frequency transient and steady state

variation characteristics depend on the total of the energy imbalance and the frequency

control action of the generating stations. A power system is stable with respect to the

network frequency if it is able to retain the nominal frequency after a severe disturbance

that leads to a significant imbalance between the demand and production of energy.

Frequency instability may appear in the form of continuous frequency fluctuates, ending

up in the disconnection of generators and/or loads. The response time of the devices,

that are introduced to react to the frequency deviation, ranges from fractions of seconds

to several minutes. For example, during the generation shortage, the under-frequency

load shedding relays and generator protection system fulfil their duties in less than a

22

couple of seconds, while the same responsibility takes the boilers in a thermal power

plant 10 to 20 minutes to meet the new demand. So, frequency instability can be

categorized into a long-term or a short-term phenomenon. An example of long-term

instability is the inability of the boilers to provide enough steam for the turbine to keep

up with the new energy demand. In this case, the frequency is re-established, but the

network is not capable of restoring it to the reference value. On the other hand, an

unplanned islanding may lead to frequency decay and picking up of under-frequency

protection relays within a few seconds [14].

Some national standard codes have been enforced on wind farm owners by grid

directors with regard to frequency support responsibilities. Generally speaking, all wind

farms should be sufficiently capable of complying with the frequency control

requirements for primary and secondary responses. In England, in the case of a

frequency fall of 0.5Hz, wind farms are meant to increase the output power within a

time frame of 0-10sec, which is sustainable for 20sec, and to stabilize the frequency, the

generators must maintain output power between 30sec and 30min [15]. It is mentioned

in [16] that the wind farms in Quebec which have a rating higher than 10MW, must

participate in reducing the transient frequency excursion of the power system. In Spain,

some plans have been developed to involve the wind farms in frequency and power

control by modifying the control loops of power converters [17]. According to the grid

codes in Germany, the transmission system operator (TSO) is permitted to ask the wind

farms to reduce their output power in cases where the frequency becomes higher than

50.5Hz. It is also stipulated that, for a frequency deviation of -0.2Hz, the generating

units must be able to increase the energy production equal to the +2% of the rating

power [18].

Grid frequency control action by generating units can be classified into two stages:

primary control and secondary control [11], [19]. Primary control consists of two

components: the first component is kinetic energy, which is a natural response to

frequency variation. It is released from the rotating mass of the generators, and is in

charge of reducing the frequency rate of change and arresting the frequency nadir. The

second component is the automatic power alteration due to the reaction of the

generating unit controller, such as the droop control system that is supposed to stabilize

the network frequency within 30sec. However, the stabilized frequency is normally

different from the nominal frequency. Secondary control is in charge to reset the

23

frequency to the reference value from 30sec to 30min by tuning (increasing or

decreasing) the droop characteristics of the generators. The new values of the droops are

determine by the operators or automatic generation control (AGC). A frequency rise or

fall has two main characteristics: frequency rate of change and the frequency nadir.

These two factors are influenced strongly by the inertia of the whole system, time

constants of the generators, the severity of the disturbance and kinetic energy stored in

rotating masses. The characteristic of the load frequency dependency is important in

frequency stability analysis. However, the main target in chapter 4 of this thesis is

comparing the frequency control characteristic of DRWT and SRWT as the generating

units. Since the loads are assumed to be the same for DRWT and SRWT, then they did

not included in the literature review. Meanwhile, in simulation results the load

frequency dependency is included (dP/df=5%).

2.3.3.3 Voltage Stability

The voltage stability margin is defined as the strength of a power network to resume

the voltage of all buses to an acceptable level when the system is exposed to a

disturbance. A main factor causing the voltage instability is the voltage drop across the

impedances of the transmission networks when active and reactive power flow through

them [20]. This phenomenon defines some limits in the voltage stability margin of the

network. Voltage support is further limited when some of the generators or the DFIG-

based or FCR-based wind farms hit their field or armature current capability limits.

Voltage stability is classified into large-disturbance voltage stability and small-

disturbance voltage stability; the former denotes the ability of the system to maintain the

nominal voltage after large disturbances such as a system fault, a trip-out of high-

capacity generating units, or heavily loaded transmission lines. In contrast to angle

stability, short circuits near load centres are important [14]. This ability is specified by

the characteristics of the network, the loads and the control systems. Small-disturbance

stability is defined as the capability of the network to maintain nominal voltage when it

is subjected to small disturbances such as an incremental growth of energy demand or

tap changer operations on the transformers.

With regard to the time frame of voltage instability, both large and small disturbance

stabilities are categorised into long-term voltage stability and short-term voltage

stability. The former involves slower acting equipment, such as a constant power load,

and generator stator/field current limiters. The frame needed to study this phenomenon 24

may lengthen to tens of minutes. Long-term stability is normally due to the saturation of

the controlling equipment rather than the severity of the disturbance. For example, when

there is a sustained growth of the load while the reactive power suppliers of the local

network, such as synchronous generators, FACTS and/or DFIG-based wind farms, are

operating at their capacitive reactive power limits, then the voltage starts to fall due to

the lack of reactive power support. The voltage fall will be progressive if the load

exceeds and the voltage reaches to the nose of the P-V curve.

Short-term voltage stability involves quicker-acting load components like large

induction motors. When a group of induction motors are connected to the distribution

network, then they all want to accelerate to nominal speed, and consequently, they draw

a large amount of reactive power from the network during the short period of

acceleration, which may lead to voltage collapse. For example, according to the

operation manuals for power stations, the internal induction motors which are used to

drive the boilers and other components should be switched in one by one to avoid any

short-term voltage instability. As another example when a fault occurs in a network that

includes induction generators, the electromagnetic torque drops dramatically and,

consequently, the generators accelerate. When the fault is removed and voltage is

recovered, then a large amount of reactive power is absorbed by the induction

generators, which may cause further voltage drop and eventually lead to short-term

voltage instability if the generators decelerate reasonable more slowly than the rate at

which the terminal voltage is restored [20].

2.3.3.4 Sub-synchronous Resonance

Increasing the power transfer capability of the transmission lines that connect the wind

farms to the grid is achievable by adding new parallel transmission lines [21]. This

method is quite expensive and shouldn’t be considered as a cost effective solution for

this issue. A good solution is to employ a series capacitor for connecting the high-rating

power wind farms [22], [23], [24]. Although series compensation is the most

economical way to enhance the transmission line power transfer capacity, it may result

in sub-synchronous resonance (SSR) [25], [26], [27]. The recent occurrence of SSR in a

number of wind farms, which lead to the disconnection of and damage to wind turbines,

has indicated that some mitigation measures should be taken to damp the oscillations

due to the SSR when the compensation level is increased [28], [29].

25

According to [11], sub-synchronous resonance phenomenon is categorized as a part of

the stability study of power system. In Fig. 2.8, the location of the sub-synchronous

resonance (SSR) is highlighted by a cloud. Consequently, it is worth investigating the

impact of DRWT-based wind farms on the SSR risk factor of the network.

The objective of this subsection is to: i) analyse sub-synchronous phenomenon and the

subcategories of this sort of instability; ii) review the impact of different wind turbine

technologies on the SSR; iii) review the approaches introduced for damping the

oscillations sourced from SSR and reducing its risk.

26

Fig. 2.8 Classification of power system [11]

2.3.3.5 Sub-synchronous Resonance Definition and Types

For studying the local oscillations or inter-area oscillations of the power grid, the

turbine is assumed to be a lumped mass (Single-mass), which presents enough accuracy.

In reality, turbines are composed of different components, i.e. such as an exciter,

couplings, and different stages of turbine, which are connected to each other through

shafts possessing specific degrees of stiffness. The shafts are normally modelled by

springs and dampers.

Therefore, there are relative torsional oscillations between parts of the generator-turbine

rotor in case of any disturbance in the grid. Problems concerning these torsional

frequencies are mentioned, as follows [11]:

- Sub-synchronous resonance with series capacitor on the interface transmission line.

- Torsional fatigue cycle because of network switching.

- Torsional interactions with control systems in network.

SSR is categorized into four groups:

- Induction generator effect (IGE)

- Torsional interactions (TI)

- Torsional amplification (TA)

- Torsional interaction between closely coupled units (TICU)

Each subcategory of SSR will be discussed briefly as follows:

2.3.3.5.1 Induction Generator Effect

A series capacitor installed on a transmission line introduces a natural resonance

frequency (fn) given by (2.1):

𝑓𝑛 = 𝑓0𝑋𝑐𝑋𝐿

𝐻𝑧 (2.1)

where, f0 is the synchronous frequency in Hz. XL is the Thevenin reactance of the

transmission line and Xc is the impedance of the capacitor. Consequently, apart from the

slip regarding the synchronous frequency, there is another slip for the natural frequency.

It is defined by (2.2):

27

𝑠1 = 𝑓𝑛−𝑓𝑚𝑓𝑛

(2.2)

where, fm is the rotational speed of the generator. s1 is negative because normally fn <

fm. There is an electrical circuit of the DFIG regarding the sub-synchronous slip, s1,

which can demonstrate the effect of s1. Its configuration is presented by Fig. 2.9.

Fig. 2.9. DFIG Equivalent circuit under sub-synchronous frequency

The resistance viewed from the rotor is negative due to the negative value of s1. If the

equivalent resistance of the induction generator viewed by the grid becomes negative,

then the system has negative resistance for the natural frequency and the armature

current with the natural frequency fn rises in a sustained or oscillatory manner. This

phenomenon is known as IGE. The IGE is strongly influenced by s1. From (2.2), s1 is

related to fn and fm, where they are respectively influenced by the level of compensation

and the wind speed. As the wind speed goes up, the fm rises up as well which leads to

higher values of s1. As a result, the negative amount of equivalent rotor resistance is

reduced and there is less risk of IGE. On the other hand, as the level of compensation

grows, fn increases as well. Consequently, s1 is reduced and the negative value of the

equivalent resistance becomes higher. So, at high levels of compensation, the risk of

IGE is increased. In [30], the effects of the wind speed and the amount of compensation

on the resonance mode is investigated and given here in Fig. 2.10.

Fig. 2.10. Network resonance mode at various wind speed and level of compensation [30]

28

2.3.3.5.2 Torsional Interactions (TI)

In the presence of a series capacitor, an extra component of voltage and current will

be introduced to the grid, the frequency of which is fn. So the current flowing in the

stator of the induction generator has two components – Is1 and Is2. Is1 is at synchronous

frequency fs, and Is2 is at natural frequency fn. The rotor current also has two

components – Ir1 and Ir2. Ir1 is at a frequency of (fs-fm), and Ir2 is at frequency of (fn-fm).

Four electromagnetics torques are produced due to the interactions between the stator

two- current components (Is1, Is2) and the rotor two current components (Ir1, Ir2). The

amplitude of the torques resulting from interactions between Is1 and Ir1, Is2 and Ir2 is

constant and its frequency of oscillation is zero. Meanwhile, the amplitude of the

torques due to the interactions between Is1 and Ir2, Is2 and Ir1 are oscillating at a

frequency of (fs-fn). Hence, the total electromagnetic torque has an element at a

frequency of (fs-fn). The magnitude of this element may be magnified if one of the

torsional frequencies of turbine-generator rotor coincides with the (fs-fn). Since they

have the same oscillating frequency, the torsional mode is excited and receives energy

from the network.

2.3.3.5.3 Torsional Amplifications (TA)

TA is mostly due to severe disturbances in the power system, like short circuits or line

switching. These types of events cause an abrupt change in the power supply, which

results in current oscillations in a series of compensated lines. The oscillations introduce

a current frequency spectrum. If the frequency of the oscillations coincides with one of

the generator shaft torsional frequencies, then the oscillation of that specific torsional

mode will be magnified. The SSR due to the TA can cause severe shaft torsional

fluctuations. This phenomenon may cause fatigue loss for the shaft. The IGE and TI can

be studied through the small signal method. However, the TA cannot be investigated

through the small signal approach due its non-linear nature [31], [32].

All sorts of SSR impose tensions on the turbine-generator rotor system. So

oscillations due to the SSR should be limited to some safe levels; otherwise, it may

cause serious damage to the mechanical and electrical equipment. Pressure and stress

leads to the fatigue of the mechanical components. Fatigue is defined as a change in the

structure of the materials. Sometimes, it causes a full fracture after a number of

oscillations. The life-expectancy of the turbine-generator shaft will be reduced when it

29

is exposed to severe disturbances. The loss of life-expectancy is a cumulative effect.

This means if there is, progressively, a fatigue factor of twenty precent, then fifty

precent and finally thirty precent, the shaft will break [33].

2.3.3.5.4 Torsional Interaction between Closely Coupled Units

For a single generating unit of N rotating masses, there are N oscillating modes. One

of the modes is called the ‘system mode’ or the ‘rigid-body mode’; in this mode, all

components fluctuate in phase with almost the same amplitude. The rest of the modes

are torsional modes and they oscillate in different frequencies. In cases the number of

parallel generating units is M, the shaft system of these units are coupled via the power

system. The torsional modes may be excited by the interaction between the shaft

systems of the parallel generators. This phenomenon was recognized for the first time in

the field tests of the Mohave power plant [34].

2.3.3.6 Methods for Analysing the SSR

Thus far, some approaches have been introduced by researchers for analysing the SSR.

In the following section, the most popular methods are mentioned:

- Frequency scanning [35], [36]

- Eigenvalue analysis [37], [38], [39]

- Electromagnetic transient analysis [40]

The first two methods are based on linear models of power systems, machines and

generator shafts, and are accurate enough for IGE and TI studies. The third method is

adequate for investigating the TA.

2.3.3.7 Risk of SSR for Different Wind Turbine Technologies

Technology of the horizontal wind turbines are classified into three main groups:

- Fixed speed wind turbines (FSIG)

- Double-fed induction generators (DFIG)

- Fully-rated converter (FRC)

2.3.3.7.1 SSR Risk in Fixed Speed Wind Turbines

FSIGs are directly coupled to the network, as presented in Fig. 2.11.

30

Fig. 2.11 Fixed speed wind turbine with induction generator

SSR may occur in FSIG wind turbines that are connected to the grid through a series

capacitor. This is mainly due to the IGE. In some cases, TI can be the source of the SSR

[28], [29], [41].

2.3.3.7.2 SSR Risk in Fully-rated Converter Wind Turbines (FRC)

To have full control over the active and reactive flow of the wind turbines, FRC is

employed [42]. The element arrangement of this technology is presented in Fig. 2.12. In

this technology, the turbine is isolated from the grid through a back-to-back converter

system. Consequently, the transients by the grid are blocked and are not transferred to

the turbine. Therefore, FRC is protected against the SSR [43]. The same is true for

HVDC when it is operating in inverter mode. Conversely, there is a risk of SSR for

HVDCs when they are operating as rectifiers, due to the likelihood of negative damping

in this mode [44].

Fig. 2.12 Fully-rated converter wind turbines

2.3.3.7.3 SSR Risk in Double-fed Induction Generators

A typical DFIG is presented in Fig. 2.13. Previously, it was believed that there is no risk

of SSR for DFIG, just like FRC. This assumption was made on the basis of the

perceived capability of the DFIG control system in regulating the torque. In October

2009, SSR was the main reason for the disconnection of the Zorillo Gulf wind farm

from the local grid [45], [46].

31

Fig. 2.13. DFIG wind turbine

2.4 Method to Assess the Impact of DRWT on the Transient Stability

This section includes four subsections to review the methods that have been reported

in the literature for investigating the degree of impact of the wind farms on,

respectively: transient angle stability; transient frequency stability; transient voltage

stability; and, sub-synchronous resonance.

2.4.1 Method to Assess the Impact of Wind Farms on Transient Angle Stability

Recently, the application of squirrel-cage rotor induction generators has increased

dramatically. The main reason is that this type of generators possesses some advantages

over synchronous generators, such as less maintenance, smaller dimensions at the same

rating power and cheaper purchase price. This type of generator is used in the majority

of wind farms; however it has also been employed in medium-size thermal and hydro

plants [47], [48], [49]. So, it is worth investigating the impact of the induction

generators on the grid operation for steady state and transient conditions. One of the

most important aspects of the feasibility study regarding the massive employment of

induction generators is the exploration of the effect of the transient response of the

network during the faults. Throughout short circuits, induction generators accelerate to

higher speeds due to the sudden decrease in electromagnetic torque. The generator may

not be able to resume its rating speed and the amplitude of the speed thus rises

progressively. Consequently, the induction generators draw too much reactive power

from the network, resulting in terminal voltage reduction and further acceleration of the

generator [50]. Therefore, the transient stability of the induction generators can be

evaluated through assessing the transient response of the rotor speed during the faulty

condition [51]. There are three main methods for analyzing the large-disturbance

stability margin of an induction generator including: dynamic simulations [47], [48],

[50], [52], [53], [54], [55]; experimental checks [56]; and the use of an analytical 32

method [57]. The current popular analytical approach to assess the margin of the

transient angle stability of the induction generators is the ‘critical rotor speed’. The

criterion is the minimum distance between the stable and unstable operating points

during the transient period. The two operating points are achievable from the cross

sections between the electrical and mechanical torque-speed curves for normal

conditions. The higher the distance between the two operating points, the greater the

transient margin is given by the induction generator [57]. This method was initially

introduced to evaluate the stability margin of the induction motors [58].The validity of

the method can be verified by numerical simulations. The electrical, mechanical and

aerodynamic transient performance quality of the wind turbine is very important in

terms of the absorption of as much energy as possible from the wind. Following this

direction, a new wind turbine generator system (WTGS) has recently been introduced. It

is called a dual-rotor wind turbine (DRWT) and has two sets of rotor systems;

importantly, it is more efficient than the conventional single rotor wind turbine (SRWT)

from an energy extraction point of view [2].

At the time of the current writing, the authors could trace [59] as the only reference to

the dynamic performance of the dual-rotor system. Multi-body dynamics is the

employed approach. Although, in this reference, a model is provided to present the

detailed procedures used to show the dynamic and aerodynamic performance of this

system, however the authors did not compare the dynamic response of the dual-rotor

wind turbine with a single-rotor wind turbine. According to [59], the commercial types

of dual-rotor wind turbines are able to generate power up to one megawatt so far.

Even though, at the same wind speed and environmental conditions, the efficiency of

the dual-rotor is higher, it does not signify that the transient performance of the DRWT

is better than that of the SRWT. Obviously, the transient behaviours of the dual-rotor

and single-rotor wind turbines are different, because, in the dual-rotor system, the

number, type and arrangement of the components are different.

The objective of this investigation is to compare the synchronizing and damping

torque introduced to the network by the DRWT and SRWT. To reach this stage, both

types of wind turbines were set up in PSCAD software. The drive train method was

employed for modelling the mechanical system of the DRWT and the SRWT. The

electrical characteristics of generator, transformer, transmission line and power system

used for the DRWT and the SRWT are identical in order to make a fair comparison.

33

The synchronizing torque is mostly dominated by the electromagnetic torque

imposed by the electrical side. The damping factor of the generating units is mostly

influenced by their control system mode and natural damping characteristic, imposed by

the mechanical drive. To assess the transient response of the DRWT and the SRWT,

when they are operating in constant speed mode, a temporary three-phase short circuit

was applied to the power system and post-fault fluctuations of the variable of interest

were recorded and compared. The validity of the time domain simulation, the damping

factor of the DRWT and the SRWT is modelled analytically. For each turbine the

damping factor is approximated by its speed droop characteristic and natural damping

coefficient.

To evaluate and compare the fault-ride through capability of the DRWT and the

SRWT, the maximum short circuit periods for which both generating units are able to

keep their stability are checked while the droop controllers are activated and both the

DRWT and the SRWT are rotating at variable pitch angle. To verify the simulation

results regarding the damping factor, an eigenvalue analysis is employed using

MATLAB software. The real portion of eigenvalues is a good measure for assessing the

damping factors of the systems.

Finally, to confirm the analysis discussed above, the ‘critical rotor speed’ method

applicable to the induction generators for evaluating the stability margin, is employed to

assess the DRWT angle stability margin versus the angle stability margin of the SRWT.

Additionally, in calculating the aerodynamic torque, the stream tube effect behind the

auxiliary rotor disk was neglected in [59]. This simplification can affect the accuracy of

the simulations negatively. In this research, the stream tube effect is incorporated into

the dual-rotor aerodynamic model, which enabled the exactness of the aerodynamic

model to be more realistic.

The issue will be argued in greater detail in Chapter 3.

2.4.2 Method to Assess the Impact of Wind Farms on Transient Frequency

Stability

The application of a large number of converter-based wind turbines, such as the double-

fed induction generator (DFIG) and fully-rated converter (FRC) reduces the system

inertia [60]. To overcome this problem, Inertia Control and Droop Control loops have

been integrated into the interface power converters to enhance, respectively, the inertial

and primary responses of the variable speed wind turbines [19], [61]. To ensure a safe 34

operation for the converter-based wind turbines during the transient period, the energy

generation of the grid-side converter must be coordinated by the energy production

capability of the turbines. For instance, to prevent the rotational speed of the wind

turbines from falling too low, the energy which is introduced by the generating unit of

the wind turbine should match the energy which is delivered to the network by

converters, with only a short delay, during the transient frequency deviation; otherwise,

the turbine may stall or the voltage of the dc-link capacitor may be considerably out of

limit due to energy imbalance [19], [62]. So, the parameters of the control loops (in

either the inertia or droop control system) have to be designed and tuned based on the

capability of the generating units of the wind turbines with regard to the supply of

energy throughout the frequency deviation [62], [63], [64].

In almost all previously reported approaches, the amount of kinetic energy (KE)

released by the body mass of the generating unit has been considered as the only factor

for the calculation of the gain of the inertia controller loops integrated into the

converters [62]- [64]. In this investigation, in addition to KE, two other factors are

introduced, which are also influential in the production of energy during the transient

period. These factors are: the method that is employed for de-loading the wind turbine;

and, the range of excursion of the operating point on the aerodynamic characteristic

curve during the transient period. [65] and [66] believe speed variations up to 25% are

acceptable for VSWTs. So, as the boundaries of excursion become wider, the influence

of the movement of the operating point cannot be ignored, in comparison to KE.

The dual-rotor wind turbine (DRWT) has been introduced to the market to increase

the aerodynamic efficiency of wind turbines. This technology, which has two sets of

turbines, is able to introduce a higher amount of active power compared to the

conventional single-rotor wind turbine (SRWT) at the same wind speed [59]. The

dynamic performance of the DRWT has been studied in [59]. It was shown that the fault

ride through capability of the DRWT is also higher than that of the SRWT [67].

One of the objectives of this research is to assess the impact of the DRWT on the

transient frequency excursions of the network when FRC is employed as the technology

for connecting the wind turbines to the grid. The criterion for comparison is the total

amount of energy which is delivered by the corresponding generating units to the

generator-side converter during the frequency deviation. The study is accomplished for

three de-loading methods, including pitch control mode, sub-optimal mode and a

35

combination of both the methods [59], [64]. Variable speed wind turbines (VSWTs)

normally operate at optimal point to obtain as much energy as possible from the wind in

order to maximize their revenue and utilization factors. However, they cannot provide a

long-term increase in active power to stabilize frequency and return it to the nominal

network frequency as their primary and secondary responses, which conventional plants

are able to do. Therefore, to have enough reserve for introducing primary and secondary

responses, it is essential for VSWTs to work in a non-optimal operating point, called a

‘de-loaded operation’. De-loading mode can be achieved mainly through two methods;

in the first method, the wind turbine is working with pitch angles that are higher than

the minimum blade angle. In case of any growth in demand, the blade angles should be

reduced to increase their aerodynamic efficiency and raise the active power production.

Another method for de-loading is operating on the sub-optimal curve in the under-speed

area on the left side of the MPPT curve and on the sub-optimal curve on the right side

of the MPPT curve. The operation on the sub-optimal curve in the under-speed area

raises some issues regarding its poor dynamic stability; therefore, in this study, the sub-

optimal curve in the over-speed area is considered for this investigation. In this method,

the speed of the turbine is controlled so as to be higher than the speed corresponding to

the optimal operating point, while the pitch angle is fixed at its feasible minimum. Since

the slope of the aerodynamic curve is negative in the over-speed area, then, the power

delivered by the sub-optimal curve, for each specific wind speed and blade angle, is less

than its corresponding optimal point. In sub-optimal mode, in the case of a drop in

frequency, extra active power is drawn from the generator by the generator-side

converter, forcing the turbine to slow down. So, the turbine is settled down at a new

operating point with lower speed and higher aerodynamic power in the over-speed

portion of the aerodynamic curve [64], [68].

It is obvious that the effect of power system inertia on the frequency control

characteristic is much higher than for a single wind farm; however, to highlight the

impact of the DRWT on the frequency control performance of the power system, the

inertial of the local network is chosen to be at the same level as that of a DRWT-based

wind farm. The issue will be further argued in detail in Chapter 4.

2.4.3 Method to Assess the Impact of Wind Farms on Transient Voltage Stability

In this section, a brief reference to the literature is provided with regard to the

influence of the induction generator-based wind farms on the short-term voltage 36

stability margin of the power system. This will facilitate the investigation of the impact

of DRWT-based wind farms on the large disturbance short-term stability of the

network, whether it is beneficial or detrimental. An approach was introduced by [51]

which proposed the evaluation of the short-term voltage stability margin of the FSIG-

based wind farm. This method is normally called the ‘critical rotor speed’ approach, and

was initially adopted to assess the transient angle stability margin of the induction

generators [69]. The criterion is the minimum distance between the stable and unstable

equilibrium points during the transient period. In case of fault, the wind turbine

accelerates and the stable operating point travels toward the unstable point. According

to [51], if during the excursion, the operating point exceeds the critical speed (unstable

point), then the generator accelerates to high values and the machine draws a

considerable amount of reactive power, leading to transient voltage instability. The

approach was also followed by [70] to investigate the effect of the mechanical and

electrical parameters of the wind farm on voltage stability. In [70] , it was claimed that

the value of each of the parameters has a different impact on the voltage stability margin

of the induction generators. For example, increasing the stator resistance and stator

leakage inductance reduces the margin, while increasing the mutual inductance

enhances the stability margin. The critical rotor speed method was modified in [71] by

including the network electrical parameters, in addition to the generator parameters. In

[71], it was claimed that the practical transient voltage margin is less than the margin

predicted by [51]. This was due to the extra voltage drop across the Thevenin equivalent

impedance of the network.

However, there are some drawbacks identified associated with the critical rotor speed

method, which lead these authors to the point where the voltage instability analysed in

[51], [70], [71] is in fact an oscillatory voltage collapse originating from the angle

instability of the induction generator, rather than the real voltage instability. So, this

approach is not chosen for assessing the effect of the DRWT on the short-term voltage

stability margin of the network. As a reaction to the disturbances that threaten the short-

term voltage stability, such as big load switching or sudden disconnection of a

generating unit, the local generators increase their power rapidly to arrest the frequency

fall and support the network frequency stability. On the other hand, [14] claims that, as

the flow of apparent power rises during the transient period to support the frequency,

there is more voltage drop across the network impedance and, consequently, the voltage

37

stability margin is reduced. This response of the generating units to the frequency fall

can be a threat to the voltage stability.

Based on the notion of [14], a criterion is introduced as a tool for comparing the

relative impact of the DWRT and SRWT on the short-term voltage stability margin. The

benchmark introduced here, is the maximum apparent power delivered by the DRWT

during the transient period, compared with the corresponding quantity of the SRWT. So,

the relative impact of the FSIG-based DRWT and the FSIG-based SRWT are compared

with respect to their capabilities of current injection to the network throughout the

transient period. The higher the transient power generation capability, the less the short-

term voltage stability margin is considered for the wind turbine in this dissertation. For

the next step, the same study is repeated on the both SRWT and DRWT when they are

equipped with DFIG technology. The impact of DFIG-based DRWT on the voltage

stability is investigated for two scenarios. For the first scenario, the generating unit is

able to deliver the required reactive power and bring back the network voltage to the

nominal value. In this scenario, the study is based on the capability curve of the DFIGs

[72], [73], [74], where the area inside the capability curve is shared between the

inductive and capacitive areas fairly equally (the area of the capacitive area is less than

inductive area around only 10%). An example of the DFIG capability curve addressed

by some of the references is given in Fig. 2.14.

Fig. 2.14. DFIG capability curve [72]

For the second scenario, the reactive losses of the interface electrical equipment are

included in the study. For this scenario to be made more realistic, the capability curve of

the connection bus of the wind farm is considered as the reactive power characteristic of

the DFIG-based wind farm. The capacitive area of the capability curve is reduced

38

dramatically in the nominal rating power of the wind farm. So, it signifies that the

failure of the wind farms to deliver the required reactive power at the rating powers, and

that the capacitive reactive power limit is hit during the transient period is likely. The

detailed models of FSIG-based and DFIG-based for the DRWT and the SRWT are set

up and implemented in PSCAD/EMTDC. The issue will be further argued in detail in

Chapter 5.

2.4.4 Method to Assess the Impact of Wind Farms on Transient Sub-synchronous

Resonance

A cost-effective way to transfer generated power, without putting the angle stability

in jeopardy, is to employ series capacitors for high-capacity wind farms. An SSR

feasibility study for each wind farm to be connected to the national grid is of profound

importance, since sub-synchronous resonance (SSR) is likely for wind farms. Besides,

capacitors in different lines excite torsional modes in the adjoining generators in that

area [75]. For instance, ABB2 Company investigates the risk of SSR for wind farms as

the second step of the feasibility study procedure.

Although both the steady state and transient responses of the DRWT are enhanced

compared to the SRWT, no investigation could be traced that compares the risk of SSR

for both systems. Through studying the types of SSR in section 2.3.3.5, it can be seen

that three forms of SSR –TI, TA and TICU– are strongly influenced by the number of

torsional frequencies. The higher the number of the torsional frequencies, the higher the

likelihood of SSR occurrence can be imagined for the generating units. In other words,

TI, TA and TICU are more likely for the drive train systems, which present more

torsional frequencies. The first perception regarding the risk of SSR is the higher risk

factor for the DRWT system. The main reason for this is the higher number of rotating

components in comparison to the single-rotor wind turbine. This is due to the extra

masses introduced by the auxiliary turbine, which impose an additional number of

torsional frequencies in comparison to the SRWT. This issue will be explored in the

present study.

During SSR, there is too much tension on the rotating components in the wind

turbine. This issue reduces the life expectancy of the components, which results in more

down time for the wind turbine and consequently less profit for the energy companies.

2 http://www.ercot.com/content/meetings/rpg/ABB_RPG_presentation.pdf 39

http://www.ercot.com/content/meetings/rpg/ABB_RPG_presentation.pdf

Some methods have been presented to prevent or damp the oscillations of the SSR and

to keep it within accepted levels. These methods are discussed below:

- Parallel compensation: a simple method could be the proper selection of the level of

compensation – that is, to select the natural frequency of the grid as not close to the

torsional frequencies of the mechanical system. Such a method is not feasible due to

the uncertainty of the XL in (2.1). Since the configuration of the power system is

changing all the time, the thevenin equivalent reactance of the grid XL viewed the by

generating unit is a variable quantity. This method can be replaced by the

application of parallel compensation [27].

- Pole-face damping windings: the employment of the pole-face damping winding in

the generators reduces the negative resistance of the generator at the grid natural

frequency. It is worth mentioning that it is not possible to install the winding on the

old machines. This method aims to reduce the risk of IGE and doesn’t have any

mitigation effect on the TI and TA [27].

- NGH-damping scheme: The NGH (N.G. Hingorani) approach is presented in Fig.

2.15. The system is composed of back-to-back thyristors, resistance and a capacitor.

The functionality of this component is described by [76], [77]. The risk of SSR

caused by TI and TA can be reduced by this method.

Fig. 2.15. Diagram of linear NGH damper

- Blocking filters: it is possible to block out the components of the transmission line

current that corresponds to the torsional frequency of the generator shaft system. As

illustrated in Fig. 2.16, this method is feasible through installing a blocking filter in

the neutral of the step up transformers. Each of the filters is designed for a specific

torsional frequency [78], [79].

40

Fig. 2.16. Three-phase transformer with blocking filters

- FACTS devices: one of the most effective methods for assessing the SSR is the

integration of associated controllers into the Flexible AC Transmission Systems

(FACTS) [28], [29], [80], [81], [82]. Series and parallel FACTS devices are both

effective in damping the oscillations resulting from SSR. Usually the series devices

are more effective than the parallel devices. This was revealed through comparing

the SSR damping performance of Static Var Compensator (SVC) and Thyristor

Control Series Capacitor (TCSC) by [28], [80], [83]. For all cases, the generator

speed has been used to be fed into the auxiliary controller. A typical control system

which has been used for SVC is given in Fig. 2.17.

Fig. 2.17. SSR damping controller implemented in SVC [28]

- Controlling the converters of DFIG: Installing the FACTS devices just for damping

the SSR is not cost-effective for wind farm owners. DFIGs can be used for

decreasing the power grid and inter area oscillations [84], [85], [86], [87].

According to [39], [88], the rotor side converter (RSC) is suitable to be used for

limiting SSR oscillations. On the other hand, the performance of the grid side

converter (GSC) is quite similar to that of the static compensator (STATCOM) and

is appropriate and adequate for reducing the SSR oscillations [36], [38].

Although several approaches have been introduced for making the amplitude of

oscillations smaller during SSR, we believe that regarding the dual-rotor systems, at the

first stage, some mitigation methods must be adopted to reduce the risk of SSR, instead

of trying to damp the oscillations during the SSR. The foci of the previous introduced

41

methods have been on the electrical side of the SSR phenomenon, and different

controller systems have been designed to limit the oscillations originating from the SSR.

At the time of writing, no research could be traced which has worked on the mechanical

side to reduce the risk of the SSR.

In each specific drive train system, the number of oscillating modes is fixed and it is not

possible to reduce them because of the fixed number of rotating elements. It is proposed

here that it is possible reduce the risk of the SSR through appropriate selection of the

mechanical parameters in the design phase. From [11] it is confirmed that the practical

level of compensation is somewhere from 20% to 70%. So, this study predicts that the

grid natural frequency is mostly placed somewhere between 22Hz and 42Hz, and this

frequency range is identified here as the high-risk range. To reduce the risk of the SSR,

it is recommended here to avoid coincidence between the torsional frequencies and the

grid natural frequency. By using genetic algorithm (GA), the mechanical parameters of

the DRWT are optimized in such a way as to delimit the torsional frequencies from the

high-risk frequency range. In other words, the target is to make the high-risk range

empty of the torsional frequencies. Obviously, the chosen parameters must fall into the

ranges specified by the designers. To achieve this, the search space of the GA for each

control variable is limited, based on the technical boundaries of manufacturing. Each

chromosome that meets the fitness function should be approved by two constraints to

make the output of the GA more practical. First, the distance between the individual

torsional frequencies achieved from the best chromosome should not be less than a

certain level, otherwise the pair of torsional frequency may superpose each other

(reinforce the oscillations) and the damping factor of the DRWT may be decreased.

Second, the other torsional frequencies located as already outside the high-risk range

shouldn’t move in due to the performance of the GA.

42

: Impact of DRWT on Transient Angle Stability of Chapter 3Network

3.1 Introduction

The main contributions of this chapter for evaluating the fault-ride-through capability

of the DRWT against that of the SRWT will be discussed in this paragraph. To have a

valid study, the comparison is carried out through three methods including: eigenvalue

analysis, critical rotor speed method and numerical simulation. The dynamic model of

the mechanical drive of the DRWT has been designed using the multi-objective method.

The dynamic model of the three-shaft bevel gear is also developed here. Through

employing eigenvalue analysis, it is recognized that the natural damping factor of the

DRWT is greater than that of the SRWT. The critical rotor speed method is used to

compare the transient angle stability margin of the DRWT and the SRWT. The tool of

comparison of this method is the speed rate of change of the wind turbines. The rate of

speed change for the DRWT is discovered to be less than for the SRWT due to its

higher momentum inertia. So, based on the critical speed method, the DRWT is

recognized to have a higher transient angle stability margin compared to the SRWT.

The degree of damping effect for the DRWT and the SRWT, introduced by the

integration of the droop loop into the pitch system, is compared. It is seen that the droop

system of the DRWT provides more damping than that of the SRWT due the auxiliary

turbine droop system. The stream tube effect of the auxiliary turbine is included in the

aerodynamic modelling of the DRWT, which makes this model more accurate in

comparison to the previously reported model.

This chapter is organized as follows: in section 3.2, the multi-objective method is

described and used for developing the drive trains of the DRWT and SRWT; dynamic

mechanical models of different components of the DRWT and SRWT are presented in

section 3.3; in section 3.4, state space equations of the turbine generator set have been

derived for eigenvalue analysis; the ‘critical rotor speed’ approach for assessing the

transient angle stability of induction generators is explained in detail in section 3.5 as a

tool for comparing the DRWT and the SRWT; the impact of the droop loop, integrated

into the pitching system, on the fault-ride through strength of the DRWT and the SRWT

is investigated in section 3.6; an aerodynamic model is introduced in section 3.7 for the

DRWT to show the stream tube effect of the auxiliary turbine on the main turbine more

43

precisely; and finally, computer simulation results are conducted in section 3.8.

3.2 Drive Train Modeling through Multi-objective Method

In any drive train model, based on the multi-objective method, each component of the

mechanical system is individually presented by a body-mass, connected to the adjacent

components through parallel combinations of spring and damper. Springs representing

the stiffness of the interface shaft and dampers are employed to simulate the damping

factor of the system due to the friction. A simple general form of an N-mass drive train

model is given in Fig. 3.1, as follows:

Fig. 3.1 General form of an N-mass drive train

where DB1, DB2, …., DBN represent the torque loss of each individual mass due to the

external damping such as friction. K12, K23, …., K(N-1)N are the elasticity of the interface

shafts which connect the adjacent masses. The effect of mutual damping between the

masses which are next to each other is modelled by D12, D23, …., D(N-1)N. And finally J1,

J2, …. , JN are the inertia of the masses. The quantities which are essential for assessing

the dynamic response of a mechanical prime mover are the angular velocity of the

components which can be presented by ωB1, ωB2, …. , ωBN; angular position of the

elements are shown by θB1, θB2, …. , θBN ; and the acting torque on each element is TB1,

TB2, …. , TBN. Usually four models of drive train are introduced for studying the wind

turbines, as follows [89]:

• One-mass or lumped model

• Two-mass shaft model

• Three-mass drive train model

• Six-mass drive train model

As the mechanical system is modeled with a higher number of masses, the accuracy of

the study is presumed to be higher. However, for some studies this accuracy is

unnecessary. The models with a higher number of masses are appropriate for transient

DB1 DB2 DB3 DB4 DB(N-1) DBN

d12 d23 d34 dN(N-1)

K(N-1)N K12 K23 K34

J1 J2 J3 J4 JN-1 JN

44

studies, while for dynamic and steady state conditions, this accuracy is pointless and

somewhat time-consuming for numerical solvers. Since the main focus of this research

is on the transient response of the wind turbine, the most appropriate model is the six-

mass model. The masses that represent the components of the turbine are mentioned

below:

• Blades are modelled by two mass

• Hub of the turbine is modelled by one mass

• Gear box is modelled by two masses

• Generator is modelled by one mass

Since, in a complicated drive train, all elements are linked to each other two by two, the

first and essential step in learning how to obtain the dynamic model of a drive train is to

develop a mathematical model of a two-mass combination. Subsequently, this model

can be extended for the general N-mass system. Fig. 3.2 shows a basic mechanical

system [90]. The shaft is modelled by a damper and a spring. The rotors are presented

by their corresponding masses and dampers.

Fig. 3.2. Mechanical elements of a two-mass system

Where:

Tls , Trs , Tem , Ta : Torques in different sides of the system.

dls : Mutual damping between the masses.

dr , dg : Self-damping of each body.

Kls : Shaft stiffness.

ωg , ωr : Angular velocity of the masses.

The first-order differential equation, which is famous as the ‘swing equation’, is

employed to demonstrate the dynamic of each mass:

45

(3.1)

The dynamic model by (3.1) is for two individual masses that, because they are

rotating independently, cannot interact with each other, while practically, they are

connected together through a shaft. To overcome this problem, the equation which

describes the interfacial effect of the shaft should be worked out as follows:

(3.2)

Through using (3.1) and (3.2) simultaneously, it is possible to model the transition of

the dynamics from one mass to the other. Now there are enough materials for

developing the dynamic model for an N-mass drive train. The extended model in given

by (3.3):

𝐽1. 1 = 𝑇𝑀 − 𝑇1 − 𝐷𝐵1.𝜔1

𝑇2 − 𝑇1 = 𝐾12. (𝜃1 − 𝜃2) + 𝑑12. (𝜔1 − 𝜔2)

𝐽2. 2 = 𝑇2 − 𝑇3 − 𝐷𝐵2.𝜔2

𝑇3 − 𝑇2 = 𝐾23. (𝜃2 − 𝜃3) + 𝑑23. (𝜔2 − 𝜔3)

𝐽3. 3 = 𝑇3 − 𝑇4 − 𝐷𝐵3.𝜔3

… … … … ….

𝐽𝑁−1. 𝑁−1 = 𝑇𝑁−1 − 𝑇𝑁 − 𝐷𝐵(𝑁−1).𝜔𝑁−1

𝑇𝑁 − 𝑇𝑁−1 = 𝐾(𝑁−1)𝑁 . (𝜃𝑁−1 − 𝜃𝑁) + 𝑑(𝑁−1)𝑁 . (𝜔𝑁−1 − 𝜔𝑁)

𝐽𝑁 . 𝑁 = 𝑇𝑁 − 𝑇𝐸 − 𝐷𝐵𝑁 .𝜔𝑁

(3.3)

where TM is the torque given to the mechanical drive by the main source of energy, such

as aerodynamic torque captured from wind, or the torque that is captured from the

ocean waves by wave farms. TE is the torque delivered to the destination by the

rrlsarr dTTJ ωω .−−=

ggemhsgg dTTJ ωω −−=

)()( rslslsrslslshsls dKTT ωωθθ −+−=−

46

mechanical drive. For instance, it can be the electromagnetic torque of the generator in

the case of modelling the drive train of generating units.

3.3 Dynamic Model of DRWT and SRWT Components

In this section, the dynamic models of different components of single and dual-rotor

wind turbines are discussed. Fig. 3.3a and Fig. 3.3b show the element arrangement of

the single and dual-rotor wind turbines, respectively.

a) Single-rotor wind turbine

b) Dual-rotor wind turbine

Fig. 3.3. Element arrangement of SRWT and DRWT

3.3.1 Gear Box

Normally, the rotational speed of the turbines is much lower than the speed of the

generators. Gearboxes in wind turbines are essential to adapt the speed of the turbines to

that of the generator.

In the SRWT, the shafts of the turbine and generator are in parallel so the spur gear

can be a good choice for speed adaptation. Spur gears are the most common type of

gears. They have straight teeth, and are mounted on parallel shafts. Sometimes, many

47

http://science.howstuffworks.com/transport/engines-equipment/gear2.htm

spur gears are used at once to create very large gear reductions. The common figure of

this component is given in Fig. 3.4a. For the DRWT, due to the lack of space, the

generator is located in the tower of the wind turbine, while the main and auxiliary

turbines have the same position as in the SRWT. So, the shaft of the generator and the

shafts associated with the turbines are perpendicular. Bevel gears are useful when,

torque is transferring between the shafts that have angle more than 0 with each other.

They are usually mounted on shafts that are 90 degrees apart, but can be designed to

work at other angles as well. The teeth on bevel gears can be straight, spiral or hypoid.

Bevel gears with straight teeth actually have the same problem as spur gear with straight

teeth, because, as each tooth engages, it impacts the corresponding tooth all at once. A

typical bevel gear is presented in Fig. 3.4b.

a) Spur gear b) Bevel gear

Fig. 3.4. Employed gears in SRWT and DRWT

In the present study, the spur gear is considered as a two-mass object. Dynamic

models for spur gearbox in the SRWT and the bevel gearbox employed in the DRWT

are presented in Fig. 3.5 and Fig. 3.6, respectively.

Fig. 3.5. Dynamic model of one stage spur gear box

48

Fig. 3.6. Dynamic model of the 2 stage bevel gear with 3 shafts

By considering a zero backlash for the transmission mechanical system, the spur

gearbox dynamic model, which is the interface between two parallel shafts, is given by

(3.4), as follows [91]:

(3.4)

where the definitions of parameters in Fig. 3.5 are as follows (please refer to Appendix

C):

J1 Inertia of pinion disc in spur gear;

J2 Inertia of wheel disc in spur gear

r1 , r2 Radiuses of pinion and wheel in spur gear;

d1, d2 Damping coefficients of spur and bevel gears;

d12 Mutual damping between the pinion and wheel

K12 Stiffness functions of contact point of the spur gear;

T1 Torque at the connection of shaft and pinion;

T2 Torque at the connection of shaft and wheel;

θ1 Rotational angle of pinion in spur;

θ2 Rotational angle of wheel in spur; The average radius in the bevel gear can be obtained as follows:

1112211121221112111 .][][. θθθθθθ dTrrdrrrKrJ −=++++

2222211122221112222 .][][. θθθθθθ dTrrdrrrKrJ −=++++

γsin.11 Ldr av −=

)90sin(.22 γ−−= Ldr av

49

where:

L : Face width.

d1 : Outside diameter of the pinion.

d2 : Outside diameter of the wheel.

γ : Pitch cone angle.

Through comparing Fig. 3.5 and Fig. 3.6, there are a number of dissimilarities

between the gearboxes used in single and dual-rotor wind turbines. The differences are

due to two main reasons. The major cause is the difference between the number of

components connected through gearboxes, since, according to Fig. 3.3a, in a single rotor

system, there are two rotors, while in Fig. 3.3b, the gearbox is connecting three shafts in

the dual rotor system. So, the configurations of the differential equations that define the

dynamic behaviours of the gearboxes are different. The minor one is related to the

structural dissimilarity of the spur and bevel gears that are supposed to transmit the

torque [92]. The spur gear transfers the power to two parallel shafts, while the bevel

gear transmits the power between two perpendicular shafts. Thus, the formations of the

teeth in the spur and bevel gears are different. This issue affects the methods for

obtaining the coefficients of the differential equations of gearboxes. In other words, the

configurations of the differential equivalents determining the response of the gearboxes

that connect only two shafts are similar. There are some differences in the values of the

coefficients in equation (3.4).

Referring to Fig. 3.6, we have derived the bevel gearbox dynamic model, which links

three shafts. It is presented in (3.5) as follows:

(3.5)

J1: Inertia of the first pinion disc in bevel gear;

J2: Inertia of wheel disc in bevel gear;

J3: Inertia of the second pinion disc in bevel gear;

1112211121221112111 .]..[]..[.. θθθθθθ dTrrdrrrKrJ av −=++++

222332232333222333311123221112322 .][][].[.][. θθθθθθθθθθ dTrrdrrrKrrrdrrrKrJ −=++++++++

3332233233223323333 .][][ θθθθθθ dTrrdrrrKrJ −=++++

50

r1av: Average radius of the first pinion in bevel gear;

r2av: Average radius of the wheel in bevel gear;

r3av: Average radiuses of the second pinion in bevel gear;

d1: Damping coefficient of the first pinion in bevel gear;

d2: Damping coefficient of the wheel in bevel gear;

d3: Damping coefficient of the first pinion in bevel gear;

d12: Mutual damping between the first pinion and wheel;

d23: Mutual damping between the second pinion and wheel;

K12: Stiffness functions of contact point between the first pinion and wheel;

K23: Stiffness functions of contact point between the second pinion and wheel;

T1: Torque at the connection of shaft and first pinion;

T2: Torque at the connection of shaft and wheel;

T3: Torque at the connection of shaft and second pinion;

θ1: Rotational angle of the first pinion in bevel gear;

θ2: Rotational angle of wheel in bevel gear;

θ3: Rotational angle of the second pinion in bevel gear;

The stiffness of the contact point is a time variable quantity depending on the number

of teeth engaged with each other. The stiffness variation for each cycle can be

considered to be a minimum value when one pair of teeth are engaged and a maximum

value when there are two contact points. The profile of the stiffness is shown in Fig. 3.7

[93].

Fig. 3.7. The profile of the stiffness

In [93], an exact equation has been formulated for modeling the vibratory effects of

the spur gearbox. However, this accuracy is pointless for evaluating the dynamic

performance of the whole wind turbine system. So, it is possible to calculate the average

value of the stiffness and employ it as a constant for the purposes of simplicity, without

51

its affecting the validity of our studies. The average stiffness is given in (3.6):

(3.6)

Where:

ε : Contact ratio

Kmin : Minimum stiffness

Kmax : Maximum stiffness

Trd : Stiffness cycle time

The minimum and maximum stiffness can be calculated by (3.7), as addressed in [91]:

(3.7)

where

E : Young Module

L : Length of Tooth Contact

υ : Poisson Coefficient

From equation (3.8), it is obvious that, at the same length of contact, minimum and

maximum stiffness are only influenced by the material of the gear. So the contact

stiffness of the spur and bevel gears are the same if they are made by the same material

and method. The contact ratio of the spur gear is presented in [94]:

(3.8)

where:

Ra : Wheel base circle radius;

Rb : Wheel external radius;

ra : Pinion base circle radius;

rd

rdrdav T

TKTKK )2.().1.( minmax εε −+−=

)1(4..

2min υπ−

=LEK

minmax .2 KK =

απα

εcos..

sin).(2222

g

baabab

mRRrrRR +−−+−

=

52

rb : Pinion external radius;

rc : Distance between the centers of two base circles;

α : Pressure angle;

mg: Module of the gear;

One conventional method for analyzing the bevel gear is to obtain its related

equivalent spur gear [94] . The contact ratio of the bevel gear can be well described by

(3.9):

(3.9)

where: Raeq : Wheel base circle radius of an equivalent spur gear;

Rbeq : Wheel external radius of an equivalent spur gear;

raeq : Pinion base circle radius of an equivalent spur gear;

rbeq : Pinion external radius of an equivalent spur gear;

Rvp : Pinion back cone distance;

Rvw : Wheel back cone distance;

With:

where:

απ

αε

cos..sin).(2222

g

vwvpaeqbeqaeqbeqbevel m

RRrrRR +−−+−=

γcos.2p

vp

dR =

)90cos(.2 γ−= w

vwd

R

avwbeq hRR +=

avpbeq hRr +=

αcos.vwaeq RR =

αcos.vpaeq Rr =

53

ha : Addendum

dp , dw : Pinion & wheel pitch diameter respectively.

So, given the same mass and geometry, the contact ratio of the gears would be different

from each other. The damping coefficient of the spur gear is given by (3.10), as

addressed in [94]:

(3.10)

We may rewrite the equation (3.10) for the bevel gear:

(3.11)

Where ξ is the damping rate which varies between 3% to 17%, depending on the material type.

3.3.2 Blade Bending Model

New designs of wind turbine continue to increase the rotor size in order to extract

more power from wind. As the rotor diameters increase, the flexibility of the rotor

structure increases, as does the influence of the mechanical drive train on the electrical

performance of the wind turbine. When the length of the rotor blades increases, the

frequencies of the torque oscillations reduce, and these oscillations may then interact

with low frequency modes of the electrical network. These oscillations must be taken

into account when analysing the dynamic performance of FSIG wind turbines for

transient stability. So, it is important to consider the effect of the blades in any

investigation of the dynamic behaviour. The combination of the hub and blades can be

presented by a two mass model. Fig. 3.8 shows the drive train model of the blades. In

this method, the blades are divided into two parts [95].

Equation, (3.12) defines the dynamic behaviour of the two mass model of the hub and

blade combination. Jflex presents the momentum inertia of the flexible part of the blade

and Jrig shows the momentum inertia of the rigid part of the blade. Two masses are

coupled together by the blade stiffness Kblade.

(3.12)

221

212

21min

.....

..2rJrJJJK

d spurz +=

εξ

22

21

min

.....

..2avpavg

gpbevelm rJrJ

IIKC

+=

εξ

)()( hubbladebladehubbladebladembladeflex dKTJ ωωθθθ −−−−=

54

where:

θblad, θhub,θsh are the rotating angle of the blades, hub and shaft, respectively.

ωblad,ωhub,ωsh are the rotating speed of the blades, hub and shaft, respectively.

By having the fundamental equations of each mechanical element, such as shaft, rotor,

gearbox and blades, it is possible to obtain the dynamic model of the whole mechanical

system for dual and single-rotor wind turbines. The all-inclusive plot which shows the

relationship between mechanical elements in both types of turbines is given in Fig. 3.9,

where the torques and speeds with index “s” and “d” indicate the variables in single and

dual-rotor systems, respectively.

a) Three blades connected to the hub

b) Equivalent torsional representation

Fig. 3.8 Two-mass model of the blades

3.3.3 Shaft and Rotor System

As shown in Fig. 3.3, dual and single-rotor wind turbines have different arrangements

of shafts, rotors and blades. The role of the shafts as interfaces is well described as a

general concept in section (3.2). The method can be extended for the DRWT and

SRWT. The drive train element arrangement of both technologies is presented in Fig.

3.9.

)()()()( shhubshshhubshbladehubbladebladehubbladehubrig dKdKJ ωωθθωωθθθ −−−−−−−−=

55

Fig. 3.9. General mechanical block diagram of variable speed wind turbine

Wind Speed

Auxiliary Rotor Aerodynam

icStream

Tube Effect Calculation

Wind Speed on the M

ain Blades

High Speed Shaft

Main Rotor

Aerodynamic

1dT

1dω

Bevel Gear2d

T

2dω

ω

Generator Shaft

6dT

6dω

Generator

7dT

7dω

Rotor Aerodynam

icLow Speed

Shaft1S

T

1Sω

Spur Gear

3S

T

3 Sω

2S

T

2Sω

Dual Rotor Turbine

Single Rotor Turbine

Auxiliary Blade BendingDynam

ic

0dT

Main Blade

BendingDynam

ic

0ST

0Sω

0dω

56

3.4 SRWT and DRWT State Space Model

To prove the validity of our studies regarding the transient response of the dual and

single-rotor wind turbines, the natural damping characteristic of the DRWT and SRWT

can be evaluated through eigenvalue analysis. The location of the eigenvalues in each

system is a powerful aid to predict the damping factor of the system. In order to get to

this stage, the state space of the induction generator must be combined with the SRWT

and DRWT separately. Both dual-rotor and single-rotor systems, as well as the

induction generator, should be linearized over the operating point.

3.4.1 Induction Generator Model

A 4th order dynamic model of an induction generator (IG) is presented by (3.13):

(3.13)

where

The expressions of AG and BG are listed in appendix D [96].

3.4.2 Turbine Model

The state space model associated with the dual and single-rotor wind turbines is

presented in (3.14), as follows:

(3.14)

where state and input variables in dual-rotor systems can be identified based on Fig. 3.9.

(3.15)

GGGGG uBxAx .. +=

TdrqrdsqsG vvvvu ],,,[=

TdrqrdsqsG iiiix ],,,[=

TTTTT uBxAx .. +=

TddddDRWTx ],.....,,,......,[ 7070 δδωω=

57

The same is true for state and input variables for single-rotor wind turbines:

],.....,,,......,[ 4040 SSSSSRWTx δδωω=

(3.16)

Mechanical state variables (xs , xd) and electrical state variables (xG) should be linked

together through electromagnetic torque provided by the generators (TS4 ,Td7). The

normal format of the electromagnetic torque (Te) is shown in (3.17):

(3.17)

Equation (3.17) is a nonlinear equation and cannot be used as a state variable. The linear

format of Te is composed of electrical state variables, as follows:

(3.18)

with idr0, iqr0, ids0, idr0 as the initial values of the generator stator and rotor currents. TS4 in

us and Td7 in ud must be replaced by their linear format, which is presented in (3.18).

Mechanical parameters for both the SRWT and DRWT can be estimated with high

accuracy through the equations suggested by [97]. The estimation is made feasible by

identifying the obvious and easily accessible parameters, such as the length of the main

and auxiliary blades, the rating power, gearbox ratio, etc.

3.5 Critical Rotor Speed of Induction Generator

Throughout the faulty condition in the grid, generating units accelerate to higher speeds

due to the lack of electromagnetic torque. The electrical torque is proportional to the

square of the terminal voltage and since, during the short circuit, the voltage falls down,

the electrical torque is reduced dramatically in this period, while the input mechanical

torque remains constant. Therefore, the transient stability of an induction generator can

be assessed through investigating the time domain response of the rotor speed after

removing the short circuit. The stability margin is strongly influenced by the fault

clearing time and potential of the generator for acceleration. To reach the stability

during the post fault period, the fault must be cleared before the speed of the generator

reaches the critical speed. The idea of critical speed was initiated for the first time by

TdddDRWT TTTu ],,[ 730=

TSSSRWT TTu ],[ 40=

)..( dsqrqsdrme iiiiLT −=

)( 0000 qrdsdsqrdrqsqsdrme iiiiiiiiLT ∆−∆−∆+∆=

58

[71] and the methodology was further investigated in [50]. A brief description of the

method is given below.

The critical speed method is developed based on the torque speed characteristic curve of

the induction machine. To obtain the mathematical form of the curve, the steady state

electrical circuit of the induction generator is required. This is given in Fig. 3.10.

Fig. 3.10. Induction generator circuit model

In Fig. 3.10, XR, XS, RR and RS are, respectively, rotor reactance, stator reactance, rotor

resistance and stator resistance. ‘s’ is the rotor slip and VT is the terminal voltage. The

electrical torque TE can be calculated based on the machine impedances and terminal

voltage. The equation is given by (3.19):

𝑇𝑒(𝜔𝑟) =𝑉𝑇2

𝜔𝑟.

𝑅𝑇(𝜔𝑟)𝑅𝑇2(𝜔𝑟) + 𝑋𝑇2(𝜔𝑟)

(3.19)

where RT(ωr) and XT(ωr) are, respectively, the Thevenin resistance and reactance of the

induction generator with [51]:

𝑅𝑇(𝜔𝑟) = 𝑅𝑆 +

𝑅𝑅𝜔𝑟 − 1𝑋𝑀

2

( 𝑅𝑅𝜔𝑟 − 1)2 + (𝑋𝑀 + 𝑋𝑅)2

𝑋𝑇(𝜔𝑟) = 𝑋𝑆 +𝑋𝑀( 𝑅𝑅

𝜔𝑟 − 12

+ 𝑋𝑅(𝑋𝑀 + 𝑋𝑅))

( 𝑅𝑅𝜔𝑟 − 1)2 + (𝑋𝑀 + 𝑋𝑅)2

If the mechanical prime mover is assumed to be constant with respect to rotational

speed, then, through sketching the torque-speed characteristic curve of the mechanical

drive and induction generator in the same plot, it is possible to find the equilibrium

points. The layout is given in Fig. 3.11.

Rs Xs Xr

Xm Rr/s VT

59

Fig. 3.11. Torque speed characteristic curve

As can be seen from Fig. 3.11, there are two cross-sections between the torque-speed

characteristic curves of the mechanical drive and the induction generator. Thus, there

are two operating points for the combination of the drive and the generator; one of them

is unstable dynamically and the other is stable. The stable operating point is ωst, which is

located in that area of the torque speed plot where the slop is positive. The main reason

that this operating point is considered stable is the proper response of the electrical

toque to the speed variation. For instance, in the case of any speed drop, the electrical

torque is reduced as well and this response assists the generating unit to be stabilized in

a new operating point. However, on the other hand, the ωcr is unstable due to the

inappropriate response of the electrical torque. For the area of the torque speed curve

where the slop is negative, the speed drop results in the growth of the electrical torque.

Since, during the transients the mechanical curve stays constant, the growth of the

torque causes further reduction of the speed. Due to this interaction between the

rotational speed and electrical torque, the speed fall may become progressive and the

generator may stall. It is possible to calculate the value of the speeds that correspond to

the stable and unstable operating points. During the steady state, there is no fluctuation

for the generator speed and its derivative is zero. So, according to the swing equation,

which is given by (3.20), the mechanical torque (TM) and electrical torque (TE) should

be the same.

𝑑𝜔𝑅𝑑𝑡

=1

2𝐻(𝑇𝑀 − 𝑇𝐸)

(3.20)

60

So, in a steady state condition, the TE in (3.19) can be replaced by TM. Through taking

the TM to the right side of the (3.19) and equating them to zero, a second order

algebraic equation is achievable. The equation is presented by (3.21) which provides

enough material to obtain the value of the speeds for stable and unstable operating

points [71].

(𝑅𝑇𝐻2 + (𝑋𝑇𝐻 + 𝑋𝑅)2). 𝑠2 + 2.𝑅𝑇𝐻.𝑅𝑅 − 𝑅𝑅 . 𝑉𝑇𝐻2

𝑇𝑀 . 𝑠 + 𝑅𝑅2 = 0 (3.21)

The answers for slip ‘s’ are illustrated by (3.22) :

sst=

(3.22)

𝑅𝑅 .(𝑉𝑇𝐻4

𝑇𝑀2− 𝑋𝑅2 − 2.𝑋𝑟.𝑋𝑇𝐻 − 𝑋𝑇𝐻2 − 𝑅𝑇𝐻.𝑉𝑇𝐻

2

𝑇𝑀) + 0,5.𝑅𝑅 .𝑉𝑇𝐻

2

𝑇𝑀− 𝑅𝑅 .𝑅𝑇𝐻

(𝑅𝑇𝐻2 + 𝑋𝑅2 + 2.𝑋𝑟 .𝑋𝑇𝐻 + 𝑋𝑇𝐻2 )

scr=

−𝑅𝑅 .(𝑉𝑇𝐻4

𝑇𝑀2− 𝑋𝑅2 − 2.𝑋𝑟 .𝑋𝑇𝐻 − 𝑋𝑇𝐻2 − 𝑅𝑇𝐻.𝑉𝑇𝐻

2

𝑇𝑀)− 0,5.𝑅𝑅 .𝑉𝑇𝐻

2

𝑇𝑀+ 𝑅𝑅 .𝑅𝑇𝐻

(𝑅𝑇𝐻2 + 𝑋𝑅2 + 2.𝑋𝑟 .𝑋𝑇𝐻 + 𝑋𝑇𝐻2 )

where slips sst and scr are associated, respectively, with stable speed ωst and unstable

operating point ωcr.

According to this, if, during a short circuit, the induction generator accelerates to speeds

higher than the critical speed, the generator then becomes unstable. Conversely, when

the clearing fault is relatively short and does not let the speed reach beyond the critical

speed, then the generator resumes the stable condition.

Fig. 3.12 shows two cases: in the first case, the fault was cleared when the rotational

speed was still less than the critical speed; in the second, the speed exceeded the ωcr and

then the fault was removed. It can be seen from Fig. 3.12a, that at the time of the fault,

the electrical torque drops down to almost zero (point ‘B’) and the generator speeds up

to point ‘C’. At point ‘C’, the fault is removed and the operating point jumps to point

‘D’. Since, at point ‘D’, the electrical torque is higher than the mechanical torque (TM),

then, according to (3.20), the generator experiences a negative acceleration and slows

61

down to point ‘A’ again. It is assumed that the pre and post-disturbance characteristic

curves are the same. The track of the operating point excursion is highlighted by arrows.

For the second case in Fig. 3.12b, the short circuit is removed when the speed is already

higher than the critical speed. The operating point jumps to point ‘D’ while there is still

a lack of electrical torque. Since, at point ‘D’, the mechanical torque is higher than TE,

then based on (3.20) the rise in speed would be progressive and stability would be lost

[50], [71].

Now it is time to investigate which turbine –the DRWT or the SRWT– accelerates to

higher values during the fault. The one that has a higher rate of acceleration, introduces

a less transient angle stability margin during the fault. In this section, an analytical

solution is proposed to pave the way for comparing the acceleration rate of the SRWT

and the DRWT throughout the faulty condition.

a) Removing the fault before reaching to the critical speed

b) Removing the fault after passing the critical speed

Fig. 3.12. The post-fault excursion of the operating point

62

The total momentum inertias of the generating units in the SRWT and DRWT are

given in (3.23) and (3.24), respectively:

𝐽𝑆𝑅 = 𝐽𝐺 +𝐽𝑀𝑇𝑛𝑚2

(3.23)

𝐽𝐷𝑅 = 𝐽𝐺 +𝐽𝑀𝑇𝑛𝑚2

+𝐽𝐴𝑇𝑛𝑎2

(3.24)

where JG, JMT, JAT are, respectively, the momentum inertia of the generator, the main

turbine and the auxiliary turbines; and nm, na are the ratio of the gearboxes connecting

the main and auxiliary wind turbines to the generator, respectively. The total mechanical

inertia momentum of DRWT (JDR) is higher than the value of the total mechanical

inertia momentum for SRWT (JSR), which is due to the auxiliary turbine in the drive

train of the DRWT.

Given the same amount of change in the electromechanical or mechanical torque

(ΔT), the change in the angular velocity of the generator in SRWT and DRWT are

achievable, respectively, through (3.25) and (3.26):

∆𝜔𝑆𝑅 =1𝐽𝑆𝑅

∆𝑇.𝑑𝑡𝑇𝑁𝑎𝑑𝑖𝑟

0 (3.25)

∆𝜔𝐷𝑅 = 1𝐽𝐷𝑅

∫ ∆𝑇.𝑑𝑡𝑇𝑁𝑎𝑑𝑖𝑟0 (3.26)

where, TNadir is the time taken by the power system frequency to approach its minimum

value (Nadir). The ratio between the rotational speed change in SRWT (ΔωSR) and

DRWT (ΔωDR) can be obtained by dividing (3.25) over (3.26):

∆𝜔𝑆𝑅 = 𝐽𝐷𝑅𝐽𝑆𝑅

.∆𝜔𝐷𝑅

(3.27)

For the same time period of TNadir, since JDRWT is always higher than JSRWD, the

angular velocity of the SRWT reaches higher values compared to those reached by the

DRWT during the fault. So, theoretically, it can be concluded that, based on the critical

rotor speed criterion, the SRWT has less margin with the critical speed and

consequently is more susceptible to loss of stability.

63

3.6 Damping Effect of Droop Control System

The conventional blade pitch angle control strategies are categorised mainly as: i)

Generator power control: and, ii) Generator rotor speed control. During the fault, the

electrical power drops down to a very low value and the generators accelerate.

Throughout the fault, constant speed wind turbines increase the pitch angle in order to

reduce the captured aerodynamic power and keep the speed constant. On the other hand,

constant power wind turbines decrease the pitch angle to restore the electric power and

this reaction pushes the generator to accelerate more, which is harmful for the stability

of the generator. During normal operation, a wind turbine is usually working in constant

power mode to extract maximum energy from the wind. In the case of any faulty

condition, the control system is switched to constant speed mode [98]. The block

diagram of pitch control in constant speed mode is presented in Fig. 3.13.

Fig. 3.13. Generator constant speed pitch control system

The dynamic of the generator speed can be described as (3.28):

(3.28)

where Dr is the rotor damping factor, Tm is the torque from the prime mover and Te is

the electromagnetic torque. To simplify the analysis of the damping effect of the speed

mode on the wind turbine, the control system in Fig. 3.13 is replaced by a droop

characteristic curve of torque versus speed. In this study, both main and auxiliary

turbines vary the captured aerodynamic torque for regulating the power in the DRWT

through the droop curve. Based on this scheme, the simplified relationship between the

generator speed and mechanical torque over the operating point can be illustrated as

follows:

rremr DTTH ωω ∆−∆−∆=∆ ...2

64

rauxrmainm KKT ωω ∆−∆−=∆ .. (3.29)

where, Kmain and Kaux are the droop factor of the main and auxiliary turbines

respectively.

According to (3.30), an electromagnetic torque (Te) can be decomposed into a

damping torque and a synchronizing torque [95]:

(3.30)

By substituting (3.29) and (3.30) in (3.28), it is possible to obtain the damping factor of

the DRWT, which is the coefficient of Δωr. The damping factors for the DRWT are

presented by (3.31), as follows:

(3.31)

If the same procedure is followed for the SRWT, then the damping factor is:

(3.32)

Kaux is equal to zero for the SRWT. The main purpose of the equations (3.29) and (3.30)

is the relative comparison of the damping factors of DRWT and SRWT rather than

precise calculation of their damping factors. So, for more simplicity the delay of the

servomotor system is assumed to be small and ignorable. Meanwhile, the equation is for

steady state condition when the system is settled down regardless the amount of delays.

On the other hand momentum inertias do not introduce enough impact of the overall

damping of the system. The inertia of a rotating mass and mechanical damping are two

distinct parameters. To summarize, excluding the delay of the servo system doesn’t

reduce the accuracy of the study enough. Meanwhile the servo system delays are

considered in the simulations.

From (3.31) and (3.32), it is obvious that, in speed control mode, the DRWT

presents, in faulty conditions, more damping compared to the SRWT under the same

condition. To verify the validity of the simplifications made in this section, both DRWT

and SRWT must be modelled and the transient responses of the wind turbines must be

simulated to assess their damping characteristics.

rdse TTT ωδ ∆+∆=∆ ..

)( auxmainrdDRWT KKDTD +++=

)( mainrdSRWT KDTD ++=

65

3.7 Aerodynamic Model for DRWT

The aerodynamic model of the DRWT is, to some extent, different to that of the

SRWT. Since the wind flowing through the main turbine in the DRWT is disturbed by

the auxiliary turbine, the stream tube effect must be included in the aerodynamic torque

calculations for the DRWT. The aerodynamic torque introduced by the blades is, via

(3.33), as follows:

(3.33)

with, R as the blade radius, λ the tip speed ratio, ρ the air density and ωM the mechanical

speed of the rotor. Cp can be calculated through (3.34) as follows [65]:

(3.34)

with, β as the pitch angle.

The same method can be followed for the main and auxiliary turbines. Tip speed

ratios for the main and auxiliary turbines are calculated through (3.35) & (3.36),

respectively.

(3.35)

(3.36)

where, V1 is the wind speed on the auxiliary wind turbine and VM is the speed of the

unified wind on the main turbine. So, the essential element for calculating the tip speed

ratio is the wind speed on the main and auxiliary turbines. Obtaining the wind speed on

the auxiliary turbine is straightforward; however, the calculation of the wind speed on

the main turbine requires further investigation.

In the dual-rotor wind turbine, the auxiliary rotor disk faces the upstream wind first

and the main turbine is normally hit by the wind disturbed by the auxiliary turbine. The

downstream wind velocity distribution on the main blade is assumed to be composed of

two parts – the disturbed and undisturbed portions with the speed of V’2 and V1,

respectively. This phenomenon is shown in Fig. 3.14a. In [59], the expansion of the

stream tube behind the auxiliary rotor disk was neglected. To obtain more precise

325 ....5.0 λωπρ MPM CRT =

λβλ

βλ λ 0068.0).54.0116(517.0),(21

+−−=−

ieCi

P

1035.0

08.011

3 +−

+=

ββλλi

1. VRAuxAuxAux ωλ =

MMainMainMain VR.ωλ =

66

results, calculations performed in [59] must be revised by considering the expansion

effect, as shown in Fig. 3.14b.

According to the mass flow rate theory by (3.37):

(3.37)

Therefore, to obtain the area of the disturbed wind (A’2) on the main turbine disc, it is

necessary to calculate the disturbed wind velocity V’2 immediately next to the main

turbine. Based on (3.38), it is possible to estimate the amount of the wind speed at any

point between the auxiliary and the main blades [99].

(3.38)

where, V’2 is the speed of the disturbed wind next to the main blade and ‘x’ is the

distance between the main and auxiliary turbines. By substituting the obtained V’2 into

(3.37), the area of A’2 at any performance of the auxiliary rotor (different CP) is

achievable.

a) Stream tube of the auxiliary turbine is neglected

12'2

'2 .. AVAV =

)).41

.21(2

111(

21'

2x

xCVV p

++

−−−=

67

b) Stream tube of the auxiliary turbine is included

Fig. 3.14. Stream tube effect of the auxiliary turbine on the aerodynamic performance of the main turbine

One approach for the examination of the dynamic functioning of the dual-rotor is to

analyse the rotors that, aerodynamically, are independent of each other. In other words,

the authors of [59] treated the flow entering the auxiliary and main rotors as two

independent uniform flows, with the speed of V1 and VM, respectively. This matter is

shown in Fig. 3.15. The equivalent uniform flow entering the main rotor (VM) produces

the same aerodynamic torque obtainable from the summation of the disturbed (V’2) and

undisturbed (V1) winds.

Fig. 3.15. Two rotors are aerodynamically independent

Equation (3.39), by employing A’2 and V’

2, delivers the value of the uniform wind speed

(VM) on the main rotor at any performance of the auxiliary rotor [59]:

(3.39)

VM from (3.39) should be replaced in (3.36) for calculating the main turbine tip speed

ratio.

23'

223

1'2

3'2 ..).(... AVAAVAV Mρρρ =−+

68

3.8 Simulation Results

The objective of this study is to investigate and compare, from different aspects, the

dynamic behaviour of the dual and single-rotor wind turbines. Both dual and single-

rotor wind turbines are set up in PSCAD software. To facilitate, a simple power system

has been chosen and is shown in Fig. 3.16. The dynamic models of the wind turbines

are established, based on the component models presented in previous sections. The

generators are connected to the power system through a step-up transformer and a

100km transmission line. The parameters of the generator and mechanical systems are

listed in Appendix A. The pitch angle control employed in this investigation regulates

the speed of the wind generator. The following simulation results compare the

capabilities of the dual and single-rotor wind turbines in the context of transient angle

stability performance.

The behaviour of the wind turbines, when pitch angle control is in operation, are

simulated following a grid three-phase short circuit of 0.3 sec at t =120 sec on the

secondary side of the step-up transformer.

a) SRWT

b) DRWT

Fig. 3.16. Simple power grid connected to either single-rotor or dual-rotor wind turbine

The information provided by Fig. 3.17 is an overview of the responses of the

variables during the fault and post-fault period. Responses of variables relative to the

SRWT are made distinct by bolded lines with circles standing on them. The squares are

SRWTWindGen

#1 #2PI

COUPLED

SECTION

ABC->G

TimedFaultLogic

RL

DRWT

GenWind #1 #2

PI

COUPLED

SECTION

ABC->G

TimedFaultLogic

RL

69

standing on the variables of the DRWT. Fig. 3.17a signifies that the damping torque

introduced by the DRWT is higher than that of the SRWT. From the latter figure, it is

clear that the amplitude of the first swing of the speed of both generators is identical.

The reason for this is that the synchronizing torque introduced by the electrical network

– which helps keeping the generators stable during their first swing – is the same for the

two systems. This is because electrical quantities are identical in the DRWT and SRWT

prior to the occurrence of the fault. From Fig. 3.17b, it is clear that the voltage in the

DRWT system can recover faster than can the SRWT. The reason is the difference

between the generator speed settling time in SRWT and DRWT systems.

According to Fig. B.1 in appendix B, terminal voltage is inversely proportional to the

slip. The generator speed in the SRWT takes longer to be recovered to its nominal

value, as does the terminal voltage in the single-rotor system. Fig. 3.17 confirms the

effect of the constant speed mode of the pitch control on the level of damping,

investigated in Section 3.6. As can be seen from Fig. 3.17c and Fig. 3.17d, active and

reactive powers also have larger oscillations during the post-fault period for the SRWT,

in comparison with the DRWT.

a) Generator angular speed

b) Terminal voltages

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175

y

Generator_Speed_Single_Rotor Generator Speed Dual Rotor

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

y

Line Voltage Single Rotor Line Voltage Dual Rotor

70

c) Active power received by the network

d) Reactive power received by the network

Fig. 3.17. Dynamic response of the wind turbine to a three phase grid short circuit when system stays stable

The pre-fault operating point of the active and reactive power influences the post-

fault oscillations; thus, to make a fair comparison we assume that the active and reactive

powers generated by the dual-rotor and single-rotor systems are the same as before the

fault. The pitch angle control mode is not the only reason for the higher damping level

of the DRWT. The simulation is rerun at the same fault duration when the pitch control

system is disabled and the wind turbine is rotating at constant pitch angle.

Fig. 3.18 shows that the SRWT becomes unstable after removing the fault. According

to Fig. 3.18a, the generator speed swings and the generator encounters over speed. On

the other hand, the DRWT resumes its variables to the pre-fault levels and continues its

power generation. In case of any large generator speed oscillation, the auxiliary turbine

acts as a flywheel. The main purpose of the flywheel in mechanical systems is to

smooth out the destructive oscillations. On the other hand, the flywheel damping effect

of the auxiliary turbine, due to the friction between the blades and wind, on the post-

fault generator speed oscillations is another reason for the higher damping torque in the

DRWT. If the fault lasts more than 0.39 sec, then the dual-rotor system is also unstable.

Therefore, the DRWT is more resistive against network disturbances, which confirms

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

y

Active Power Single Rotor Active Power Dual Rotor

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-2.00

-1.50

-1.00

-0.50

0.00

0.50

y

Reactive Power Single Rotor Reactive Power Dual Rotor

71

the quantitative analysis in section 3.5. In this investigation, the fault ride through

capabilities of the DRWT and SRWT are calculated based on the parameters in tables

A.1 to A.3. To verify the results displayed in Fig. 3.18, the eigenvalue is a very handy

solution. MATLAB function (eig) is the tool used to calculate the eigenvalues of a state

space model of the DRWT and SRWT. The initial values of the generator currents are

achievable through abc-dq0 transformation function in PSCAD, after running the time

domain simulation.

Table 3.1 illustrates the eigenvalues of both the DRWT and SRWT systems. The data

presented in this table reveals that the number of eigenvalues introduced by the DRWT

is more than that of the SRWT. Since the mechanical parameters of the main turbines in

the DRWT and SRWT are the same and the generators are identical, some of the natural

frequencies in both systems are quite close together.

a) Generator angular speed

b) Terminal voltages

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

0.950

1.000

1.050

1.100

1.150

1.200

1.250

y

Generator_Speed_Single_Rotor Generator_Speed_Dual_Rotor

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

y

Line_Voltage_Single _Rotor Line Voltage Dual Rotor

72

c) Active power received by the network

d) Reactive power received by the network

Fig. 3.18. Instability of single-rotor wind turbine when pitch angle control is disabled

For the pairs of natural frequencies presented by the DRWT and SRWT, which are

close enough, the real part of the eigenvalue in the DRWT is more negative. This matter

signifies that installing the auxiliary turbine in the DRWT increases the damping factor

of the natural frequencies common to the DRWT and SRWT. For example, the natural

frequency in the third row in the DRWT and the natural frequency in the second row of

the SRWT are almost the same. However, since the real part for the DRWT is higher,

the oscillations caused by this natural frequency would be damped faster, compared to

the SRWT. This verifies the simulation results presented in the previous sections. The

effect of the controller is excluded from the modal analysis. The modal analysis is only

based on the dynamic model of the generator and the turbines. Since the focus was on

the influence of the auxiliary turbine on the overall damping factor of the DRWT, the

type of the modes hadn’t been included in the study.

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

y

Active_Power_Single _Rotor Active_Power Dual Rotor

Main : Graphs

115.0 120.0 125.0 130.0 135.0 140.0 145.0 150.0 155.0

-2.00

-1.50

-1.00

-0.50

0.00

0.50

y

Reactive_Power_Single _Rotor Reactive_Power_Dual_Rotor

73

Table 3.1

Eigenvalues presented by DRWT and SRWT

DRWT SRWT

1 -0.9228 ±j 0.2737 -0.4583 ±j 0.3435

2 -0.1440 ±j 0.8543 -0.3748±j1.6603

3 -0.4211 ±j 1.6606 -10.6351 ±j 4.9604

4 -1.2357 ±j 2.6463 -9.5895 ±j190.6651

5 -11.3546 ±j 5.0993 -8.0281±j312.67

6 -10.8854 ±j 190.6042 0

7 -9.1439±j316.34 -11.6401

8 0 -673.2330

9 -3.6866 -1271.6

10 -11.3882

11 -689.5295

12 -1292.6

13 -2131.4

The number of states in DRWT mechanical drive is 16 (equation (3.15) and Fig. 3.9)

while its 10 in case of SRWT. The number of state variables is correctly mentioned as 4

as it is used in simulation coding. So, the number of eigenvalues should be 20 and 14

for DRWT and SRWT respectively. If you count the number of eigenvalues in DRWT

and SRWT in table 3.1, they are 20 for DRWT and 14 for SRWT which are the same as

the number of the turbine and generator set in terms of number of eigenvalues.

Regarding the dimension of the AG in appendix D it is worth mentioning that in the

appendix three elements of the currents of rotor and stator including d, q, 0 are

mentioned while in simulations only d and q components have been implemented and

zero element is ignored.

74

3.9 Conclusions

In this chapter the transient angle stability margin of the DRWT has been evaluated and

compared against that of the SRTW. The stream tube effect was included in the

aerodynamic torque calculation, a feature ignored in the previous literature. So, the

aerodynamic model of the DRWT in this chapter was more accurate compared to

previous investigations. A multi-objective method was adopted to obtain the dynamic

model of each component and to create the drive train of the DRWT. Through

eigenvalue analysis, it was seen that, after inserting the state equations of the auxiliary

turbine into the state space model of the combination of the main turbine and the

generator (the control system state equations were excluded), the real part of the some

of the eigenvalues is moved leftward, which shows that the natural damping of the

DRWT is higher than that of the SRWT. The ‘critical rotor speed’ approach was used to

compare the margin of the transient angle stability of the DRWT against that of the

SRWT. The acceleration rate of both systems is a key factor to identify which one is

more susceptible to transient angle instability. It was seen that the acceleration rate of

the SRWT is higher than that of the DRWT in the same situation. This was because of

the extra momentum inertia produced by the auxiliary turbine in the DRWT. It signifies

that the transient angle stability margin of the DRWT is higher than that of the SRWT.

The application of droop loops is able to improve the damping factor of the wind

turbines. It was shown that the integration of the droop loop into the pitch control

system introduced a higher damping degree to the DRWT rather than to the SRWT.

This was due to the functioning of the droop loop of the auxiliary turbine.

75

Investigating the Impact of Dual-Rotor Wind Turbine Chapter 4on the Transient Network Frequency

4.1 Introduction

In this chapter, the principal target is to compare the transient frequency control

capability of the DRWT with the SRWT for different de-loading modes. The main

contributions of this chapter to achieve the target are as follows: The current practice to

enhance the short-term frequency control ability of the converter-based wind turbines is

to configure the inertia and/or droop loops of the power converters, based only on the

potential of the turbine for releasing kinetic energy (KE) via the body mass. However,

in this chapter it is revealed, through sensitivity analysis, that the transient variation of

the aerodynamic energy is also influential in determining the practical energy

production potential of the turbine. The transient aerodynamic power is the function of

the transient excursion of the operating point along the aerodynamic curve. So, the

inertial response of the wind turbines should not be treated in the same way as the

synchronous generators. The new factor that is identified demonstrates that the

influence of both the area where the excursion of the operating point occurs as well as

the range of excursion needs to be taken into consideration. In case of any transient

operating point excursion, its impact on the inertial response capability of the turbine

will be ‘boosting’ in the case of an over-speed operation and ‘weakening’ in the case of

an under-speed operation. It is revealed that the SRWT releases only slightly more KE

than the DRWT at the same frequency drop. Therefore, the amount of aerodynamic

transient variation during the frequency transient deviation is acknowledged to be the

key factor in understanding which of the DRWT or SRWT is more capable in arresting

the frequency nadir. In pitch de-loading mode, the transient frequency support

capability of the DRWT is better than that of the SRWT, due to the greater weakening

effect of the operating point excursion on the inertial response in the SRWT. In sub-

optimal de-loading mode, the SRWT is more effective in comparison to the DRWT

because of the higher boosting effect of the operating point transient displacement in the

SRWT. It is shown that, in the combination mode, only through application of a well

tunned droop controller of the auxiliary turbine, is it possible to enhance the frequency

control performance of the DRWT to be better than that of the SRWT.

The chapter is organized as follows: in section 4.2, the transient variation of

76

aerodynamic energy is proposed, in addition to the released kinetic energy (KE), as a

factor that also has an influence on the energy generation capability of the wind turbine

during the transient period; in section 4.3, the strength of the SRWT and the DRWT is

investigated with respect to the release of KE throughout the transient period; in

section 4.4, the transient frequency characteristic of the DRWT and SRWT is assessed

and compared for three different controlling modes according to wind velocity. The

impact of the integrated droop controller on the frequency support capability of the

SRWT and the DRWT is also studied in this section; and, simulation results in section

4.5 assist in verifying the validity of the analytical claims discussed in the previous

sections.

4.2 Impact of Aerodynamic Energy versus Kinetic Energy

To ensure a smooth and controllable energy production from wind farms, power

electronics are employed. However, due to the application of power converters, the

wind turbines are fully or semi-isolated from the network and are not able to react to the

network frequency deviation automatically, which causes poor frequency controllability

of these technologies of wind turbines. To overcome this drawback, the inertia control

loop and/or droop control loop are integrated into the power flow control loop of the

grid-side converter to allow the wind farms to contribute in regulating the grid

frequency. The related control loops emulate the hidden kinetic energy stored in the

rotating mass of the turbine; they are supposed to release energy to the network to arrest

the frequency nadir, stabilize it and, in the case of having enough reserve active power,

returns the frequency as close as possible to the nominal frequency. The gain of the loop

should be chosen with consideration to the short-term frequency stability of the network

on the one side, and the potential of the transient energy production of the wind-

generating unit on the other. If the gain is set higher than it should be, then the rotor

speed becomes less than its bottom limit and the wind turbine may stall. On the other

hand, if the gain is smaller than it should be, then the potential of the generating unit for

limiting the frequency drop isn’t used efficiently and the frequency may not, to some

extent, be well controlled. Some approaches have been introduced in [63], [65], [100]

for calculating the gain of the inertia control loop. Since the energy introduced by the

grid-side converter should be met and backed up by the generating unit, one of the

initial steps for designing the gain is to evaluate the potential of the mechanical prime

77

over with respect to releasing the kinetic energy stored during the deceleration period.

The higher this potential is, the higher the value of the gain that can be chosen without

the risk of stalling.

According to the approach by [64] , the gain of the inertial controller integrated into the

interface converters can be calculated through equating the required KE by grid with the

maximum KE that the generating unit is able to provide throughout the transient

frequency deviation. The method is described by (4.1):

(4.1)

where ωsn, ωrn are the rated synchronous and rotor speeds, 𝜔𝑟and 𝜔𝑟0are the present and

initial per-unit of the wind turbine rotor speed, and 𝜔𝑠 and 𝜔𝑠0are the present and initial

per-unit of the synchronous generator speed. Two main points can be concluded from

(4.1). First, the designing of the loop gain of the converters should be based on the

maximum capability of the related generating unit with respect to energy production

during the transient period. Second, the gain is obtained only according to the kinetic

energy released by the body mass of the drive train. However, besides the released KE,

the total energy delivered by the generating unit during the transient period is also

influenced by the route between initial and final locations that the operating point

travels along during the transient period.

4.2.1 Kinetic Energy

The maximum KE released from a rotating mass is given by (4.2), which is addressed

by [101]:

(4.2)

where, J is the inertia momentum of the rotating mass, ω0 is the initial angular velocity

and Δω is the maximum amount of change in angular velocity. The total power which is

discharged to the grid by the KE released from the body mass is given by (4.3):

∆𝑃𝐾𝐸 .𝑑𝑡 = ∆𝐸 (4.3)

20

2

20

2

2

2

2ss

rr

rn

snHKωωωω

ωω

−−

××=

)).((21 2

02

0 ωωω −∆+=∆ JE

78

According to [65], the induction generators are loosely coupled to the network, which is

an advantage compared to conventional thermal plants, with regard to the kinetic energy

released by the generating unit. In [65] it is claimed that the rotational speed of an

induction generator can drop by up to 0.75p.u, while the speed drop in synchronous

generators is limited to 5% by under-speed relays. However, in [64] it is mentioned that

the speed variation of a wind turbine during the transient period of the primary

frequency control is less than 0.1p.u. A conservative choice for the speed drop in the

present study can be 10% of the initial operating speed. The required parameters for the

comparison are given in [64].

Table 4.1 gives the amount of released ΔPKE of a typical fixed speed wind turbine for

different aerodynamic curves (different wind speeds and initial pitch angles). For each

scenario, the initial operating point corresponds to the maximum power at the optimal

point. It is assumed that Δω = 0.1p.u, dt = 3 sec.

β

V 00 100 200 300

7 m/s 0.0425 0.0294 0.0197 0.0122

11 m/s 0.0802 0.0557 0.0371 0.023

15 m/s 0.1493 0.1035 0.0687 0.0425

19 m/s 0.2148 0.1493 0.0988 0.0611

Table 4.1. Released PKE for 10% turbine speed drop in p.u.

4.2.2 Aerodynamic Transient Response According to Area of Operation

The next step is to obtain the total change in the aerodynamic power and compare it

with the kinetic power released under the same conditions. Generally, the transient

excursion of the operating point is determined by two factors. The first factor is the

command fed to the pitch control system to adjust the output power by altering the

blade angles to meet the new demand and to specify the new location of the operating

point. The second factor is the response of the rotational speed to the disturbance that

results in either deceleration or acceleration of the generator. Deceleration shifts the

operating point to the left on the aerodynamic curve, while acceleration drags it to the

right. The second factor causes the deviation of the operating point from the track that is

79

determined by the pitch control and which exists only during the transient period. Fig.

4.1 presents the track of the transient excursion of the operating point between the initial

(ωOP1, POP1) and final (ωOP2, POP2) operating points, for three cases. For the first case,

the effect of the disturbance is ignored and the movement of the operating point is

influenced only by the pitch control, where the rotational speed deviation is negligible.

The track is shown by a solid line on which arrows stand. The second case illustrates the

transient excursion route of the operating point in the presence of a disturbance that

leads to the deceleration of the generator. The path is presented by a dashed line with

arrows located on the very left side. And finally the displacement path of operating

point at the presence of a disturbance which results in the acceleration of generator is

given for the third case. The course is presented by a dashed line with intermittent

arrows, which is placed on the far right side. Since the main focus is on the transient

response of the wind turbine, the initial final operating points should be the same for all

three cases. In this example, the aerodynamic curve is raised from the aerodynamic

curve Aero1 to the aerodynamic curve Aero3. The Aero1 is obtained from the initial

blade angle and the Aero3 corresponds to the final angle that the blades settle in. To

investigate the impact of the acceleration or deceleration of the turbine on the

aerodynamic power production of the turbine during the transient period, a snapshot of

the aerodynamic curve is required, as an intermediate curve (Aero2) when it is traveling

from the initial curve to the final one. The cross sections of the Aero2 with cases1, 2

and 3 are, respectively, Ppi , Pdec , Pacc , where Pacc < Ppi < Pdec. This fact is valid for all

other snapshots of intermediate curves. Although the starting point (ωOP1 , POP1) and

destination (ωOP2 , POP2) of all three cases are the same, however throughout the

transient period (especially the first swing), the amount of power which is introduced by

the cases is different. It can be predicted that the curve envelope of active power

associated with the disturbances which lead to the acceleration of the generators should

be located below the corresponding one for the disturbances which result in the

deceleration of the generator.

To confirm the results achieved from interpreting Fig. 4.1, a sensitivity analysis can

be a good choice for investigating the impact of the operating point excursion on the

aerodynamic power coming from the blades throughout the transient period. The

aerodynamic power introduced by the blades is given by (4.4), as follows:

80

𝑃𝐴𝐸 = 0.5𝜌𝜋𝑅5𝐶𝑃𝜔𝑚2 /𝜆3 (4.4)

with R the blade radius, λ the tip speed ratio, ρ the air density and ωm the angular speed

of the rotor. Cp can be calculated by (4.5), as addressed by [65]:

(4.5)

Where:

where β is the pitch angle.

Fig. 4.1 Impact of generator speed variation on the transient aerodynamic generation

Since both the pitch control system and turbine speed deviation are in effect

simultaneously to determine the transient aerodynamic power, to calculate the total

change of the power (PAE), the sensitivity of the aerodynamic energy to the speed and

blade angle variations should be achieved through taking the partial derivative of PAE in

(4.4) with respect to ωm and β. It is illustrated by (4.6), as follows:

∆𝑃𝐴𝐸 =𝜕𝑃𝐴𝐸𝜕𝜔𝑚

|𝜔𝑚0.∆𝜔𝑚 +

𝜕𝑃𝐴𝐸𝜕𝛽

|𝛽0 .∆𝛽 (4.6)

where the first term shows the impact of speed deviation (Δωm) on the PAE alteration.

λβλ

βλ λ 0068.0).54.0116(517.0),(21

+−−=−

ieCi

P

1035.0

08.011

3 +−

+=

ββλλi

VRm .ωλ =

81

The second term presents that portion of the aerodynamic power influenced by variation

of pitch angle. The components of the terms are given as follows:

𝜕𝑃𝐴𝐸𝜕𝜔𝑚

|𝜔𝑚0=

0.5𝜋𝜌𝑅2𝑉3

𝜔𝑚02 [𝑒𝑈1 .𝑈2 − 𝑈3]

−𝜔𝑚0 . 60.𝑅. 𝑒𝑈1

𝑈4− 0.007

𝑅𝑉

+21.𝑅. 𝑒𝑈1 .𝑈2

𝑈4

and

𝜕𝑃𝐴𝐸𝜕𝛽

|𝛽0 = −0.5𝜋𝜌𝑅2𝑉3

𝜔𝑚0𝑒𝑈1 . (𝑈6 + 𝑈5.𝑈2)

with:

𝑈1 = 0.735𝛽03+1

− 21

0.08𝛽0+𝑅.𝜔𝑚0

𝑉

,

𝑈2 = 0.207𝛽 −60

0.08𝛽0 +𝑅.𝜔𝑚0𝑉

+2.1

𝛽03 + 1+ 2.585

𝑈3 = 0.007𝑅.𝜔𝑚0

𝑉

𝑈4 = 𝑉. (0.08𝛽0 +𝑅.𝜔𝑚0

𝑉)2

𝑈5 =1.68

(0.08𝛽0 +𝑅.𝜔𝑚0𝑉 )2

−2.2 ∗ 𝛽02

(𝛽03 + 1)2

𝑈6 = 2.86.𝑈5 + 0.207

To have a better understanding of the various impact of Δω and Δβ on the ΔPAE, the

outputs of some sensitivity analysis are given in Table 4.2 and Table 4.3, using the

materials which are already provided by (4.6). Table 4.2 gives the amount of increase in

the aerodynamic power when only the pitch control is in effect and the rotational speed

deviation due to the disturbance is ignored (Δβ= -50 and Δω=0). For Table 4.3, the same

82

procedure is repeated, where the maximum transient speed reduction of 10% is included

in the study, while the pitch angle remains constant during the transient period (Δω= -

0.1p.u and Δβ= 00).

β

V 50 100 150 300

7 m/s 0.0637 0.0496 0.0389 0.0293

11m/s 0. 1198 0.0938 0.0758 0.0555

15 m/s 0.2228 0.1745 0.1366 0.1032

19 m/s 0.3208 0.2513 0.1960 0.1486

Table 4.2. Aerodynamic power growth due to the reduction of blade angles by 50 while speed remains constant

β

V 50 100 150 300

7 m/s -0.0171 -0.0077 -0.0062 -0.0046

11m/s -0.0308 -0.0143 -0.0108 -0.0081

15 m/s -0.0598 -0.0251 -0.0206 -0.0153

19 m/s -0.0718 -0.0381 -0.0304 -0.0232

Table 4.3. Aerodynamic power reduction due to the 10% drop off of rotational speed in under-speed area while pitch angle remains constant

β

V 50 100 150 300

7 m/s 0.0103 0.0089 0.0078 0.0064

11m/s 0.0206 0.0173 0.0164 0.0131

15 m/s 0.0366 0.0346 0.0297 0.0238

19 m/s 0.0544 0.0465 0.0414 0.0322

Table 4.4. Aerodynamic power increase due to the 10% drop off of rotational speed in over-speed area while pitch angle remains constant

83

From the results provided in Table 4.3 and Table 4.4, it can be seen that the excursion

of the operating point in the under-speed area has a subtractive impact on the transient

total aerodynamic energy generation, while this effect is additive when the excursion is

in the over-speed area. So, in this paper, the total variation of the power generation of

the generator during the transient period is proposed to be decomposed into three

elements that are sourced by the KE, the control system and the generator speed

deviation. The mathematical presentation of the above-mentioned statement is

demonstrated by (4.7):

ΔPT = ΔPKE + ΔPAE (4.7)

with:

ΔPAE = ΔPEX + ΔPPI

where: ΔPT is the total change in the power produced by the generator; ΔPKE is the total

released kinetic power; ΔPAE is the portion of total power delivered by the blades of the

turbine; ΔPPI and ΔPEX are the components of ΔPAE, where ΔPEX is the amount of

change in the delivered aerodynamic energy due to the excursion of the operating point

which corresponds to the first term in (4.6); and ΔPPI is the portion of ΔPAE enforced by

the pitch control system or, generally speaking, the controlling system. The second term

of (4.6) represents the ΔPPI.

4.3 Kinetic Energy Released by SRWT and DRWT

In this section, an analytical solution is proposed to pave the way for comparing the

amounts of kinetic energy released by the body mass of the SRWT and the DRWT

throughout the frequency deviation. In this study, the dynamic model of the

components, such as spur and bevel gears [91]- [92], blades [95] and the set of the

shafts and hubs are used to model the drive trains of these two types of wind turbine.

There are two influential factors operating on KE; these are the total momentum inertia

of the generating unit and the maximum rotational speed deviation. Their effects are

contradictory. The higher the momentum, the less the speed deviation is introduced by

the rotating mass. This scenario is valid for both the DRWT and the SRWT. From

(3.23) and (3.24), the total momentum of the DRWT is higher than for the SRWT,

while, conversely, based on (3.27), the speed deviation of the SRWT is higher than that

84

of the DRWT. In the next sub-section, arguments will be put forward as to which one of

these factors is more dominative of KE.

4.3.1 Impact of Momentum on KE versus Speed Variation

From (4.8) it is obvious that the kinetic energy of a rotating mass is a function of the

inertia momentum and angular velocity. As to which one is more influential will be

explored in this section.

𝐸 = 12𝐽𝜔2 (4.8)

To investigate this matter, an abrupt increase in the momentum inertia is recommended

followed by a check as to whether the stored kinetic energy increased immediately

afterwards or it was reduced. If the amount of the rotating mass is suddenly increased,

then the speed drops down at constant torque. But the question is whether or not the

kinetic energy of the mass is going to increase or decrease. Firstly, it should be clear as

to which – the momentum or the angular velocity – is more influential in terms of the

energy. To investigate this matter, sensitivity analysis is applied to (4.8), as follows:

(4.9)

The relationship between torque and angular momentum is presented in (4.10):

(4.10)

where L=Jω.

The linear format of angular momentum is given in:

(4.11)

Equation (4.10) is equal to zero at constant torque.

By substituting (4.11) into (4.10) and equating the result to zero (at constant torque

ΔT=0), it is possible to calculate the change in speed according to the amount of change

in momentum of inertia:

ωωω ∆+∆=∆ ....21

0020 JJE

tLT∆∆

=∆

JJL ∆+∆=∆ .. 00 ωω

85

(4.12)

The total kinetic energy variation is achieved, based purely on ∆J, by replacing ∆ω in

(4.9) by its equivalent from (4.12). Equation (4.13) shows that, as the inertia is

increased (Positive ΔJ) the amount of kinetic energy decreases (Negative ΔE). This is

due to the drop in angular velocity. This fact signifies that the kinetic energy is more

influenced by angular velocity than the momentum of inertia.

(4.13)

Therefore, the kinetic energy of the rotating masses is more dominated by their speed

variations (Δω) than by their momentum of inertia (ΔJ). From the above theoretical

analysis, it can be predicted that the SRWT can release more kinetic power in

comparison to the DRWT. However, it is still not clear by how much the SRWT is able

to release more kinetic energy. Or, in other words, what is the ratio between the KE

released by the SRWT and the DRWT? This matter will be explored in the next

subsection.

4.3.2 KE Ratio between SRWT and DRWT

It has already been discussed by (3.27) in section (3.5) that the rate of acceleration of

the SRWT is higher than the DRWT for the ratio of JDR/JSR, in which JDR is the total

momentum inertia of the DRWT and JSR is the momentum inertia of the SRWT.

Through adopting (4.2) for the SRWT and the DRWT, and replacing ΔωSR by its

equivalent from (3.27), it is possible to obtain the maximum available KE delivered by

the SRWT and the DRWT, at that moment when the network frequency reaches its

nadir, as follows:

∆𝐸𝑆𝑅 =𝐽𝐷𝑅2

. (𝐽𝐷𝑅𝐽𝑆𝑅

.∆𝜔𝐷𝑅2 + 2.𝜔0.∆𝜔𝐷𝑅)

(4.14)

∆𝐸𝑆𝑅 =𝐽𝐷𝑅2

. (∆𝜔𝐷𝑅2 + 2.𝜔0.∆𝜔𝐷𝑅) (4.15)

where always:

JJ

∆−=∆ .0

0ωω

JE ∆−=∆ ..21

0ω

86

JDRJSR

> 1

By comparing (4.14) and (4.15), it is clear that the maximum KE released by the

generating unit of the SRWT is higher than the energy released by the corresponding

unit of the DRWT at the same frequency excursion and regardless of the mode of

control. The part of active power generated, relatively, by the SRWT and the DRWT,

sourced from KE, is achievable, respectively, through (4.16) and (4.17):

∆𝑃𝑆𝑅_𝐾𝐸 .𝑇𝑁𝑎𝑑𝑖𝑟 = ∆𝐸𝑆𝑅 (4.16)

∆𝑃𝐷𝑅_𝐾𝐸 .𝑇𝑁𝑎𝑑𝑖𝑟 = ∆𝐸𝐷𝑅 (4.17)

Through dividing (4.16) by (4.17), it is possible to calculate the ratio between the

∆PSR_KE, and ∆PDR_KE. The ratio is given in (4.18):

∆𝑃𝑆𝑅_𝐾𝐸

∆𝑃𝐷𝑅_𝐾𝐸=

𝐽𝐷𝑅𝐽𝑆𝑅

.∆𝜔𝐷𝑅2 + 2.𝜔0.∆𝜔𝐷𝑅∆𝜔𝐷𝑅2 + 2.𝜔0.∆𝜔𝐷𝑅

(4.18)

Equation (4.18) signifies that the kinetic power released by the SRWT is more than that

released by the DRWT. However, the ratio between the momentum inertia of the

DRWT and the SRWT is always less than 1.2; thus, the amount of difference in released

kinetic power is not that significant. For instance, if JDR/JSR and ω0 are assumed to be

1.2 and 1p.u respectively, then for the maximum speed drop of 10% for the DRWT, the

ratio in (4.18) is almost 1.01.

Fig. 4.2. A set up for evaluating the impact of momentum of inertia on kinetic energy generation

1.01

StoT

I M

W

S

TMotor

W1_

Dua

l

1.01

HydroGovernor

Tm w Tm0SP 1.0

If1_Dual Ef1_Dual

Tm01_DualTm01_Dual

VabcIf Ef

VrefExciter (SCRX)

STe

Tm

Tm0Tm w

EfIf

P+jQ

P+jQ

BR

K_Load

0.47Input Torque

87

To verify the theoretical discussion in this section regarding the impact of the value of

the momentum of inertia on the capability of the generation of kinetic energy, during

the transient frequency excursion, a test has been carried out in which only the effect of

the kinetic energy is included. The test set up is presented in Fig. 4.2 where an induction

generator is connected to a synchronous generator as the infinite bus. To have only

kinetic energy in effect during the transient, the influences of the pitch control and

aerodynamic energy should be left out in this test. To achieve this, on the one hand, the

mechanical input torque from the prime over is kept constant during the transient

period, and on the other, since the induction generator is not connected to any wind

turbines, the impacts of aerodynamic energy and pitch control system are ignored. So

the mechanical drive of the induction generator is selected to be a diesel engine which

provides a constant mechanical torque of 0.88p.u. A local load of 20 + j5 MVA is

suddenly switched in at t=100s. In this study, the test was repeated twice where the

inertia constant of the combination of the mechanical drive and generator was set, in the

first case, to H=4s and, in the second, to H=3s.

a) Speed deviation of induction generators

b) Extracted kinetic power from induction generator

Fig. 4.3. Responses of induction generator to sudden rise of energy demand when it is driven by diesel engine

Generator : Graphs

390 400 410 420 430 440 450 460

0.950 0.960 0.970 0.980 0.990 1.000 1.010 1.020 1.030 1.040 1.050

y (p

u)

Speed (H=4) Speed (H=3)

Generator : Graphs

395.0 400.0 405.0 410.0 415.0 420.0 425.0 430.0 435.0

-0.040

-0.020

0.000

0.020

0.040

0.060

0.080

y (MW

)

Kinetic Power_H=4 Kinetic_Power_H=3

88

The responses of the rotational speed and kinetic power extracted from the rotor of the

generator are given in Fig. 4.3a and Fig. 4.3b respectively. The responses for the second

case are represented by a thin green curve on which squares stand. From Fig. 4.3a, the

result by (3.27) – that, in the case of any disturbance, the speed deviation in the lighter

generating unit is more than that of the heavier one – is confirmed. The kinetic energies

discharged by the induction generators are presented in Fig. 4.3b which concurs with

the analytical conclusion from (4.14) and (4.15). The extracted power of the lighter

generating unit reaches a higher peak compared to the heavier generator.

4.4 Frequency Control Capability of SRWT and DRWT Based on Control Mode

In this section, the main concentration is on the impact of the initial location and

transient excursion range of the operating point on the transient aerodynamic power

generation capability of the DRWT and SRWT. To make it feasible for wind farms to

play a part in primary frequency control, the wind turbines should have some reserved

active power in order to deliver during the transient period. According to [64], the de-

loading mode is achievable in one of following three ways: operating on sub-optimal

curves; down-regulating the active power by pitch control through operating with blade

angles higher than zero; and, a combination of pitch control and sub-optimal curve. The

sub-optimal curves for de-loading purposes can be located on either the left side (in the

under-speed area) or the right side (in the over-speed area) of the MPPT curve.

However, operating in the under-speed area raises some dynamic stability issues due to

the positive slope of the power-speed aerodynamic characteristic curve in the under-

speed region. In cases where the initial operating point of a wind farm is located on the

left side sub-optimal curve, at a time of sudden load growth, the generators decelerate.

Since, in sub-optimal curves, the blade angle are locked at zero, the slowing down of

the generator should lead to a reduction of aerodynamic torque, resulting in further

reduction of generator speed and mechanical torque, which may cause progressive

speed drop and the stalling of the generator. Conversely, in the over-speed area, the

slope of the aerodynamic curve is negative and for the sub-optimal curve in this region,

as the rotational speed drops due to the abrupt load increase, the output of aerodynamic

power of the turbine increases; this response helps to arrest the speed nadir and to

stabilize it. This means the wind turbine has a much higher stability margin in the over-

speed area, while the impact is negative in the under-speed area. So de-loading in the 89

under-speed area is left out in this study due to the pure dynamic stability margin.

Adoption of de-loading methods is mostly influenced by the wind speed [64], [102]. For

example according to [64], for low wind-speed modes, the 90% sub-optimal operation

is chosen, while the pitch angle is fixed at zero and does not change during the rise or

fall of active power. When frequency falls, the active power is increased just through

decelerating the turbine through the control system of the rotor-side converter in DFIG,

or the generator-side converter of the FRC. Regarding high-speed winds, operating in

the over-speed area is impossible because it hits the speed limit, which is 1.2p.u. So de-

loading in high wind-speed mode is achievable only through the pitch angle control

system. The active power is initially set to 0.9p.u. It is possible to arrest the network

frequency fall and stabilize it by decreasing the pitch angle. For medium wind-speed

cases, the combination of the pitching system and sub-optimal de-loading presents the

best performance. In this mode, since in the sub-optimal method less tear and wear is

imposed to the blade in comparison to pitch angle control system, the functioning of the

speed control system is superior to the blade angle control system. However, because of

the rotor speed upper limit, the pitch control system is supposed to take necessary action

to keep the speed in permitted limits.

In reality, both ∆PPI and ∆PEX are simultaneously involved in the determination of the

output of active power. The excursion track of the operating point during the transient

period for the above-mentioned three control mode scenarios includes: pitching control

(high speed winds); sub-optimal curve (low speed winds); and a combination of these

(medium-speed winds). These are presented in Fig. 4.4 (section 4.4.1), Fig. 4.5 (section

4.4.2) and Fig. 4.6 (section 4.4.3), respectively. Most of the disturbances that threaten

frequency stability are due to the sudden growth of active power demand, which leads

to the deceleration of the generators; therefore, in this study the disturbance results in

generator deceleration. In this investigation, the main focus is on the transient responses

of the SRWT and the DRWT. So, to have a comprehensive study and apple-to-apple

comparison, the initial (ω1, P1) and final (ω2, P2) operating points are considered to be

the same for the DRWT and the SRWT.

4.4.1 SRWT and DRWT in Pitch Control Mode

In Fig. 4.4, the power regulation is taken over only by the pitch angle control loop. So,

on the one hand, the operating point drifts to left due to the rotational speed fall and, on

90

the other hand, the aerodynamic curve is shifted up to increase the output power due to

the reaction of pitching system and reduction in blade angle. Since the range of speed

deviation of the DRWT is less than that of the SRWT, the route of the transient

excursion for the DRWT is located on the left hand side of the SRWT (DRWT is

denoted by a dashed line). The other two snapshots of the aerodynamic curves are

illustrated between the initial and the final curves, which are a kind of sample to

illustrate how the initial aerodynamic curve travels from the initial to the final curve.

The cross-sections of the transient excursion routes of the SRWT and the DRWT and

the two intermediate aerodynamic curves can assist us in predicting the relative

aerodynamic power generation of the DRWT and the SRWT. For the first intermediate

aerodynamic curve, the cross sections for the SRWT and the DRWT are respectively

PSPI1 and PDPI1 which PSPI1 < PDPI1. The same is true for the second aerodynamic curve,

where PSPI2 < PDPI2. So, from Fig. 4, it can be concluded that, in pitch control mode, the

curve envelope of the aerodynamic power produced by the DRWT should be located

above the aerodynamic power curve of the SRWT.

Fig. 4.4. Transient excursion of operating points of SRWT and DRWT for high wind speeds

To mathematically investigate which of the SRWT or the DRWT is able to deliver more

power to the grid during the transient period, when their power production is controlled

by the pitching system, the (4.7) should be extended for both systems, based on the

control mode, and the results should be compared.

∆𝑃𝑆𝑅 = ∆𝑃𝐾𝐸_𝑆𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝑆𝑅 +𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝑆𝑅0 .∆𝜔𝑆𝑅 (4.19)

91

∆𝑃𝐷𝑅 = ∆𝑃𝐾𝐸_𝐷𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝐷𝑅 +𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝐷𝑅0 .∆𝜔𝐷𝑅 (4.20)

where first, second and third terms on the right side of the (4.19) and (4.20) represent,

respectively, the ΔPKE, ΔPPI and ΔPEX in (4.7). ΔPPI is the command by the pitch control

system and determines the spot where the new operating point is going to settle down.

Since, in this technology, increasing the ΔPPI is possible only through a reduction of

pitch angle, the sign of the second term in the following equations should be negative to

present the effect of the pitch angle response correctly. In the under-speed area, the sign

of the derivative of active power (P) with respect to rotational speed (𝜔𝑚) is positive,

while the sign of ∆𝜔 is also negative in case of any seep reduction. When the initial

operating point is located in MPPT, the reduction of the rotational speed of the turbine

leads to a decrease of ΔPEX. So the sign of the last term in (4.19) and (4.20) is chosen to

be positive to correctly show the effect of the transient speed reduction on the total

power generated during the transient period.

The main focus is on the transient response of active power; therefore, the pre- and

post-disturbance operating points should be the same. To make this possible, ΔPPI is

assumed to be the same for both systems. For example, both the SRWT and the DRWT

are commanded to raise the power for ΔPPI = 0.1p.u through the pitch control system.

In (4.19), the ∆PKE_SR can be substituted by its equivalent, ∆PKE_DR, which is given by

(4.18). Also, ∆ωSR in (4.19) can be replaced by its equivalent from (3.27). So the

updated version of (4.19) is given as follows:

∆𝑃𝑆𝑅 = 𝐾𝐾𝐸 .∆𝑃𝐾𝐸_𝐷𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝐷𝑅 +𝐽𝐷𝑅𝐽𝑆𝑅

.𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝑆𝑅0 .∆𝜔𝐷𝑅 (4.21)

where

𝐾𝐾𝐸 =𝐽𝐷𝑅𝐽𝑆𝑅

.∆𝜔𝐷𝑅2 +2.𝜔0.∆𝜔𝐷𝑅

∆𝜔𝐷𝑅2 +2.𝜔0.∆𝜔𝐷𝑅

Since all the quantities of the SRWT in (4.21) are associated with the DRWT, it is

possible to compare (4.21) with (4.20) to reveal which of the SRWT or DRWT is able

to transiently deliver more active power to the grid or the generator-side converter,

depending on the energy conversion technology. In order to make a valid comparison, 92

the corresponding components of the total power should be compared. The component

regarding the ΔPKE in the SRWT is higher than the DRWT, due to the factor KKE, which

is multiplied to ∆PKE_DR in (4.21). However, on the other hand, the amount of reduction

of ∆PEX in the SRWT system is also higher than in the DRWT because of the factor

JDR/JSR; this fraction is always higher than 1. To explore which of KKE or JDR/JSR is more

influential, an evaluation of the magnitude of these factors is recommended From

section 3, it was recognized that this factor is relatively close to one which can be

rounded down to one. So, the element regarding the kinetic energy can be assumed to be

the same for both the SRWT and the DRWT. On the other hand, according to the

mechanical equations, the ratio between the momentum inertias of the DRWT and the

SRWT (JDR/JSR) cannot be more that 1.2 due to the ratio between the length of blades of

the auxiliary and main turbines in the DRWT [103]. So, it can be concluded that the

impact of ∆PEX is more significant than ∆PKE for determining which of the SRWT or the

DRWT is able to generate more power during the transient period. Therefore, in pitch

control mode, due to the higher reduction of ∆PEX in the SRWT system, which is due to

the higher speed reduction, it can be deduced that the DRWT introduces higher energy

to the grid during the transient period, while both systems will settle down in the same

final operating point. In other words, the curve envelope of the DRWT should be

located above the related curve for the SRWT throughout this time frame. To check the

validity of the above theoretical discussion regarding the relative impact of ΔPEX and

ΔPKE, a simple way is to suddenly increase the active power demand on the SRWT and

the DRWT while the pitch control is locked into its initial value. In this way, the second

term of (4.20) and (4.21) turn to zero and only the ΔPEX and ΔPKE are specifying the

output active powers of the SRWT and the DRWT. Through comparing table1 and

table3, it can be seen that the amount of increase of ΔPKE is more than the reduction of

ΔPEX during the slowing down. Therefore, the output power is expected to rise during

the transient period and, since the pitch angle is locked, it returns to its initial value

when the transients are gone.

4.4.2 SRWT and DRWT in Sub-optimal Mode

The generator-side converter performs two essential duties. The first duty is to maintain

the interface DC-link voltage in an acceptable level through adjusting the power drawn

from the generator. Therefore whenever the grid-side converter increases the power 93

generation, the generator-side converter should follow the same procedure after a small

time constant to avoid any dramatic fall in the voltage of the interface capacitor. The

second duty is to make it possible for the wind turbine to operate in a sub-optimal curve.

In sub-optimal mode (in the over-speed area), the generator side converter draws

enough power from the generator to force it to slow down and settles down the new

operating point in a spot with a lower speed and a higher delivered aerodynamic power.

Fig. 4.5 shows the excursion of the operating point when only the sub-optimal control

system is in charge of regulating the output power. In this scenario, the pitch angle is

fixed to zero and there is only one aerodynamic curve where the operating point can

move up or down on it. The only alternative method of arresting the frequency fall is to

increase the active power demand from the generator-side converter, which leads to

long-term deceleration of the generators. Since the turbine is operating in the over-speed

region, the generating unit should settle down in lower speeds while permanently

delivering a higher amount of power. From the transient response point of view, the

minimum speed reached by the SRWT during the transient period is less than the

corresponding speed for the DRWT, based on (3.27), in the sub-optimal control mode.

This means that the maximum aerodynamic power touched by the SRWT is more than

the maximum transient power delivered by the DRWT. Therefore, from Fig. 4.5, it can

be seen that, in sub-optimal control mode, the application of the SRWT is more

beneficial for increasing the margin short-term frequency stability. The operating points

corresponding to the minimum speed for both the SRWT and the DRWT are,

respectively, (ωS_Mi , PS_Ma) and (ωD_Mi , PD_Ma), illustrated by the multiplication sign(×).

To exemplify the situation, the excursion track of the operating point of the DRWT is

represented by some arrows.

Fig. 4.5. Sub-optimal control mode for low speed winds

94

To mathematically investigate the validity of the conclusion that was taken from

interpreting Fig. 4.5, the (4.20) and (4.21) must be updated according to the new mode

of control. The updates should comply with the facts that the variation of the pitch angle

is zero (Δβ=0), since the pitch angle is fixed at its minimum. Then, an element should

represent the alteration of the active power due to the change of speed set point dictated

by the generator-side converter. The variation of active power delivered by the

generating units of the SRWT-FRC and the DRWT-FRC to the generator-side converter

is given, respectively, in (4.22) and (4.23) in sub-optimal control mode.

∆𝑃𝑆𝑅 = 𝐾𝐾𝐸 .∆𝑃𝐾𝐸_𝐷𝑅 − 𝐾𝑆𝑢𝑏𝑜𝑝𝑡 .∆𝜔𝑆𝑢𝑏𝑜𝑝𝑡 +𝐽𝐷𝑅𝐽𝑆𝑅

.𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝑆𝑅0 .∆𝜔𝐷𝑅 (4.22)

∆𝑃𝐷𝑅 = ∆𝑃𝐾𝐸_𝐷𝑅 − 𝐾𝑆𝑢𝑏𝑜𝑝𝑡 .∆𝜔𝑆𝑢𝑏𝑜𝑝𝑡 +𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝐷𝑅0 .∆𝜔𝐷𝑅 (4.23)

The term Δωsub_opt is the steady state amount of the shift of speed dictated by the

converter. Like the command by the pitch control system (Δβ), the command by the

converter (Δωsub_opt) should also be the same for the SRWT and the DRWT, in order to

make a fair comparison. KSub_opt represents the ratio between the change in aerodynamic

power and the changing of the rotational speed reference point. Since any reduction of

speed in the over-speed area leads to an increase in the aerodynamic power, then the

sign of the term regarding the speed control command given by the converter is

negative. The slope of the aerodynamic curve in the over-speed area and the speed

reduction are both negative, so the sign of the last term is chosen to be positive to

correctly show the relationship between the speed deviation and the ΔPEX in the region

of operation. The definition of KKE is already given in (4.21).

Based on the theoretical discussion in section (4.3.2), PKE can be assumed as being

equal to one in this comparison, so this quantity is the same for both the SRWT and

DRWT. Again from (3.27), the speed deviation of the SRWT, as a response to the

power demand drawn by the converter, is higher than the DRWT. So, the SRWT

reaches higher locations in the over-speed area in comparison to the DRWT (refer to

Fig. 4.5). Through comparing (4.22) and (4.23), it can be seen that the element which

represents the ΔPEX (third term) for the SRWT is higher than the corresponding element

for DRWT; this is due to the JDR/JSR fraction, which signifies that the curved envelope

of active power generated by the SRWT should be located above the corresponding

95

curve of the DRWT during the transient period, until they settle down in the new

operating point.

4.4.3 SRWT and DRWT in Combination Mode

Fig. 4.6 illustrates the transient excursion track of the operating point of the SRWT and

the DRWT for medium wind speed scenarios, where both the pitch control and sub-

optimal curve control system are enacted to increase the output power of the wind

turbine. On the one hand, pitch control lifts the aerodynamic curve by reducing the pitch

angle, and on the other hand, the generator side converter tries to increase the loading of

the generator to force it to decelerate. The dashed line represents the route travelled by

the DRWT, and the solid line shows the corresponding route travelled by the SRWT.

Again, due to the higher deviation of the SRWT, its related route is placed on the left

side of the DRWT. The cross-sections of the routes with intermediate aerodynamic

curves in Fig. 4.6 signify that during the transient period the curve envelope of active

power of the SRWT should be located above the curve envelope of the DRWT for

medium wind speed scenarios, when both control modes are engaged to increase the

power. This means, just as in the sub-optimal mode, for the latter scenario, the SRWT is

more supportive with regard to the transient frequency stability of the local network.

Since the aerodynamic efficiency of the DRWT is higher than that of the SRWT and

additionally its margin of transient angle stability is also higher, it can be foreseen that

this technology will having a remarkable share in the market in the close future. Thus, it

is also worth improving the frequency control performance of the DRWT in the sub-

optimal and combination modes.

It is possible to enhance the ability of the wind turbines, with respect to the transient

energy production, through integrating a proper designed droop loop into the pitch

control system. It will assist the mechanical prime-over to get through the transient

period without experiencing any abnormal situation [63], [68]. To ensure the generating

unit delivers a safe and optimum performance, during the network frequency transient

deviation, the gain of the droop controller integrated into the pitching control system

should be coordinated with the gain of the droop control loop merged into the grid-side

converter. In this study, it is preferable to implement the droop control loop, rather than

the inertial loop, into the grid-side converter.

96

Fig. 4.6. Combination of pitch control mode and over-speed mode for medium speed winds

In the case where frequency falls, the Δf becomes negative and the generating unit is

supposed to increase the mechanical torque; conversely when there is frequency rise

(Δf>0), the mechanical torque should be reduced. Thus, the droop output signal should

be added to the summation junction of the pitching system through a negative sign.

The all-inclusive view of the controlling loops of different components of the SRWT-

FRC and the DRWT-FRC are illustrated in Fig. 4.7a and Fig. 4.7b, respectively. The

input signal selected to be fed into the droop control loop of the pitching system is the

rotational speed of the generator. In this study, each of the main and auxiliary wind

turbines have their own related droop loops. RSR and RAUX are the droop coefficients,

respectively, for the main and auxiliary turbines of the DRWT. The droops for the main

turbine in the DRWT and the turbine for the SRWT are assumed to be the same, in

order to have a fair comparison. For the SRWT and the main turbine of the DRWT, the

desired quantity to be controlled by the pitching system is the mechanical torque on the

interface shaft between the turbine and the gears (TMa). The mechanical torque delivered

to the bevel gear in the DRWT by the auxiliary turbine (TAux) is regulated in the same

way. In the presence of the droop controller, the turbine is able to regulate its output

power to keep up with the energy production of the grid-side converter.

97

a) SRWT

b) DRWT

Fig. 4.7. Control diagram for the SRWT and DRWT coupled to the grid through FRC

98

For presenting an inclusive study regarding the capabilities of the SRWT and the

DRWT in the presence of the FRC, the contribution of the pitch system integrated with

the droop loop and sub-optimal control mode should be examined. The steady state

change of active power (it is assumed that transients are damped) is specified by the

pitch control (blade angle control) and the speed control system of the generator-side

converter.

To mathematically investigate whether the SRWT or the DRWT is able to deliver

more active power to the grid during the transient period, with the above-mentioned

circumstances, (4.19) and (4.20) should be updated by including two extra elements.

One of the elements is the signal fed by the droop loop into the pitch control system,

and the other is the portion of the power change resulting from the new set point of the

rotational speed enforced by the generator-side converter in sub-optimal control mode.

For example, the generator-side converter controls the generator to slow it down and

consequently, the active power will settle down into higher locations on the

aerodynamic curve in the over-speed area as long as the new command is initiated.

However, the speed set point adjustment cannot match the new demand and,

consequently, the speed keeps decreasing until the pitch control takes necessary action

to compensate for the lack of power by reducing the pitch angle. The modified version

for the SRWT and the DRWT is given, respectively, in (4.24) and (4.25):

∆𝑃𝑆𝑅 = ∆𝑃𝐾𝐸𝑆𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝑆𝑅 +𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝑆𝑅0 .∆𝜔𝑆𝑅 −1𝑅𝑆𝑅

.∆𝜔𝑆𝑅

−𝐾𝑆𝑢𝑏_𝑜𝑝𝑡.∆𝜔𝑆𝑢𝑏_𝑜𝑝𝑡

(4.24)

∆𝑃𝐷𝑅 = ∆𝑃𝐾𝐸𝐷𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝐷𝑅 +𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝐷𝑅0 .∆𝜔𝐷𝑅 −1𝑅𝑆𝑅

.∆𝜔𝐷𝑅

− 1𝑅𝐴𝑢𝑥

.∆𝜔𝐷𝑅 − 𝐾𝑆𝑢𝑏_𝑜𝑝𝑡.∆𝜔𝑆𝑢𝑏_𝑜𝑝𝑡

(4.25)

where RSR, RAux are the contributions of the droop system to the torque set point,

respectively, by the main and the auxiliary turbine in the DRWT. Since the turbine in

the SRWT is quite similar to the main turbine in the DRWT, a droop system with the

same characteristic is used for it. 99

In the same way as was done to (4.19), it is possible to replace the elements of (4.24)

with their corresponding equivalent quantities from the DRWT. This action paves the

way for a comparison of (4.25) and (4.26).

∆𝑃𝑆𝑅 = ∆𝑃𝐾𝐸_𝐷𝑅 −𝜕𝑃𝑀𝜕𝛽

|𝛽0 .∆𝛽𝐷𝑅 + 𝐽𝐷𝑅𝐽𝑆𝑅

. 𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝐷𝑅0 .∆𝜔𝐷𝑅 −𝐽𝐷𝑅𝐽𝑆𝑅

. 1𝑅𝑆𝑅

.∆𝜔𝐷𝑅 −

𝐾𝑆𝑢𝑏_𝑜𝑝𝑡.∆𝜔𝑆𝑢𝑏_𝑜𝑝𝑡 (4.26)

Through comparing (4.25) and (4.26), it can be seen that the droop gain of the auxiliary

turbine plays an integral role in determining whether the DRWT or the SRWT is able to

introduce more power to the grid during the transient period in sub-optimal control

mode. If the value of the droop gain of the auxiliary turbine fulfils the following

inequality by (4.27), then the DRWT could also be more efficient than the SRWT in the

combination mode. It is worth remembering that the first, second and last elements are

the same in (4.25) and (4.26). So, they are cancelled out by both sides of the, inequality.

1𝑅𝐴𝑢𝑥

> 𝐽𝐷𝑅𝐽𝑆𝑅

− 1 (𝜕𝑃𝑀𝜕𝜔𝑚

|𝜔𝐷𝑅0 +1𝑅𝑆𝑅

) (4.27)

Therefore, through proper tuning of the droop characteristic of the auxiliary turbine, the

DRWT-FRC-based wind farm would be more effective for limiting the frequency nadir,

compared to the SRWT-FRC-based wind farm, when the method for under-regulating is

the combination of the sub-optimal and pitch control modes. Additionally, higher values

of droop provide a better damping for frequency oscillation [84].

To confirm the analytical calculations in sections 5 and 6, simulation studies need to be

performed.


The main focus of this section is the assessment and comparison of the SRWT-based

and the DRWT-based wind farms with respect to the frequency control capability for

different de-loading methods. For the first test, the de-loading is done through the pitch

control system, the theoretical discussion of which has already been provided in section

(4.4.1) and exemplified in Fig. 4.4. In the latter section, it was claimed that the DRWT

has a better frequency control performance in comparison to the SRWT when they are

under-regulated at a non-optimal operating point through the pitching system and the 100

energy conversion system is FSIG. For the second test, the de-loading is achieved only

by operating in the over-speed area and by placing the initial operating point on the sub-

optimal. Related theoretical issues are discussed in section 4.4.2 and depicted in Fig. 4.5

where it was concluded that in the over-speed mode (where the pitch angle is fixed) the

SRWT has a higher capability for limiting the transient frequency deviations. In the

final test, which was set up as for the second test, a droop loop is integrated into the

pitch control system was. The impact of the droop system on the frequency support

performance of the SRWT and the DRWT has also been discussed in section 4.4.3 and

Fig. 4.6, where it was concluded that, in the presence of a well-tuned droop loop for the

auxiliary turbine in the DRWT, it is possible for the DRWT to regulate the frequency

better than the SRWT, even in the over-speed area. To make it feasible for the wind

turbine to operate in the over-speed region, in the second and third test, the FRC system

has been used as the energy conversion technology. The criterion of the capability is the

amount of the maximum transient active power delivered to the generator side converter

by the generating units of the SRWT and the DRWT. The higher the delivered energy to

the converters, the more capable is the FRC-based wind farm in limiting the network

frequency fall.

Simulations were carried out using PSCAD/EMTDC software. The power system

referred to in the case study is shown in Fig. 4.8; it includes a 300MVA thermal power

station, a 0.69/16 kV transformer, a 100km length transmission line, a 300MVA local

load and a 150MVA wind farm. The required data for modeling the DRWT and SRWT

wind farms is provided in [67]. To apply a frequency deviation to the system, an

unscheduled load of 60MW and 10MVar is suddenly switched in. Each test has been

conducted separately for the SRWT and the DRWT and results will be sketched in the

same plot. The pitch angle rate limit is set to dθ/dt=10 (º/s) to avoid mechanical tension

on the blades. The aerodynamic model presented by [67] has been used to calculate the

pitch angle of the blades in the DRWT. To avoid any voltage instability during the

transient period, the excitation limiters of the generators are set to 5p.u to inject enough

reactive power to the grid to support the network voltage. The mechanical torque

produced by the turbine is the desired quantity to be controlled by the pitching system

for all the tests.

Based on the recommendation in the last paragraph of section 4.4.1, the initial

operating point for both the DRWT-FSIG and the SRWT-FSIG are placed at the

101

optimum point for the first test. To have a 20 precent reserve for active power, the initial

pitch angle of the SRWT and the DRWT are locked at 50 and 80, respectively and they

do not altering during the transients.

Fig. 4.8. Employed power system for the tests

The DRWT introduces a higher torque than the SRWT at the same blade angle and

wind speed. On the other hand, to have a fair comparison, both technologies should

generate the same torque for the same wind speed. So, at the same wind speed and

torque, the DRWT operates with higher blade angles in comparison to the SRWT. Both

wind turbines generate 0.8 p.u active power. The unscheduled load is connected to the

network at t=570s to let the network experience frequency excursion. According to Fig.

4.9a, the minimum speed reached by the generators in the SRWT and the DRWT,

respectively, is 0.999 p.u. and 1.007 p.u (from the initial value of 1.023 p.u) and finally

settles down at 1.02 p.u.; since the pitch control is locked, the new operating point is

rotating at a lower speed than the initial speed. The angular velocities of the generators

are recovered after almost 30 seconds. As the angular velocity drops, a part of the KE

stored in the rotor is accordingly injected into the network.

The test result illustrated by Fig. 4.9a confirms the proposed quantitative approaches

in section 4.4.1 and denotes the higher ability of the SRWT, compared to the DRWT, to

discharge KE. However, the difference is not that significant.

Synch 1

#1 #2PI

COUPLED

SECTION

Load Bus

BRK3

BRK2

BRK1

SRWTWindGen

DRWT

GenWind

300MVA Thermal Plant

1 [mH]0.6 [ohm]VA

300 MVA Local Load

6 [mH]3.8 [ohm]VA

60 MVA Local Load

BRK7

Synch 2

Synch 3

102

a) Generator speed response

b) Output active power

c) Power system frequency

Fig. 4.9. Responses of generating units of SRWT and DRWT to the load switching in pitch control mode

The responses of the wind turbine’s active powers are presented in Fig. 4.9b. According

to (4.7), during the transient period, the total active power injected into the network by

FSIG includes kinetic power and the aerodynamic power delivered by the turbine. Since

the pitch angle is locked (ΔβSR=ΔβDR=0) in the first test, for this test, the only indecisive

factor is the determination of the transient response of the ΔPAE is ΔPEX. In other words,

Main : Graphs

560 570 580 590 600 610 620

0.9950

1.0000

1.0050

1.0100

1.0150

1.0200

1.0250

1.0300

y

SRWT_Generator_Speed DRWT_Generator_Speed

Main : Graphs

560 570 580 590 600 610 620

0.790

0.800

0.810

0.820

0.830

0.840

0.850

y

SRWT_Active_Power DRWT_Active_Power

Machine Speed,Main : Graphs

565.0 570.0 575.0 580.0 585.0 590.0 595.0 600.0 605.0

48.50

48.75

49.00

49.25

49.50

49.75

50.00

50.25

50.50

50.75

y (ra

d/s)

Frequency SRWT Frequency DRWT

103

ΔPPI=0 for this test. Fig. 4.9b shows that, during the transient period, the active power

generated by the DRWT is higher than that of the SRWT when only the pitch control

system is in charge for de-loading.

The result by Fig. 4.9b confirms the conclusion by (4.20) and (4.21). Since the blade

angles remain unchanged for both the DRWT and the SRWT, the active power resumes

its initial value when the transients fade away. Due to the positive slope of the power-

speed curve in the under-speed area, the oscillations of the output power are almost in

phase with the generator speed. For example, it is possible to trace the related power

and speed curves, for the DRWT and the SRWT, between the local extremum points at

t= 580s and t= 582s.

The frequency excursion of the power system connected to FSIG-based wind farms is

illustrated in Fig. 4.9c. It signifies that the DRWT is more successful in limiting the

network frequency fall. Since the excursion occurs in under-speed region, then, due to

the higher speed reduction of the SRWT, the total active power of the SRWT is less

than that of the DRWT during the transient period. Consequently, the frequency fall in

the power system connected to the DRWT-FSIG-based is less than the power system

that includes the SRWT-FSIG-based in MPPT mode.

The second test is carried out to investigate the frequency capability of SRWT-FRC-

and DRWT-FRC-based wind farms in terms of the load switching when de-loading is

achieved only through operating in sub-optimal mode. The pitch angle of the SRWT

and DRWT is locked at 00 and 20, respectively. The generator side converter regulates

the rotational speed to 1.172p.u. At this speed and pitch angle, wind turbines deliver 0.8

p.u active power. As a response to the growth in demand, the generator-side converter

of the FRC reduces the rotational speed for 0.02p.u in order to increase the power

production of the turbine by 0.05p.u to keep up with the power delivered to the network

by the grid-side converter.

The responses of the quantities of the generating units of the SRWT and DRWT to the

speed reduction have been presented in Fig. 4.10. In Fig. 4.10a, the rotational speed

follows the command by the converter to reduce its value by 0.02p.u. The minimum

speeds reached by the SRWT and DRWT are 1.124 p.u. and 1.112 p.u (from the initial

value of 1.712 p.u), respectively. So in the sub-optimal mode, similar to the pitch

control mode, the amount of drop in the rotational speed of the SRWT is higher than for

the DRWT.

104

As shown in Fig. 4.10b, the curve envelope of total active power delivered to the

generator-side converter of the SRWT is located above the DRWT, which verifies the

theoretical result by (4.22) and (4.23) regarding the better frequency control

performance of the SRWT in sub-optimal control mode. It implies that the control loop

integrated into the grid-side converter of the SRWT can be tuned to deliver active power

to the grid with higher overshoot during the transient period, in contrast to the DRWT in

this mode in the over-speed area. The obtained result from the sub-optimal mode (test2)

is in contrast to the result of the pitch control mode (test1). The main reason is the slope

of that portion of the aerodynamic curve that the operating point travels along during

the transients. Since the initial position of the operating point is located at the optimum

point, in the case of any speed reduction, the slope of that portion that the operating

point excurses is negative. In the negative area, as the speed falls further, ΔPEX becomes

more negative. For the sub-optimal mode, since the initial operating point is placed in

the over-speed area, the slope of that part of the aerodynamic curve is positive and, as

the speed drops further, the ΔPEX rises. By carefully tracing the oscillations of the

generator rotational speed and the active power output, it is recognized that the phase

difference between their oscillations is almost opposite. For example, between t= 575s

and t= 580s, the speed of the DRWT is increasing, while, in this period, its active power

is decreasing. This is also valid for the SRWT. This matter signifies that the operating

point is located in the over-speed area (the slope is negative).

The response of the power system frequency is demonstrated in Fig. 4.10. The

frequency nadir of the system coupled to the generating unit of SRWT-FRC wind farms

is located above the nadir of the power system connected to the DRWT-FRC by a value

of 0.1Hz. This is due to the higher peak of active power reached by the SRWT-FRC

during the frequency transient deviation. From the second test, it can be deduced that, in

sub-optimal mode, the grid-side converter of the SRWT is able to produce a higher

amount of energy and be more effective than the DRWT-FRC for limiting the frequency

fall; this is so when de-loading is performed by controlling the speed of the generator

through the generator-side converter (sub-optimal mode).

105

a) Generator speed

b) Output active power

c) Power system frequency

Fig. 4.10. Responses of generating units of SRWT and DRWT to the load change for sub-optimal mode

The third test is performed in order to check the validity of the claim already made in

section 4.4.3 regarding the impact of the droop loop integrated into the pitching system

on the performance of the SRWT-FRC and the DRWT-FRC systems through

comparing (4.25) and (4.26). The amount of droop for the turbine of the SRWT and the

main turbine of the DRWT is chosen to be 12. The droop factor of the auxiliary turbine

is set to 2.3 to satisfy the inequality in (4.27). In this test, both the speed controls, from

the generator side converter and the pitch control system, are in action simultaneously

Generator : Graphs

570 580 590 600 610 620

1.120

1.130

1.140

1.150

1.160

1.170

y (p

u)

SRWT_Gen_Speed DRWT_Gen_Speed

Main : Graphs

560 570 580 590 600 610 620

0.760 0.780 0.800 0.820 0.840 0.860 0.880 0.900 0.920 0.940

y

SR_Active_Power DR Active Power

Machine Speed,Main : Graphs

560 570 580 590 600 610 620

49.30

49.40

49.50

49.60

49.70

49.80

49.90

50.00

50.10

y (ra

d/s)

Frequency SRWT Frequency DRWT

106

to increase the power generation by the turbine. The set point of the speed of the

generator is commanded by the converter to be reduced by 0.01p.u, and rest of the

power requested by the converter is supplied by reducing the pitch angle. The new

speed reference is also applied to the droop controller of the pitching system.

The rotational speed of the generators is outlined by Fig. 4.11a. The speed tends to

settle down in the new reference point determined by the speed control loop of the

generator-side converter to meet its share in terms of increasing the power production.

However, the amount of demand by the converter is more than the rise in power due to

the displacement of the new operating point with a lower speed. So, the speed keeps

falling until the droop controller reacts to the fall and reduces the pitch angle and,

thereby tries to bring the speed back to the new reference point of the rotational speed.

The simulation results in Fig. 4.11b confirm the theoretical discussion in section 4.4.3 –

that the equivalent droop characteristic for the DRWT is higher than for the SRWT, and

this assists the generating unit of the DRWT to deliver more energy to the generator-

side converter. Consequently, the grid-side converter of the DRWT-FRC is permitted to

generate more energy during the transient period in comparison to the corresponding

converter of the SRWT-FRC, which results in a better frequency control performance.

The frequency transient deviation is presented in Fig. 4.11c. The frequency of the power

system connected to the DRWT-FRC is reduced to 49.7Hz, while this value is 49.62Hz

for a power system that includes the SRWT-FRC.

Through measuring the amount of frequency fall in Fig. 4.9c, Fig. 4.10c and

comparing them, it is revealed that the transient frequency control performance of the

sub-optimal controlling mode is better than the pitch control mode at the same amount

of steady state growth of power. The main reason for this is the negative value of ΔPEX

when the excursion of the operating point is in the under-speed area during the

transients for the pitch control mode. However, in sub-optimal mode, any deceleration

of speed results in positive ΔPEX due to the negative slope of the aerodynamic curve in

the over-speed area.

Therefore, the transient excursion of the operating point introduces an incremental

impact on the total power coming from the blades in sub-optimal control mode, while

this impact is detrimental for the pitch control mode.

107

a) Generator speed

b) Active power

c) Network Frequency

Fig. 4.11. Responses of DRWT-FRC and SRWT-FRC with droop controller in action

Through comparing the frequency fall in Fig. 4.10c and Fig. 4.11c, it can be observed

that implementing the droop controller into the pitch control loop for the sub-optimal

mode enhances the frequency support ability of the wind turbine in this mode. The

droop loop forces the wind turbine to respond to the demand in power more quickly

and, as a result, the grid-side converter can release more power to the grid during the

Generator : Graphs

560 570 580 590 600 610 620

1.1600

1.1620

1.1640

1.1660

1.1680

1.1700

1.1720

1.1740

y (p

u)

SRWT Gen Speed DRWT Gen Speed

Main : Graphs

560 570 580 590 600 610 620

0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950

y

SRWT_Active_Power DRWT_Active_Power

Main : Graphs

560 570 580 590 600 610 620

49.550 49.600 49.650 49.700 49.750 49.800 49.850 49.900 49.950 50.000

y

SRWT_Frequency DRWT Frequency

108

transient period without risk of abnormal operation for the generating unit. In the sub-

optimal mode, the frequency of the system that includes the SRWT falls to 49.5Hz

while, by adding the droop controller to the pitch system, the frequency drops to 49.6Hz

as a response to the same disturbance.

4.6 Conclusion

In this chapter, the impact of the DRWT on the margin of the short-term frequency

stability for three different controlling modes was investigated. This frequency control

performance is compared with that of the conventional single-rotor wind turbine

(SRWT). The pitch control mode, the sub-optimal control mode and the combination of

the sub-optimal and pitching system were adopted as the de-loading mode, respectively,

for high, low and medium wind speeds. In the existing methods, the portion of the

stored kinetic energy (KE) that can possibly be released by the wind turbine is

considered as the only factor for calculating the gain of the frequency control loop

integrated into the power converters of the wind farms. However, in this chapter, it was

shown that, besides the released KE, the transient variation of the aerodynamic energy

during the transient frequency deviation was also influential in the inertial response

characteristic of the wind turbine. The amount of the aerodynamic change during the

transient period was a function of the initial location of the operating point specified by

the de-loading mode, and its range of excursion along the aerodynamic characteristic

curve was specified by the severity of the disturbance. It was recognized that the

capability of the SRWT in releasing the KE is only slightly higher than that of the

DRWT, regardless of the mode of control. Therefore, the transient variation of the

aerodynamic energy was an integral factor in understanding which one is superior with

respect to limiting the frequency nadir. It was shown that, in pitch control de-loading

mode for low-speed winds, the transient frequency control performance of the DRWT

was enhanced to a greater degree than in the SRWT. This was due to the higher

weakening effect of the SRWT operating point excursion in the under-speed area, in

comparison to that of the DRWT. For the sub-optimal de-loading mode (high-speed

wind scenario), it was found that the SRWT was more successful in arresting the

network transient frequency nadir in comparison to the DRWT. This was because of the

higher speed drop of the SRWT rotational speed in the over-speed region, which

resulted in higher boosting impact of the SRWT operating point excursion in

109

comparison to that of the DRWT. In the combination mode for the medium wind

speeds, it was seen that, through appropriate selection of the droop factor for the pitch

control system of the auxiliary turbine, the DRWT was enhanced and it was more

effective in limiting the transient frequency deviation. The main turbine in the DRWT

and the turbine in SRWT had the same droop system.

110

The Impact of DRWT on the Short-term Voltage Chapter 5Stability of the Power System

5.1 Introduction

The principal target of this chapter is to investigate the relative impact of DRWT- and

SRWT-based wind farms on the short-term voltage stability margin of the network. The

main contributions of this chapter towards achieving this target are as follows: first, the

current, most popular approach for assessing the transient voltage stability margin of

IG-based generating units –that is, the critical rotor speed method – is described.

Although this method is quite accurate for the prediction of the transient angle stability

margin of the induction generators, the view has been taken that its assessment

regarding the voltage stability margin does not include all the dominant factors of this

phenomenon. The peak of the transient apparent power delivered by the wind turbines

during the transient period is proposed here as the criterion for the evaluation of the

relative impact of the DRWT and SRWT. The higher the transient apparent power

generated by the generating unit, the less the transient voltage stability margin is

predicted for the local network by this method. To verify the validity of the proposed

method, three energy conversion scenarios will be examined, including: FSIG; DFIG in

nominal condition; and DFIG when it’s supplied reactive power hits the capacitive

limit. For all scenarios, the wind turbines are operating in maximum power point

tracking (MPPT) mode and massive load switching is selected for the type of the

disturbance that leads to the transient voltage instability. For the FSIG scenario, it is

found that the DRWT has a negative impact on the transient voltage stability in

comparison to the SRWT. This is due to the higher maximum transient apparent power

delivered by the DRWT than that of the SRWT. For the DFIG scenario under normal

operation, it is reported that there is no substantial difference between the transient

voltage support performances of the wind turbines. However, when the reactive power

supplied by the wind farm hits the reactive power limits of the wind farm’s grid

connection bus, , during the transient period, the SRWT keeps its benefit over the

DRWT for the same reason as it does in the FSIG scenario.

The following chapter is organised as follows: in section 5.2, the existing method for

analysing the short-term voltage stability margin of the induction generator-based wind

farms is explained. Then, in section 5.3, the reasons that this approach is not suitable for

111

assessing the impact of the DRWT on the short-term voltage stability margin will be

discussed. In section 5.4, for the evaluation of the relative transient voltage

supportiveness of the DRWT versus the SRWT, based on FSIG technology, a new

criterion is proposed. The validity of the proposed method is tested when both systems

are operating in optimum point tracking and the disturbance leads to generator

deceleration. The voltage regulation performance of the DRWT and the SRWT is

compared in section 5.5 where the technology of the energy conversion is DFIG. A

consideration of the capability curve of the wind farm’s connection bus, rather than the

associated curve of the individual DFIGs, is recommended. Two cases are included in

this section; in the first case, the wind farm does not hit the reactive power limit, while

for the second case, it hits the limit. Finally, in section 5.6, the theoretical claims made

in the previous sections will be checked through simulation results.

5.2 Critical Rotor Speed as the Current Approach

Large-disturbance voltage stability is the ability to maintain steady state voltage after

a large disturbance, such as system loss of generating units or faults and the tripping out

of the heavily loaded transmission lines [20]. The current approach for assessing the

voltage stability margin of wind farms equipped with fixed speed induction generators

has been presented by [51]. The methodology was adopted from [69] which was

originally introduced for analysing the transient angle stability margin of FSIG-based

windmills. This approach is discussed extensively in section 3.5 of this dissertation. A

brief description about the critical rotor speed method is given below.

Through overlaying the torque-speed characteristic curves of the generator and

mechanical drive in the same plot, two intersects are delivered. These intersects are the

operating points at the steady state condition; one of them is stable and the other one is

dynamically unstable. For example, in Fig. 5.1, ω0 and ωcr are, respectively, the stable

and unstable operating points. The speed corresponding to the unstable operating point

is called the ‘critical speed’. During the fault occurrence, the terminal voltage of the

induction generator drops dramatically and consequently the generator accelerates

because the electromagnetic torque is much less than the mechanical torque. So, the

stable operating point (ω0) accelerates toward the critical point (ωcr). During the post-

fault transient period, if the speed of the stable operating point exceeds the critical speed

(ω0>ωcr), then the wind generating unit is considered by [51] as transiently voltage

unstable. The main reason addressed by this reference is the high amount of reactive 112

power absorption by the machine during the acceleration. In other words, the more the

distance between the stable operating point and the critical speed during the transient,

the higher the transient stability margin achieved.

Fig. 5.1. Overlay of torque-speed characteristics of generator and mechanical drive

Although the above-mentioned method is accurate enough to predict the transient

angle stability margin of FSIG-based wind farms, according to the following reasons,

this approach is not precise enough to estimate the voltage stability margin of the

networks contained in wind farms. For more simplicity, this method will hereafter be

termed the ‘critical rotor speed’. The arguments will be discussed in the next section.

5.3 Analyzing the Validity of the Critical Speed Method for Voltage Stability Assessment In this section, the legitimacy of the critical rotor speed method for evaluating the

voltage stability margin of the induction generator-based wind farm is going to be

discussed in more depth. This method has traditionally been used to calculate the

transient angle stability margin of the induction motors and generators. Due to the

following reasons, the critical rotor speed method has not been chosen as the benchmark

for assessing the impact of the DRWT on the short-term voltage margin of the power

system.

5.3.1 Voltage Collapse due to the Voltage Instability or Angle Instability

Voltage collapse after a large disturbance can originate from either angle instability or

short-term voltage instability. Voltage collapse due to the angle instability has an

oscillatory nature. The voltage oscillations are due to the variable angle difference

between the two groups of generators, or a single generator and network during the 113

post-fault period. In contrast, voltage collapse due to short-term voltage instability is

progressive and settles down in a new steady state value (without periodic oscillation) in

such a way that the new voltage level is not acceptable by power system operators

according to the grid codes [14]. The progressive voltage fall is defined for the power

systems which the operating generating units are connected to the grid directly.

However, it was proposed in [104] that in the presence of the HVDCs, operating in

constant power mode, the voltage collapse due to the voltage instability may introduce

oscillatory response to the disturbance. This is due to the functioning of the control

system of the converters to fulfil the duties.

The voltage collapses in [51] and [70], which have used the critical rotor speed method

to evaluate the impact of some sort of factors on the short-term voltage stability, are

oscillatory, which is against the standard definition of transient voltage collapse by [14].

Fig. 5.2 shows post-fault responses of the induction generator terminal voltage and

speed addressed by [70] as the short-term voltage instability. So, it can be concluded

that, the voltage collapse introduced as the short-term voltage instability by the critical

rotor speed method, is actually due to the angle instability which results in oscillatory

voltage collapse. Fig. 5.2b shows the generator speed that confirms the angle instability

of the FSIG.

a) Terminal voltage

b) Generator speed

Fig. 5.2 An abnormal condition addressed as voltage unstable by [70]

114

5.3.2 Generator Speed Reaction to the Short-term Voltage Instability

According to the critical rotor speed method, short-term voltage instability must be

accompanied by transient angle instability. In other words, because, according to this

method, the short-term voltage instability occurs whenever the stable operating point

exceeds the unstable operating point, then the transient voltage instability must always

be accompanied by the transient angle instability; they would happen at the same time.

Fig. 5.2b in section 5.3.1 illustrates the response of the rotational speed to the condition

called ‘transient voltage instability’ by [70]. However, based on [14], the voltage

collapse may occur when the rotor angle is stable and has no instability problem. In this

section, a voltage instability case study is carried out to verify the statement by [14]. An

individual SRWT is connected to a local load and a big reactive load is suddenly

connected to the network to allow the grid to experience voltage instability. No

supplementary voltage control equipment is provided in order to reduce the short-term

voltage instability margin which helps to push the system toward the voltage instability,

rather than angle instability. Fig. 5.3 presents the response of the network voltage and

the wind farm generator speed when a big reactive load is connected to the weak power

system as the disturbance.

a) Terminal voltage

b) Generator speed

Fig. 5.3 Voltage collapse due to the voltage instability after a big load switching

Main : Graphs

460 480 500 520 540 560 580 600 620

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

y

BUS VOLTAGE1

115

The wind farm is operating in speed control mode. When the load is entered, the voltage

collapses and settles down in a low and unacceptable value of 0.68p.u after a short

period (Fig. 5.3a). Therefore, the system is unstable with respect to network voltage.

Fig. 5.3b indicates that the generator rotational speed of the wind farm is restored to its

initial value after elapsing the transient period and there is no concern about the angle

instability. So, there can be a voltage collapse while angle stability is not an issue.

5.3.3 Influential Factors on Voltage Stability and Angle Stability

Different types of factors influence the angle and voltage instabilities. For instance, the

generator dynamic model and short circuits close to the generating units are important

for the angle stability analysis, while to investigate the short-term voltage stability

strength, the dynamic model of loads, electrical parameters of transmission lines,

dynamic reactive compensators and the response of the excitation systems of the

generators are of most importance. For example, in some areas of operation, during the

post-disturbance period, the stable operating point does not exceed the critical speed

and, according to the critical rotor speed method, the network should be voltage stable.

However, other factors, such as transmission line parameters, the excitation limit of

adjacent conventional generators and the dynamic of reactive power compensators are

involved in determining the voltage stability margin of the power system; therefore, the

network voltage may transiently collapse while there is enough margin between the

stable and unstable operating point of an FSIG-based wind farm.

5.3.4 Disturbances which Lead to Generator Deceleration

The critical rotor speed method is only useful for analysing the short-term voltage

instabilities due to the sorts of disturbances that make the generators accelerate (Like

faulty condition). However, most other types of disturbances that have a negative

impact on the voltage stability margin force the generators to decelerate. For instance,

the sudden disconnection of neighbouring generation units, or switching on big loads

like powerful induction motors, forces the generators to slow down during the transient

period until the balance between the energy demand and production is resumed again.

Thus, this method is not able to argue the disturbances that, on the one hand, are a threat

to the voltage stability and, on the other, cause the deceleration of the generator.

For the above-mentioned reasons, the critical rotor speed method is not suitable to be

used as an approach for comparing the relative influence of DRWT and SRWT on the 116

transient voltage stability margin of the local network. In the following section, a

method will be introduced to pave the way for comparing the impacts of DRWT and

SRWT. In this method, instead of critical rotor speed, the capability of maximum

apparent power injection during the transient period is the criterion for making the

comparison.

5.4 Proposed Method for Assessing the Impact of DRWT on Large Disturbance Voltage Stability To investigate the impact of the DRWT on the short-term or transient voltage stability

of the local power network, the voltage stability strength of a network that incorporates

the DRWT should be compared with the strength of the same power network when the

wind farm is based on the SRWT. FSIG-based wind farms always absorb reactive

power from the network unless they are equipped with electronic power converters to

form the double-fed induction generator (DFIG) or fully-rated converter (FRC).

In the proposed method, the maximum transient active and reactive powers fed to the

grid during the transient period are considered as the criterion for evaluating the impact

of the DRWT-based wind farms on the short-term voltage stability margin of the

network. This method will be discussed below.

When a big load is switched in or a big generator is tripped out, then other generating

units try to increase their output powers fast enough to match the whole demand for

arresting and limiting the network frequency drop. On the other hand, because of rapid

growth in power production, a higher amount of current flow through the transmission

or distribution lines and consequently, the bus voltages decrease more and the system

approaches its short-term voltage stability limit. So, if in the first few seconds after the

disturbance the generating units increase their output power with lower rates, then the

transient bus voltage rate of drop should be less and the short-term voltage stability

margin will be higher. However, in the scenario with lower rates the network frequency

would drop to lower values in comparison to the quick increase of the apparent power

with higher rates.

In this section, in order to investigate the effect of the DRWT on the transient voltage

stability, it is recommended to examine whether the DRWT or SRWT generates less

peak of apparent power during the transient period to meet the same steady state

demand. During the transient period, if the DRWT introduces apparent power less than

the SRWT (the overshoot of the apparent power corresponding to the DRWT is less 117

than that of the SRWT), then local power systems that incorporate the DRTW-based

wind farms benefit from the higher short-term voltage stability margin, in comparison to

the networks connected to the SRWT-based wind farms. Otherwise, the transient

voltage stability margin should decrease in the presence of the DRWT technology in the

power system. To have a fair comparison, both systems settle down in the same

operating point.

The active and reactive power of an FSIG wind turbine is achievable through the

classic model of the induction generator. The electrical circuit of the induction generator

employed for calculating the active and reactive power is presented in Fig. 3.10. The

amount of active and reactive power is consumed by, respectively, equivalent Thevenin

resistance and the reactance of the induction generator shown by (5.1) as follows:

𝑃𝐼𝐺 = 𝐼𝑇2.𝑅𝑇 (5.1)

𝑄𝐼𝐺 = 𝐼𝑇2.𝑋𝑇

where XT and RT are the Thevenin equivalent reactance and resistance respectively

which have already been given by (3.19). IT is the current flowing through the Thevenin

equivalent circuit. For convenience, the Thevenin impedances are given here again:

𝑋𝑇 =𝑋𝑀(

𝑅𝑟𝑠 )2+𝑋𝑟.(𝑋𝑀+𝑋𝑟)

(𝑅𝑟𝑠 )2+(𝑋𝑀+𝑋𝑟)2

𝑅𝑇 =𝑅𝑟𝑠 .𝑋𝑀

2

(𝑅𝑟𝑠 )2+(𝑋𝑀+𝑋𝑟)2

in which Xr , XM, Rr are, respectively, the rotor leakage reactance, the magnetizing

reactance and the rotor resistance. ‘s’ represents the generator slip. To have an apple-to

apple comparison, all electrical parameters are equal for the generators used in the

DWRT-based and SRWT-based wind farms. In the following section, the transient

power generation capability of the active and reactive power of the DRWT and SRWT

will be examined.

5.4.1 Transient Response of Active Power

The active power introduced by wind turbines, regardless whether the technology is

FSIG, DFIG or FCR, is sourced from the energy captured from the wind by the blades. 118

So, pitch controllers still play an integral role in regulating the flow of active power.

Throughout the transient period, the pitch controllers try to force the under control

quantity (generator rotational speed, output power or mechanical torque) to reach the

corresponding reference. So the first perception is that, during the transient period, the

energy generation is only regulated by the blade angle alteration. However, according to

Chapter 4, there is another factor in effect during the transient period that should be

taken into account. This factor is the transient excursion of the operating point along the

aerodynamic characteristic curve of the wind turbine during the post-disturbance period.

The excursion is because of the generator speed transient rise or fall. Based on Fig. 5.4,

in steady state, the turbine is working at operating point (ωop , Pop). When a big load is

switched in or a generating unit is switched out, the generator stator current increases

and consequently the electromagnetic torque is increased beyond the mechanical torque

until the mechanical drive matches it. Then, based on the swing equation, the rotational

speed of the generator slows down to keep the balance between the mechanical and

electromagnetic torques. This generator slowdown is transferred to the turbine blades

through the gearbox. According to (3.36), the wind energy absorbed by the blades is

proportional to the cubic of rotational speed of the turbine:

So the turbine slowdown leads to the shift of the operating point towards the left from

the original point (ωop1 , Pop1) on the aerodynamic curve. The arrows in Fig. 5.4

illustrate the direction of excursion of the operating point. As a load is switched in or a

local generator is tripped out, the FSIG-based wind farms are commanded to increase

their power through adjusting the pitch angles to match the new demand. As a

consequence, the generating units of the wind farm slow down for a while and settle

down in a new operating point determined by the pitch control system. In MPPT mode,

the deceleration of the generator forces the turbine operating point to travel down along

λβλ

βλ λ 0068.0).54.0116(517.0),(21

+−−=−

ieCi

P

1035.0

08.011

3 +−

+=

ββλλi

VRm .ωλ =

119

the aerodynamic characteristic curve in the under speed area. Although the command by

the pitch control and the excursion of operating point on the aerodynamic curve are in

action at the same time, for more simplicity in Fig. 5.4, at first the command from pitch

control is applied to the wind farm which permanently shifts up the operating point

from its initial value (ωOP1, POP1) to the new final operating point (ωOP2, POP2).

Subsequently, the effect of the turbine deceleration on the output active power forces

the operating point to travel down from the final operating point to point B, which is

associated with minimum rotational speed ωmin. When the transients are damped, the

turbine again settles down to the final operating point (ωOP2, POP2). The route of the

operating point excursion is shown by a line with arrows on it. Fig. 5.4 just shows the

top-left portion of the aerodynamic curve as a zoom-in to make it more readable.

Fig. 5.4. Excursion of operating point on the aerodynamic curve

Fig.5.4 is used just to show the effect of the components of the aerodynamic power on

the output power during the transient period (∆PPI , ∆PEX ) in a simpler way and it is not

illustrating the real path of the operating point during the transient period when the

effect of the both pitch control system and generator speed deviation are involved

simultaneously. So, the straight line between the aerodynamic curve at the bottom and

aerodynamic curve at the top does not mean the reaction of the pitching system is

instantaneous. Fig. 4.4 shows the simultaneous effect of the pitch control and generator

speed deviation on the output power during the transient period.

From the above discussion, the aerodynamic power introduced by a wind turbine is

influenced by two main components: the functioning of the pitch controller that is

effective for both transient and steady state stages; and the excursion of the operating

point of the wind turbine on the aerodynamic characteristic curve, in effect only during

120

the transient phase. This matter is already explained extensively in section 4.2.2. Since,

according to section 4.3 the kinetic energy (KE) released by the DRWT and the SRWT

are almost the same, KE is omitted for this chapter. The version of the power coming

from the turbine excluding KE is presented by (5.2):

∆𝑃𝑇 = ∆𝑃𝑃𝐼 + ∆𝑃𝐸𝑋 (5.2)

where ΔPT, ΔPPI are respectively; the output active power generation by turbine, active

power regulated by pitch controller to meet the new demand. ΔPEX in Fig. 5.4 is the

change of captured power by blades due to the transient excursion of the operating point

on the aerodynamic curve.

In Fig. 5.4, ΔPPI is dictated by the pitch control system to regulate the mechanical

torque as demonstrated in Fig. 5.5. In this study, the pitch system is in charge to control

the mechanical torque coming from the turbine. The reference of the mechanical torque

is set by wind farm operators in order to track the maximum operating point. Since the

main focus on the comparison is on the transient response characteristic, the initial and

final operating points must be the same for SRWT and DRWT.

Fig. 5.5. A simple pitch angle controller for regulating the mechanical torque

At the same severity of disturbance, to have an apple-to-apple transient characteristic

comparison, the pitch control system in the DRWT and SRWT should receive the same

command from the network operators to adjust the output energy. So the portion of ΔPT

sourced by the pitch control system (ΔPPI) is the same for the DRWT and the SRWT.

Therefore, analysing the impact of ΔPAE on ΔPT is a key factor in understanding

whether the DRWT or the SRWT introduces more active power to the grid during the

transient period in the MPPT mode. To do so, it is essential to examine which of the

DRWT or SRWT introduces a higher range of speed deviation at the same severity of

disturbance. The higher the generator speed reduction, the lower spot the operating

point reaches on the aerodynamic curve, which has a weakening impact on the power

121

production capability of the generating unit. In this regard, the swing equation is

relatively handy for assessing the speed deviation rate of the DRWT in comparison to

the SRWT during the transient period. The general form of the swing equation is

presented by (5.3) as follows:

𝐽. = ∆𝑇 (5.3)

where J is the inertia momentum and should be substituted by JDRWT and JSRWT,

respectively, for the DRWT and the SRWT. ΔT is the difference between the

mechanical and electromagnetic torque considered as an input to the swing equation.

ΔT is modelled as a step change to simulate the impact of the fault on the torque

balance (ΔT(s) = s-1). Damping coefficients are neglected here (DDR , DSR = 0). The time

domain response of the rotational speed of the DRWT and the SRWT to the input step

change is presented by (5.4):

𝜔𝐷𝑅(𝑡) = 𝜔0 +𝐴𝑠𝑐 . 𝑟(𝑡)𝐽𝐷𝑅

𝜔𝑆𝑅(𝑡) = 𝜔0 +𝐴𝑠𝑐 . 𝑟(𝑡)𝐽𝑆𝑅

(5.4)

where ω0 is the initial speed before the fault and should be identical for both wind

turbines, r(t) is a ramp function, and Asc is the amplitude of the step change which is the

same for both the DRWT and SRWT to have a fair comparison. For the disturbances

that lead to an electromagnetic torque higher than the mechanical torque, the Asc is

negative, otherwise Asc is positive. In (5.4), almost all elements are the same except the

total momentum inertias (JDR , JSR). From (3.23) and (3.24), it was seen that JDR > JSR.

Therefore, the DRWT is expected to introduce a lower rate of speed change during the

disturbance. Based on (5.4), Fig. 5.6 shows the responses of the SRWT and the DRWT

to a sudden increase of electromagnetic torque (Te). The duration of simulation is set to

be tf. The rate of speed change for DRWT is 1/JDR, while this rate, for SRWT, is 1/JSR.

Since JDR> JSR always, then the rate of speed change of the SRWT is always greater

than the DRWT, and consequently, at a specific period of time, SRWT accelerates or

decelerates with a higher rate of change compared to DRWT (∆ωSR > ∆ωDR).

122

Fig. 5.6. Rate of speed change in the DRWT and SRWT as a response to electromagnetic step up (the

damping factor is neglected)

To exemplify, Fig. 5.7 presents the rotational speed oscillations of the SRWT and the

DRWT to the same disturbance leading to the deceleration of the generators. The results

confirm the theoretical claim by (5.4).

Fig. 5.7. Slowdown of generators as a response to sudden growth in load

Since, in this section, the impact of the operating point transient excursion on the power

generation is the main target, in Fig. 5.8, the blade angles are assumed to remain

constant to exclude the effect of the pitch control system on the aerodynamic

characteristic curve. Therefore, the aerodynamic curve doesn’t shift up or down during

the transient period for the time being. Fig. 5.8 shows how the operating point travels

along the aerodynamic curve when the turbines decelerate. According to the results

from (5.4) the rotational speed of the SRWT drops down to ωs, which is less than the

minimum speed reached by DRWT, ωd. Consequently, since the slope of the curve is

positive in the under-speed area, the power captured by the blades of the SRWT goes

down to PSR, which is less than the minimum power delivered by DRWT, PDR. In other

words, the amount of reduction in the energy captured by the blades in the SRWT is

more than in the DRWT (∆𝑃𝐴𝐸_𝑆𝑅 > ∆𝑃𝐴𝐸_𝐷𝑅) during the transient deceleration of the

turbine.

Main : Graphs

660 670 680 690 700 710 720 730

1.0180

1.0190

1.0200

1.0210

1.0220

1.0230

1.0240

1.0250

1.0260

y

SRWT_Generator_Speed Generator_Speed_Dual_Rotor

123

Fig. 5.8. Comparing the excursion of operating points of SRWT and DRWT on the aerodynamic curve

To summarize the reaction of the ∆𝑃𝑃𝐼 , ∆𝑃𝐸𝑋 to the deceleration of the generators:

- The pitch controller reduces the angle of the blades appropriately to increase the

output power in order to match the new demand. So the variation of ∆𝑃𝑃𝐼 is positive.

To have a valid comparison, ∆𝑃𝑃𝐼 is assumed to be the same for both the SRWT and

the DRWT.

- In the under-speed area, the ∆𝑃𝐸𝑋 is decremented and its sign is negative. The

amount of decrease of this quantity in the SRWT is more than that in the DRWT.

Now, at this stage, there are enough materials to make a decision as to whether the

DRWT or SRWT introduces a higher over-shoot of active power during the transient

period to meet the sudden growth in demand. The transient peak of the active power for

the both technologies is given in (5.5), as follows:

∆𝑃𝑀𝑎𝑥_𝑆𝑅 = 𝑀𝑎𝑥 ∆𝑃𝑃𝐼_𝑆𝑅 + ∆𝑃𝐴𝐸_𝑆𝑅

∆𝑃𝑀𝑎𝑥_𝐷𝑅 = 𝑀𝑎𝑥 ∆𝑃𝑃𝐼_𝐷𝑅 + ∆𝑃𝐴𝐸_𝐷𝑅

(5.5)

As mentioned previously, the impact of the pitch control system should be the same for

the DRWT and the SRWT. It is given below:

∆𝑃𝑃𝐼_𝑆𝑅 = ∆𝑃𝑃𝐼_𝐷𝑅

On the other hand, from Fig. 5.8, it was seen that the operating point excursion of the

SRWT has a more weakening effect on the total generated energy than that of the

DRWT. The mathematical statement of this phenomenon is given below:

|∆𝑃𝐸𝑋_𝑆𝑅| > |∆𝑃𝐸𝑋_𝐷𝑅|

124

where the sign for both ∆𝑃𝐸𝑋_𝑆𝑅 and ∆𝑃𝐸𝑋_𝐷𝑅 is negative. This is due to the reduction of

these quantities during the generator’s deceleration. Since, the total reduction of ∆𝑃𝐴𝐸_𝑆𝑅

is more than the total reduction of ∆𝑃𝐴𝐸_𝐷𝑅, from (5.5), it can be concluded that during

the transient period, the maximum peak of active power generation delivered by DRWT

(∆𝑃𝑀𝑎𝑥_𝐷𝑅) is higher than that delivered by the SRWT (∆𝑃𝑀𝑎𝑥_𝑆𝑅). This theoretical

claim is presented by (5.6):

∆𝑃𝑀𝑎𝑥_𝐷𝑅 > ∆𝑃𝑀𝑎𝑥_𝑆𝑅 (5.6)

In Fig. 5.8 the impact of ∆𝑃𝑃𝐼 on the movement of the operating point is neglected for

reasons of simplicity. However, in reality, the impacts of the both ∆𝑃𝑃𝐼 and ∆𝑃𝐸𝑋 are

simultaneously involved in the determination of the output active power. The excursion

track of the operating point of the wind turbines during the transient period is presented

in Fig. 5.9, while the effects of both ∆𝑃𝑃𝐼 and ∆𝑃𝐸𝑋 are included at the same time. In

this figure, on the one hand, due to the reduction in the blade angle, the aerodynamic

curve is shifted up to increase the output power, and on the other hand, the operating

point drifts toward the left due to the temporary fall of the rotational speed. Since the

range of speed deviation of the SRWT is higher than that for the DRWT, given the same

severity of disturbance, the route of the transient excursion for the SRWT is located on

the left hand side of the DRWT (DRWT is represented by a dashed line). The other two

aerodynamic curves are sketched between the initial and final curves as the samples to

illustrate how the initial aerodynamic curve expands to the final curve. The initial and

final operating point are respectively (ωop1, Pop1) and (ωop2, Pop2) and are considered to

be the same for both the DRWT and the SRWT. The cross-sections between the

transient excursion route of the SRWT and DRWT, with the two intermediate

aerodynamic curves, are pretty helpful in predicting the responses of the active power of

the DRWT in comparison with that of the SRWT. For the first curve, the cross-sections

for the SRWT and the DRWT are PS1 and PD1, respectively, where PS1 < PD1. The same

is true for the second intermediate curve where PS2 < PD2. So, from (5.6) and Fig. 5.9, it

can be concluded that the curve envelope of the active power of the DRWT will be

located above that of the SRWT during the transient period in MPPT mode.

125

Fig. 5.9. Transient excursion of operating points of SRWT and DRWT while the effect of both ∆PPI and

∆PEX are involved

The claim by (5.6) and Fig. 5.9 is approved by Fig. 5.10. This figure illustrates the

post-disturbance response of the active powers generated by the DRWT and the SRWT

when a fairly small load is suddenly connected to the local network at t=670s. Although

the active powers of both the wind turbines converge to the same new stable operating

point, the maximum transient active power reached by the DRWT at t=672s (solid line

in green) is higher than that of the SRWT.

Fig. 5.10. Post-disturbance responses of active power of DRWT and SRWT to the growth of load

In the next step, the transient response of the consumed reactive power, as the other

component of the apparent power, will be investigated. The target is the comparison of

the peak of reactive power consumption drawn by the DRWT and the SRWT

throughout the transient period.

5.4.2 Transient Response of Reactive Power

The reactive power consumed by an induction generator is given by (5.1) where the

detail of Thevenin reactance, XT, is presented by (3.19). In this section, an equation will

Main : Graphs

670.0 675.0 680.0 685.0 690.0 695.0 700.0

0.500

0.510

0.520

0.530

0.540

0.550

0.560

0.570

y

SRWT Active Power DRWT Active Power

126

be developed to show how the reactive power of an induction generator is a function of

the generated active power and its rotational speed. This equation is able to assist us to

predict the reactive power response of the generator during the transient period. The

procedure will be presented below.

From (5.1), it can be concluded that the multiplication of the active power and the

equivalent Thevenin resistance is equal to the multiplication of the reactive power and

the equivalent reactance. The reason is the equality of both multiplications with the

square of the terminal voltage, (𝑃𝐼𝐺 .𝑋𝑇 = 𝑄𝐼𝐺 .𝑅𝑇 = 𝐼𝑇2). Thus, it is possible to obtain

the reactive power as a function of the active power and the slip of the generator. This is

presented by (5.7):

𝑄𝐼𝐺 = 𝑄𝐹𝐴𝐶𝑇 .𝑃𝐼𝐺 (5.7)

where,

𝑄𝐹𝐴𝐶𝑇 =𝑅𝑟𝑠

2+ 𝑋𝑚.𝑋𝑟 + 𝑋𝑟2

𝑅𝑟𝑠 .𝑋𝑚

From (5.7) it can be deduced that, during the transient period, the consumption of the

reactive power by FSIG is determined by the values of the active power generation and

the slip. It is seen that the consumption of the reactive power is directly proportional to

the amount of active power generation. Therefore, if the speed is assumed to remain

constant during the transient period, then where the active power is increased to meet

the load growth, the absorption of reactive power would consequently be increased as

well. On the contrary, every time the wind turbine reduces the generation of active

power at the constant speed, then the induction generator draws less reactive power

from the network. Moreover, in addition to the active power, the variations of the slip

should also be included for investigation of the transient response of the reactive power

during the transient period. So, to assess whether the SRWT or the DRWT draws higher

reactive power during the transient period, the response of both the active power and the

QFACT factor should be traced in this period simultaneously. As concluded in section

5.4.1, the maximum peak of the active power introduced by the DRWT is higher than

the maximum power produced by the SRWT in MPPT mode. Now it is time to explore

more about the reaction of the QFACT to the generator speed deviation.

127

QFACT coefficient is composed of the slip and the electrical parameters of the induction

generator. Fig. 5.11 presents the QFACT as a function of the slip. Arrows in Fig. 5.11

show the direction of the QFACT deviation during the generator deceleration. This is

quite helpful in investigating the impact of the amount of the speed reduction on the

value of QFACT. Normally, in practical systems, the nominal values of slips of generators

in wind turbines are less than 0.15. According to Fig. 5.11, for the slips less than 0.16,

as the slip (speed) goes down, the QFACT becomes smaller. Consequently, since the

speed reduction of the SRWT is more than the speed reduction of the DRWT, the

minimum QFACT_SR reached by the SRWT (blue rectangle) is less than the minimum

QFACT_DR hit by the DRWT (red circle).

Fig. 5.11. Coefficient of active power (QFACT) as a function of slip in generating mode

At this stage, based on (5.7) and Fig. 5.11, there is enough material to assess whether

the SRWT or the DRWT absorbs higher reactive power throughout the transient period.

The argument is classified as follows:

- Regarding PIG, according to (5.6) the maximum active power delivered by the

DRWT (∆𝑃𝑀𝑎𝑥_𝐷𝑅) is greater than the peak of the active power generated by the

SRWT (∆𝑃𝑀𝑎𝑥_𝑆𝑅).

- During the rotational speed excursion, the feasible minimum QFACT for the SRWT is

less than that of the DRWT (QFACT_SR < QFACT_DR).

-0.4 -0.3 -0.2 -0.1 0

-1.2

-1

-0.8

-0.6

-0.4

Slip

Qfa

ct

Qfact-DR

Qfact-SR

128

By substituting the associated quantities of the SRWT and the DRWT in (5.7), the

following inequality is achievable:

𝑄𝐹𝐴𝐶𝑇_𝐷𝑅 .∆𝑃𝑀𝑎𝑥_𝐷𝑅 > 𝑄𝐹𝐴𝐶𝑇_𝑆𝑅 .∆𝑃𝑀𝑎𝑥_𝑆𝑅 (5.8)

It is signified by (5.8) that the maximum reactive power absorbed by the DRWT

during the transient period is higher than the maximum reactive power absorbed by the

SRWT. The post-disturbance reactive powers drawn by DRWT and SRWT, when a

normal load is suddenly connected to the local network, are presented in Fig. 5.12. The

statement by (5.8) is confirmed by this figure. Although the reactive powers of both

wind turbines converge to the same value of the new steady state operating point, the

peak of the reactive power absorbed at t=672s by the DRWT during the transient period

is higher than that of the SRWT.

Fig. 5.12. Post-disturbance response of active power of the DRWT and the SRWT to the growth of load

From (5.6) and (5.8), it is revealed that the maximum delivered active power and

absorbed reactive power of the DRWT is respectively higher than the maximum active

and reactive powers reached by the SRWT during the transient period. Although this

ability assists the DRWT to be more transiently successful in increasing the active

power to limit the power system frequency nadir, nevertheless, this matter leads to less

short-term voltage stability margin for the local network connected to the DRWT-based

wind farms. This is due to the higher voltage drop across the impedance of the

transmission or distribution lines located between the loads and the DRWT-based wind

farms. In other words, the higher the active power injection during the post-disturbance

period, the more the voltage drop happens in this time frame, which makes the network

closer to the edge of short-term voltage collapse. On the other hand, it was also

Main : Graphs

670 680 690 700 710 720

-0.2800

-0.2750

-0.2700

-0.2650

-0.2600

-0.2550

-0.2500

y


129

observed that the higher generation of the active power causes higher absorption of the

reactive power by the induction generators in wind farms, which results in further

voltage reduction of the grid connected to the DRWT.

Based on the quantitative analysis in this section, it is possible to theoretically claim

that the impact of the DRWT on the short-term voltage stability margin of the local

power system is detrimental and leads to the reduction of this margin.

5.5 Performance of DRWT and SRWT at the Presence of DFIG

Replacing the FSIG-based wind farms with the variable speed wind turbines, such as

a double-fed induction generator (DFIG) or a fully-rated converter (FRC) wind turbine,

improves the long-term and short-term voltage stability margins of the power system.

This is due to the capability of these technologies to inject reactive power to the

network, while FSIG wind turbines always absorb reactive power. The rotor-side

converter of DFIG makes it possible to control the terminal voltage of the wind turbine

by controlling the rotor current components. A typical block diagram of a rotor-side

converter is presented in Fig. 5.13. In this control diagram, the terminal voltage is under

control by regulating the d component of the rotor current. The q component of the

current is used to regulate the rotational speed [105].

Fig. 5.13. Control block diagram of DFIG wind turbine

The impact of DFIG on the voltage stability of the power system has already been

studied in [106], [107]. In the presence of DFIG, the voltage instability would occur at

higher levels of demand or disturbance. The matter is more understandable through Fig.

5.14, which shows the effect of DFIG on the PV curve of the network [107].

130

Fig. 5.14. Impact of DFIG on steady state voltage stability margin

It is worth noticing that, in analysing the short-term voltage stability, the time delay

of the control system employed in DFIG is also an important factor that must be

included in the study. In addition to the magnitude of reactive power that the wind farm

is able to deliver, the time taken by the system to react to the voltage-threatening

disturbances must be appropriately fast. Otherwise, although a wind farm is able to

supply enough reactive power to meet the demand, if it comes with a relatively high

delay, then voltage collapse may occur. In this study, it is assumed that both control

systems used in the DRWT and the SRWT are fast enough when they have the same

parameters and configurations. According to [73], in cases where the control system is

well-tuned, the DFIG is able to react fast enough to damp the negative effects of the

disturbances that are harmful for the short-term voltage stability of the network. So

DFIG is able to enhance both steady state or long-term voltage stability, and transient or

short-term voltage stability.

Although DFIG enhances the margins for long-term and short-term voltage stabilities,

there are some caps on the capacitive reactive power provided, due to the technical

limitations, by the grid-side or rotor-side converters. Thus, when the total reactive

demand enforced by the disturbance is beyond the practical reactive power limit, the

local network may experience voltage collapse even when the terminal voltage of the

wind farm is regulated by DFIGs. The voltage instability in most scenarios is generally

the result of disturbances when the system is heavily loaded and there is a lack of the

reactive power sources to support the voltage. In this situation, the reactive power

providers of the network, such as synchronous generators, shunt reactive compensators

(SVC, STATCOM, …) or variable speed wind farms are working quite close to their

reactive power generation limits [72]. It is worth paying attention to the power

capability curve of a sample DFIG in Fig. 5.15.

131

Fig. 5.15. Capability curve of DFIGs

The capability curve of each individual DFIG is almost symmetrical for both the

capacitive and inductive mode. From Fig. 5.15, it can be seen that, for the major range

of active power generation, the maximum reactive power provided by each generating

unit is almost the same. Mostly occurring between 0.85p.u to 1.0p.u of the rating active

power, the maximum reactive power capability drops due to technical limitations. So,

DFIGs are able to react to the reactive power demand, regardless of the amount of

generation of active power for the majority of operating points inside the capability

curve. As an example, the DFIG, the capability curve of which is given in Fig. 5.15, is

supposed to increase the reactive power generation from 0.3p.u to 0.6p.u to support the

voltage against a specific disturbance. This scenario is repeated for three levels of active

power generation, such as 0.5p.u, 0.7p.u and 0.9p.u. For the first two levels of active

power, the DFIG is able to match the required reactive power. Therefore, for active

power production less than 0.85p.u, the voltage support performance of the DRWT and

SRWT should be the same when they are equipped with DFIG technology. However, in

section 5.4.1, it was seen that, for the FSIG system, the DRWT consumes more reactive

power than the SRWT during the transient period, which denotes a less short-term

voltage stability margin for the DRWT. For the third case (PGEN= 0.9p.u) the required

reactive power falls beyond the reactive power capability of the generating unit and

DFIG is not able to match the demand. Then the DFIG hits the upper limit of the

reactive power and it can be modelled by an induction generator and a fixed capacitor at

the terminal of the generator. In this circumstance, in the occurrence of any further

growth of reactive power demand, the voltage starts to fall and there is no control over

132

the voltage, just as in the FSIG scenario. In this area of operation, again the SRWT is

advantageous over the DRWT with respect to the transient voltage stability margin.

The capacitive area of the capability curve covered by the DFIG-based wind farm

would be reduced significantly when it comes to the reality of the reactive losses of the

electrical interface equipment. In Fig. 5.15, the capability curve of the wind farm is

obtained from the multiplication of the capability curve of the individual DFIGs and the

number of DFIGs existing in the wind farms. However, in practice, for onshore wind

farms, the individual wind turbines are spread in a very broad area to absorb the wind

energy as much as possible. The same story is true for offshore wind farms located far

out in the ocean. To avoid voltage drop, each turbine should be equipped with its own

step up transformers and longer cables are required to connect the wind turbines to the

grid connection bus. A typical schematic of a wind tower placement in a wind farm is

given in Fig. 5.16 [108].

Fig. 5.16. Physical layout of a typical wind farm

Fig. 5.16 signifies that there is reactive power consumption by the reactance of the

step up transformers and interface transmission lines or cables. Therefore, to have a

correct image of the reactive power characteristic of the wind farm, the capability curve

of the grid connection bus should be obtained instead of the summation of the capability

of the individual curves. The capability of the grid-wind farm connection bus can be

calculated for inductive and capacitive modes through (5.9) and (5.10), respectively:

𝑄𝐺𝑟𝑖𝑑_𝐵𝑢𝑠_𝑖𝑛𝑑𝑢 = 𝑛 ∗ 𝑄𝐷𝐹𝐼𝐺_𝑖𝑛𝑑𝑢 + 𝐼𝑊𝐹2 ∗ 𝑋𝑇ℎ_𝑊𝐹 (5.9)

𝑄𝐺𝑟𝑖𝑑_𝐵𝑢𝑠_𝐶𝑎𝑝𝑎 = 𝑛 ∗ 𝑄𝐷𝐹𝐼𝐺_𝑐𝑎𝑝 − 𝐼𝑊𝐹2 ∗ 𝑋𝑇ℎ_𝑊𝐹 (5.10)

where ‘n’ is the number of the individual DFIGs; 𝑄𝐷𝐹𝐼𝐺_𝑖𝑛𝑑𝑢 , 𝑄𝐷𝐹𝐼𝐺_𝑐𝑎𝑝𝑎 are the

inductive and capacitive limits of each DFIG; 𝐼𝑊𝐹2 is the current flowing from wind

10km10km 10km

10km20km 20km

LV/M

V

LV/M

V

LV/M

V

LV/M

V

MV/HV MV/HV

Grid connection

bus

133

farm to grid; and, 𝑋𝑇ℎ_𝑊𝐹 is the equivalent Thevenin reactance of the wind farm viewed

by the grid.

It means that, in reality, the capacitive reactive power supplied to the grid connection

bus by the DFIG-based wind farms would be reduced proportional to the square of IWF.

For example, for 1p.u of the wind farm current (IWF) and 0.3p.u of the equivalent

Thevenin reactance of the wind farm (XTH_WF), the capability curve in Fig. 5.15 should

be shifted down for 0.3p.u for the curve of the connection bus at the rating active power

(PGEN=1p.u). So, the capacitive region is significantly less than was expected from the

ideal system in Fig. 5.15. Based on (5.9) and (5.10), an approximation of the capability

curve of the grid connection bus is given in Fig. 5.17. Since the reactive losses are

proportional to the square of the current amplitude, then the capability curve should be

pushed down proportional to the square of the injected current.

Just as in an ideal DFIG-based wind farm, the impact of the DRWT on the transient

voltage stability margin is going to be assessed for the second time, while the capability

curve of the connection bus defines the limits for the supplied reactive power this time.

Fig. 5.17 shows that when the active power generation of the wind farm exceeds the

0.65p.u, the wind farm is not able to match the required reactive power demands for the

second and third test. It means that the strength of the DFIG-based wind farm is reduced

dramatically as the active power generation gets closer to 1.0p.u.

Fig. 5.17. Capability curve of grid connection point

As the capacitive region of the wind farm is decreased, it is more likely for the wind

farm to approach the capacitive reactive power capability limit when the active power

generation is close to the rated value. Thus, the DFIG is not able to take proper action

to resume the voltage in the case of any further reactive power demand and the

134

difference between the voltage support characteristics of the SRWT and the DRWT

would be more significant when the DFIG hits the reactive power limit.

It can be concluded that in the presence of DFIG, the impact of the DRWT and the

SRWT on the transient voltage stability margin of the local network is the same, as long

as the required transient reactive power for controlling the voltage falls inside the

boundaries of the capability curve of the grid connection bus. As soon as the grid

connection bus hits its related capacitive reactive power limits, which based on Fig.

5.17 is likely in reality, then the DFIG has no more control over the voltage of the

connection bus and the situation is quite similar to the FSIG scenario. In other words,

for higher levels of disturbance, the transient voltage support performance of a saturated

DFIG is quite similar to the reaction of the FISG to lower levels of disturbance. This

means that, in the over-excited region of the DFIG, the short-term voltage stability

margin of the grid connected to the DRWT-based wind farm is predicted to be less than

the case when it is connected to the SRWT-based wind farm at MPPT operating mode.

The theoretical claims in this chapter will be checked and discussed in the section

detailing the simulation results.


The main focus of this section is to verify the theoretical claims already made in section

5.4 and section 5.5 with respect to the validity of the method proposed to investigate the

impact of the DRWT-based wind farm on the large disturbance voltage stability of the

power grid. The MPPT operating mode is chosen for the case study in order to verify

the prediction by the method. To reach this stage, the performance of this technology is

compared with that of the SRWT-based wind farm regarding the capability of resuming

the voltage to the nominal value after the occurrence of a severe disturbance. This

section is organised as follows: firstly, the capability of DRWT-based and SRWT-

based wind farms are compared when the technology employed is the FSIG. This

scenario is helpful in confirming the validity of the theoretical discussion in section 5.4

and gives more insight into details about the impact of the mechanical characteristics of

the prime mover on the transient voltage stability margin. The same procedure is

repeated for the second scenario when both the SRWT and the DRWT benefit from the

DFIG technology. The second scenario aims to investigate the effect of DFIG

employment on the transient voltage supporting performance of the DRWT. In the third

135

test, the effect of DFIG is evaluated when the rotor-side converter of the DFIG hits the

capacitive reactive power generation limits. In real power systems, it is likely that the

grid connection bus of the wind farm hits its corresponding capacitive reactive power

limits due to the reactive power losses of the electrical interface equipment at nominal

ratings. Each test is conducted separately for the SRWT and the DRWT.

The simulations were carried out using PSCAD/EMTDC software. The power system

used for the tests is given in Fig. 5.18. The power system contains: a 300MVA thermal

power station; a 150MVA wind farm; a 0.69/16 kV transformer; a 100km-length

transmission line; and a local load with a peak of 300MVA.

Since the short-term voltage instability and transient rotor angle instability are

intertwined, sometimes it is hard to make a distinction between these two phenomena

[109]. Because, in this chapter, only the voltage stability is desired to be studied, then

the generators are de-loaded to 50% of their nominal capacity, which helps to increase

the angle stability margin. On the other hand, the over-excitation limit (OEL) of the

excitation system of the thermal power plant is set to 1.3p.u of the nominal voltage,

which reduces the reactive support ability of the thermal plant and makes the system

closer to the edge of the transient voltage stability margin. In this way, the fairly big

load switching is more a threat for the voltage stability than for the angle stability of the

system. Although the aerodynamic efficiency of the DRWT is better than the SRWT

regarding the capturing of the energy of the wind, to have an apple-to apple-

comparison, they are supposed to deliver the same torque for each wind speed through

regulating the torque by the pitch control system. This means that the blade angles of

the DRWT should be higher than those of the SRWT for each level of wind speed and

turbine rotational speed. The speed of the wind is assumed to be constant during the

transient period. The operating point is located at the MPPT and consequently the

excursion of the operating point along the aerodynamic curve would happen in the

under-speed area as the response to the disturbance (big load switching).

To apply a disturbance as a threat to the large disturbance voltage stability of the

system, a big load is suddenly switched in. The load is of a constant power type, which

consumes related nominal power regardless of its terminal voltage and network

frequency. Right after the switching, the network frequency and the bus voltage drop

and the generating units increase their power production to support the frequency. This

136

reaction results in more voltage drop across the power system impedances and brings

the network closer to voltage collapse.

Fig. 5.18. Employed power system for the tests

Fig. 5.19 shows the response of DRWT and the SRWT-based wind farms to the load

switching of 60 MVA at t=470s. The technology of both generating units is FSIG.

According to Fig. 5.19.a, as was expected from section 5.4, the voltage of a power

system connected to the DRWT-based wind farm is reduced dramatically and settles

down to the new steady state value of 0.77p.u; this level of voltage is considered to be

unstable by network operators. Since the voltage instability occurs almost 25 seconds

after the disturbance, it can be called as short-term voltage instability for the DRWT.

Conversely, the network containing the SRWT system resumes the nominal voltage

when the transients fade away. The amount of active and reactive power produced by

the generating units is presented in Fig. 5.19b and Fig. 5.19c, respectively. For the first

few seconds after the disturbance, both systems try to increase the active power to

supply the required demand. In Chapter 4, it was seen that the DRWT is more

successful in limiting the frequency fall in MPPT operating mode.

However, this characteristic of the DRWT regarding the frequency control leads to a

higher absorption of reactive power and consequently more voltage drop occurs across

the transmission lines, resulting in the voltage collapse of the network connected to the

DRWT-based wind farm in this test. The only reason that the SRWT system was

successful to ride out the disturbance is the less transient active power generation and

consequently less reactive power consumption during the transient period. So there is

less transient voltage drop for the SRWT system.

Synch 1

#1 #2PI

COUPLED

SECTION

Load Bus

BRK3

BRK2

BRK1

SRWTWindGen

DRWT

GenWind

300MVA Thermal Plant

1 [mH]0.6 [ohm]VA

300 MVA Local Load

6 [mH]3.8 [ohm]VA

60 MVA Local Load

BRK7

Synch 2

Synch 3

137

a) Terminal voltage

b) Active power consumption

c) Reactive power consumption

d) Generator rotational speed

Fig. 5.19. Response of DRWT and SRWT FSIG-based wind turbine to the large disturbance

Main : Graphs

480 500 520 540 560 580 600 620

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

y

SRWT BUS VOLTAGE DRWT BUS VOLTAGE

Main : Graphs

480 500 520 540 560 580 600 620

-0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

y

SRWT_Active_Power DRWT_Active Power

Main : Graphs

480 500 520 540 560 580 600 620

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

y


Main : Graphs

460 480 500 520 540 560 580 600 620

0.850

0.900

0.950

1.000

1.050

1.100

1.150

1.200

y

SRWT_Generator_Speed DRWT_Generator_Speed

138

When the transients are gone, then the active power of the SRWT rises to complete its

responsibility for feeding the load. Conversely, there is a reduction in the active power

generation of the DRWT. This matter signifies that the DRWT is unstable with respect

to voltage. According to the P-V curve in Fig. 5.14, when the operating point passes the

critical point – also known as the nose point – there is a decrease in the active power of

the generating units as the demand grows. The same story is true for the reactive power

consumption of the DRWT. As illustrated by Fig. 5.19d, the rotational speeds of both

generators resume their initial reference value, which confirms that the angle stability is

not an issue in this case. So the quantity that dominates the instability of the system is

the network voltage, rather than the angle of the generators.

For the next step, it is worth checking the analytical claims which have already been

stated in section 5.5 regarding the impact of the application of DFIG on the relative

performance of the DRWT and SRWT with respect to the short-term voltage stability

margin. The operating conditions, generator parameters, mechanical parameters of the

DRWT and the SRWT, network electrical parameters, initial loading and the severity of

disturbance is exactly the same as in the first scenario, when the FSIG technology was

used. The initial reactive power working point is placed in -0.27p.u where there is

enough distance from the capacitive limit of associated capability curve, which is

0.5p.u. This ensures that the DFIG is able to afford the reactive power required by the

disturbance for supporting the voltage during the post-disturbance period without

reaching the capacitive reactive power limit. To apply the disturbance, the load of

60MVA is again connected to the network at t= 150s. The generation of the active

power is shown in Fig. 5.20a. The curve related to the active power of the DRWT is

located above that of the SRWT due to the higher weakening effect of the frequency fall

on the energy generation of the SRWT, in comparison to the DRWT.

As was seen in section 5.4 and Fig. 5.9, the higher the speed reduction, the lower the

spot that would be reached by the operating point on the aerodynamic curve during the

transient period. The technical reason has already been discussed in more detail in

section 4.4.1.The rotational speed deviation is presented in Fig. 5.20b. The speed drop

of the SRWT is more than that of the DRWT, which is confirmed by the theoretical

analysis in section 5.4.1. The response of the network voltage is presented in Fig. 5.20c.

139

a) Active power

b) Generator rotational speed

c) Terminal voltage

d) Supplied reactive power

Fig. 5.20. Response of the DRWT and the SRWT DFIG-based wind turbine to disturbance

Main : Graphs

150.0 152.5 155.0 157.5 160.0 162.5 165.0

0.300

0.320

0.340

0.360

0.380

0.400

0.420

y

SR_Active_Power DR_Active_Power

Main : Graphs

148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0

0.9550

0.9600

0.9650

0.9700

0.9750

0.9800

y

SR_Speed DR_Speed

Main : Graphs

150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0

0.930

0.940

0.950

0.960

0.970

0.980

0.990

1.000

y

SR_Terminal_Voltage DR Terminal_Voltage

Main : Graphs

150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0

-0.300 -0.250 -0.200 -0.150 -0.100 -0.050 0.000 0.050 0.100

y

SR Reactive DR Reactive

140

The transient voltage fall in the system, which includes the DRWT, is higher than the

system connected to the SRWT, while the voltages of both systems settle down in the

same final value of 0.975 p.u. This is due to the higher transient active power reached

by the DRWT, given in Fig. 5.20a. However, both DRWT and SRWT wind turbines are

successful in recovering the voltage to the set point value, while in the first case (FSIG)

the load switching resulted in the voltage collapse of the system connected to the

DRWT-based wind farm. According to Fig. 5.20d, the operating points of the reactive

powers of the SRWT and DRWT travel from the inductive mode (-0.27p.u) to the

capacitive area (0.05p.u) to support the bus voltage without any saturation. The rise

time of reactive power of the DRWT is more than that of the SRWT.

The reason is the higher amount of voltage error fed to the PI in the leg in charge of

controlling the terminal voltage in the DRWT system. Fig. 5.20 indicates that, firstly, in

the presence of the DFIG, the short-term voltage stability margin of the network is

enhanced considerably due to the injection of reactive power by the wind farm.

Secondly, as long as the demand falls in the capability limits of DFIG, the transient

voltage support capabilities of the SRWT and the DRWT are quite similar. This is a

victory for the DRWT.

According to the theoretical statement in section 5.5, it is worth investigating the

voltage support ability of the DRWT and the SRWT DFIG-based wind farms when the

capacitive reactive power limit of the DFIG is hit by the operating point during the

transient period. In this scenario, the generation of the reactive power by the DFIG is

restricted by over excitation limiters (OEL), and both wind turbines aren’t able to meet

the required reactive power demand. For getting to that stage, the wind turbines are set

to generate 1.05p.u active power, which signifies that the wind farms are operating close

to their rating power. To keep the voltage at set point value of 0.98p.u, the wind farms

have to generate capacitive reactive power to nullify the reactive losses of interface

cables and transformers. The initial operating point of reactive power of DFIG is

0.29p.u and is located pretty close to the capacitive limit of the DFIG. For the third

scenario, a load of 80MVA is switched in at t=120s. When the load is switched in, the

network voltage drops and DFIGs increase their output reactive powers to maintain the

voltage at set point value. However, the reactive power cannot get beyond the reactive

power limit and is stuck at 0.5p.u. In this scenario, the DFIGs are not able to match the

reactive power demand and consequently the network voltages are not expected to

141

resume their pre-disturbance value. In saturation mode, the DFIG can be imagined as a

combination of the induction generator and a fixed shunt capacitor at the terminal. So

the situation is quite similar to the first scenario when the FSIG technology was in use.

The main differences are the amount of active power generation, which is quite higher

for the third scenario, and the generating unit reactive power mode, which is inductive

for the FSIG and capacitive for the saturated DFIG. The DRWT generates more active

power than the SRWT during the transient period so the reactive power losses and as a

result the transient voltage drop should be higher while the saturated DFIG is not able to

make it up. So, the transient voltage stability margin of the DRWT DFIG-based wind

farm is less than that of the SRWT when both of them reach the reactive limits and are

not able to match the reactive power demand.

The response of the quantities for the third scenario is presented in Fig. 5.21. The

variables of the DRWT are represented by a thick dark line with circles standing on it.

The voltage response is shown in Fig. 5.21a. Due to the lack of reactive power supply,

the voltage of the network connected to the SRWT is not able to resume the voltage to

the set point value (0.98p.u) and the operating point settles down to the new stable

point, which is 0.93p.u. The voltage of the network that includes the DRWT is

maintained at 0.75p.u for a short period and becomes unstable after around nine

seconds. The way the reactive powers of the SRWT and DRWT react to the disturbance

is given in Fig. 5.21b. Since, right after load switching, the voltage drop is sharper in

the DRWT, consequently, the rise in reactive power of the DRWT is sharper as well in

order to arrest the voltage fall. After around less than one second, both wind farms reach

the reactive power limit and are stuck there. As soon as the voltage of the DRWT

collapses, the reactive power is reduced dramatically. The active power in Fig. 5.21c is

ordered to be increased to supply the load. However, the active power of the DRWT

also collapses as the voltage is unstable.

Just as in the first scenario, because of the higher speed deviation of the SRWT, the

curve envelope of the DRWT active power is located above the corresponding envelope

for the SRWT. As declared in 5.4.1, the more the active power generation during the

transient, the less the voltage stability margin would be considered for the generating

unit. The induction generator in the DRWT accelerates to very high values. The reason

of the acceleration is the lack of electromagnetic torque compared to the mechanical

torque after the voltage instability occurrence.

142

a) Terminal voltage

b) Reactive power

c) Active power

d) Generator speed

Fig. 5.21. Response of DRWT and SRWT DFIG-based wind turbine to disturbance when DFIG can’t match the reactive power demand

Main : Graphs

118.0 120.0 122.0 124.0 126.0 128.0

0.00

0.20

0.40

0.60

0.80

1.00

1.20

y

DR Load Bus Volt SR Load Bus Volt

Main : Graphs

120.0 121.0 122.0 123.0 124.0 125.0 126.0 127.0 128.0

0.250

0.300

0.350

0.400

0.450

0.500

0.550

y

DR_Reactive_Power SR_Reactive_Power

Main : Graphs

119.0 120.0 121.0 122.0 123.0 124.0 125.0 126.0 127.0 128.0

0.950

1.000

1.050

1.100

1.150

1.200

y

DR_Active_Power SR Active_Power

Main : Graphs

120.0 122.0 124.0 126.0 128.0

0.930 0.940 0.950 0.960 0.970 0.980 0.990 1.000 1.010 1.020

y

DR Speed SR Speed

143

Fig. 5.21 confirms that, when DFIG-wind farms are operating at their nominal active

power ratings, it is likely that they hit the reactive power limit of the grid connection

buses during the transient and their attempt toward retaining the voltage may fail. In this

situation, the SRWT is more supportive for the grid voltage and as a result it can be

concluded that the DRWT DFIG-based wind farms introduce a detrimental impact on

the transient voltage stability margin of the power system when the DFIG hits the

reactive power limits.

5.7 Conclusion

In this chapter, a method is introduced to investigate the impact of the DRWT on the

margin of the short-term voltage stability of the power grid. The critical rotor speed

method was discussed as the current popular approach for evaluating the transient

voltage stability margin of the IG-based generating units. Although this method is quite

accurate for the prediction of the transient angle stability margin of the induction

generators, it does not cover all influential factors in the transient voltage stability

margin. Therefore, this approach was rejected for comparison purposes here. A method

was proposed as the tool of comparison, the criterion of which is the maximum transient

active and reactive power generated by the wind turbines during the transient period. By

this method, as the generating unit generated higher transient apparent power, the

transient voltage stability margin of the local grid was predicted to be less. The validity

of the introduced method was tested for FSIG, DFIG in nominal condition and DFIG

when its delivered capacitive reactive power hit the practical limit. For the all three

scenarios, the disturbance which leads to transient voltage instability was chosen to be a

big load switching and the wind turbines were operating in the maximum power point

tracking (MPPT) mode. For the FSIG technology, the SRWT presented a more positive

impact on the transient voltage stability in comparison to the DRWT. The reason was

that the SRWT delivered less maximum transient apparent power than that did the

DRWT. To identify the reactive limits of the wind farm, it was suggested to calculate

the capability characteristic curve of the grid connection bus of the wind farm instead of

the algebraic summation of the ideal capability curves of the individual wind turbines

recommended by the literature. For the DFIG scenario with normal operation, it was

seen that there was no noteworthy difference between the transient voltage support

performances of both wind turbines. Conversely, the SRWT keeps its benefit over the

144

DRWT for the same reason as in the FSIG scenario, if the delivered reactive power by

the wind farm reaches the reactive power limits of the grid connection bus of the wind

farm during the transient period.

145

Impact of DRWT-based Wind farms on SSR Risk Chapter 6

6.1 Introduction

The risk of sub-synchronous resonance (SSR) is likely in wind farms connected to

power systems through a series of compensated transmission lines. This chapter shows

that the risk of TI-SSR and TA-SSR in the DRWT is higher than in the SRWT. The

main reason is higher number of torsional frequencies in the DRWT in comparison to

the SRWT due to the higher number of rotating elements. So, the likelihood of

coincidence of the complementary of grid natural frequency and one of the torsional

frequencies is higher in the DRWT system. This problem may be considered as a

serious drawback for this new technology. To stabilize the generating units against SSR,

most of the approaches have introduced mitigation techniques to damp the oscillations

originating from the SSR phenomenon. However, there is still fatigue for components

because of the mechanical tensions during the process of oscillation damping which

reduces the lifetime of the wind turbines. This study believes that it is possible to

optimize the mechanical parameters of the DRWT to delimit the torsional frequencies

from the high-risk area (22Hz < f < 42Hz). Consequently, the risk of interaction

between the torsional frequencies and the complementary of the grid natural frequency

is reduced remarkably. This method is able to assist in lowering the risk of the TI-SSR

and TA-SSR of the DRWT. Genetic algorithm (GA) has been employed as the

optimization tool.

This chapter is organised as follows: the state space model of the DRWT and the SRWT

is formed in section (6.2) as the essential material for calculation of the torsional

frequencies. In section (6.3) the proposed method is discussed in detail; in section (6.4)

the GA is configured and the associated fitness function is developed. To make the 146

method more realistic and practical, two constraints are introduced in section (6.5). The

performance of the proposed method is evaluated in section (6.6). The dynamic

response of the DRWT is tested through numerical simulation after updating the

parameters to investigate any negative impact on the damping factor of the system.

6.2 Modeling the DRWT and the SRWT for Torsional Studies

The torsional frequencies can be calculated through the modal analysis method. The

dynamic equations describing the behavior of the mechanical components should be

written in state-space form, as shown in (6.1):

= 𝐴. 𝑥 (6.1)

where the state variables (x) are rotor angle displacements ∆θi and speed deviations ∆ωi.

A sample is given by (6.2) for a general N-mass system:

(6.2)

The elements of ‘A’ matrix are defined by the values of the momentum of inertia of the

individual masses (J), the stiffness of the connections (K), the damping coefficients (D)

and the electromagnetic torque components (∆Te).

The torsional frequencies of a shaft system are the imaginary components of the

eigenvalues of matrix ‘A’. So the first step for obtaining the torsional frequencies of the

DRWT and SRWT is to calculate their corresponding matrix ‘A’. The wind turbine

mechanical system of the DRWT and the SRWT is already presented in Fig. 3.3. The

DRWT consists of the main turbine, the auxiliary turbine, a three-shaft bevel gear, the

interface shafts and a generator located in the tower. Each set of blades is presented by a

two-mass model in this study. The bevel gear in presented by 3 masses and the

generator with a one-mass model. The SRWT is composed of a turbine, which is also

given by a two-mass model, and a spur gear that introduces two masses to the torsional

studies. From section 3.3, there is the possibility of forming the state-space equation of

the DRWT and SRWT. The drive train of the SRWT and the DRWT suitable for

torsional studies is given in Fig. 6.1 and Fig. 6.2, respectively. From these figures, it is

feasible to form the state space matrices of the SRWT and the DRWT and calculate the

corresponding torsional frequencies. The linear first-order differential equations of the

SRWT and the DRWT are given in section 6.2.1 and section 6.2.2.

TNNx ]...,,,,...,,[ 2121 θθθωωω=

147

Fig. 6.1 Structure of five-mass shaft system of SRWT

Fig. 6.2. Structure of eight-mass shaft system of DRWT

6.2.1 State-space model of the SRWT

The following equations representing the dynamic response of each component in the

drive train of the SRWT to the small disturbances is given below.

The dynamic response of the flexible portion of the blade in the SRWT is given in (6.3):

𝑓𝑙𝑥 =1𝐽𝑓𝑙𝑥

𝑇𝑚 − 𝐾𝐵𝑙𝑑𝜃𝑓𝑙𝑥 − 𝜃𝑟𝑔𝑑 − 𝐷𝐵𝑙𝑑(𝜔𝑓𝑙𝑥 − 𝜔𝑟𝑔𝑑)

𝑓𝑙𝑥 = 𝜔𝑓𝑙𝑥

(6.3)

where

ωflx : is the rotational speed of the flexible part of the blades of the SRWT.

θflx : is the angle of the flexible part of the blades of the SRWT.

ωrg : is the rotational speed of the rigid part of the blades of the SRWT.

θrg : is the angle of the flexible part of the blades of the SRWT.

Jflx : is the momentum inertia of the flexible part of the blades of the SRWT.

148

KBld : is the stiffness of the blades of the SRWT.

DBld : is the damping factor of the blades of the SRWT.

Tm : is the aerodynamic torque captured by the blades.

The dynamic response of the rigid portion of the blade plus the hub of the turbine is

given by (6.4).

𝑟𝑔𝑑 = 1𝐽𝑟𝑔

−𝐾𝐵𝑙𝑑𝜃𝑟𝑔 − 𝜃𝑓𝑙𝑥 − 𝐷𝐵𝑙𝑑𝜔𝑟𝑔 − 𝜔𝑓𝑙𝑥 − 𝐾𝑀𝑡_𝑆𝑝𝜃𝑟𝑔 − 𝜃𝑆𝑝_𝑙𝑤 −

𝐷𝑀𝑡_𝑆𝑝(𝜔𝑟𝑔 − 𝜔𝑆𝑝_𝑙𝑤)

𝑟𝑔 = 𝜔𝑟𝑔 (6.4)

where

ωSp_lw : is the rotational speed of the low-speed gear of the spur gear.

θSp_lw : is the angle of the low-speed gear of the spur gear.

Jrg : is the momentum inertia of the rigid part of the blades plus the inertia of the hub.

KMt_Sp: is the stiffness of the interface shaft between spur gear and hub.

DMt_Sp : is the mutual damping factor of the interface shaft between spur gear and hub.

The dynamic response of the two-shaft spur gear is given by (6.5) for the low-speed

gear and by (6.6) for the high-speed gear.

(6.5)

(6.6)

].[.

]..[.

)( D) ( K1

______

______

rgSpr_lwMt_SprrgSpr_lwMt_Spr_

_

HghSprHghSprlwSprlwSprHghlwlwSpr

HghSprHghSprlwSprlwSprHghlwlwSpr

lwSprlwSpr

RRDRRRKR

J

ωω

θθ

ωωθθω

+−

+−

−−−−=

lwSprlwSpr __ ωθ =

]..[.

]..[.

).().(1

______

______

_____

_

HghSprHghSprlwSprlwSprHghlwHghSpr

HghSprHghSprlwSprlwSprHghlwHghSpr

GenHghSprGenSprGenHghSprGnSprHghSpr

HghSpr

RRDRRRKR

DKJ

ωω

θθ

ωωθθω

+−

+−

−−−−=

149

where

ωSpr_Hgh : is the rotational speed of the high-speed gear of the spur gear.

θSpr_Hgh : is the angle of the high-speed gear of the spur gear.

JSpr_lw : is the momentum inertia of the low-speed gear.

JSpr_Hgh : is the momentum inertia of the high-speed gear.

Klw_Hgh : is the stiffness between the low-speed and high-speed gears.

Dlw_Hgh : is the mutual damping factor between the low-speed and high-speed gears.

RSpr_lw : is the radius of the low-speed gear

RSpr_Hgh : is the radius of the high-speed gear

The dynamic response of the generator is given by (6.7).

𝐺𝑒𝑛 = 1𝐽𝐺𝑒𝑛

(6.7)

where

ωGen : is the rotational speed of the high-speed gear of the generator.

θGen : is the angle of the generator.

JGen : is the momentum inertia of the generator.

KSpr_Gen : is the shaft stiffness between the low-speed gear and generator.

DSpr_Gen : is the mutual damping factor between the low-speed gear and generator.

Te : is the electromagnetic torque, which for torsional studies should be

linearized as indicated in (6.8):

(6.8)

The state variable vector for the SRWT is given below by (6.9):

hihi ωθ =

)()( ____ HghSprGenGenSprHghSprGenGenSpre DKT ωωθθ −−−−

gengen ωθ =

)( 0000 qrdsdsqrdrqsqsdrme iiiiiiiiLT ∆−∆−∆+∆=

150

(6.9)

The overall ‘A’ matrix of the SRWT is presented by

(6.10)

Where,

, ,

,

,

, , ,

,

,

, ,

,

,

, ,

, ,

, , ,

[ ]GenHghSprlwSprrgflxGenHghSprlwSprrgflx θθθθθωωωωω ____

=

10000000000100000000001000000000010000000000100000

0000000000

00000000000000

510595554

4104948454443

383736343332

282726232221

17161211

AAAAAAAAAA

AAAAAAAAAAAA

AAAA

ASRWT

flx

Bld

JDA −=11

flx

Bld

JDA =12

flx

Bld

JKA −=16

flx

Bld

JKA =17

rg

Bld

JDA =21

rg

SprMtBld

JDD

A _22

+−=

rg

SpMt

JD

A _23 =

rg

Bld

JKA =26

rg

SpMtBld

JKK

A _27

+−=

rg

SpMt

JK

A _28 =

lwSpr

SprMt

JD

A_

_32 =

lwSpr

lwSprHghlwSprMt

JRDD

A_

2___

33

.+−=

lwSpr

lwSprHghSprHghlw

JRRD

A_

___34

..−=

lwSpr

SprMt

JK

A_

_36 =

lwSpr

lwSprHghlwSprMt

JRKK

A_

2___

37

.+−=

lwSpr

lwSprHghSprHghlw

JRRK

A_

___38

..−=

lwSpr

lwSprHghSprHghlw

JRRD

A_

___43

..−=

lwSpr

HghSprHghlwGenMt

JRDD

A_

2___

44

.+−=

HghSpr

GenSpr

JD

A_

_45 =

lwSpr

lwSprHghSprHghlw

JRRK

A_

___48

..−=

lwSpr

HghSprHghlwGenMt

JRKK

A_

2___

49

.+−=

HghSpr

GenSpr

JK

A_

_410 =

Gen

HghSpr

JD

A _54 =Gen

HghSpr

JD

A _55 −=Gen

HghSpr

JK

A _59 =Gen

HghSpr

JK

A _510 −=

151

6.2.2 State Space Model of the DRWT

The following equations, given below, represent the dynamic response of each

component in the drive train of the DRWT to the small disturbances.

The dynamic response of the flexible portion of the blade in the auxiliary turbine of the

DRWT is given in (6.11):

𝑓𝑙𝑥𝐴𝑥 =1

𝐽𝑓𝑙𝑥_𝐴𝑥[𝑇𝑚_𝐴𝑥 − 𝐾𝐵𝑙𝐴𝑥𝜃𝑓𝑙𝑥_𝐴𝑥 − 𝜃𝑟𝑔_𝐴𝑥 − 𝐷𝐵𝑙_𝐴𝑥(𝜔𝑓𝑙𝑥_𝐴𝑥 − 𝜔𝑟𝑔_𝐴𝑥)]

𝑓𝑙𝑥_𝐴𝑥 = 𝜔𝑓𝑙𝑥_𝐴𝑥

(6.11)

where

ωflx_Ax : rotational speed of the flexible part of the auxiliary blades of the DRWT.

θflx_Ax : angle of the flexible part of the auxiliary blades of the DRWT.

ωrg_Ax : rotational speed of the rigid part of the auxiliary blades of the DRWT.

θrg_Ax : angle of the flexible part of the auxiliary blades of the DRWT.

Jflx_Ax : momentum of the flexible part of the auxiliary blades of the DRWT.

KBld_Ax: stiffness of the auxiliary blades of the DRWT.

DBld_Ax: damping factor of the auxiliary blades of the DRWT.

Tm_Ax : aerodynamic torque captured by the auxiliary blades of the DRWT.

The dynamic response of the rigid portion of the auxiliary blade plus the hub of the

turbine is given by (6.12).

𝑟𝑔_𝐴𝑥 =1

𝐽𝑟𝑔_𝐴𝑥[−𝐾𝐵𝑙_𝐴𝑥. 𝜃𝑟𝑔_𝐴𝑥 − 𝜃𝑓𝑙𝑥_𝐴𝑥 − 𝐷𝐵𝑙_𝐴𝑥. 𝜔𝑟𝑔_𝐴𝑥 − 𝜔𝑓𝑙𝑥_𝐴𝑥

−𝐾𝐴𝑥_𝑏𝑣𝑙. 𝜃𝑟𝑔_𝐴𝑥 − 𝜃𝐵𝑣_𝑀𝑖𝑑 − 𝐷𝐴𝑥_𝑏𝑣𝑙. (𝜔𝑟𝑔_𝐴𝑥 − 𝜔𝐵𝑣_𝑀𝑖𝑑)]

(6.12)

𝑟𝑔_𝐴𝑥 = 𝜔𝑟𝑔_𝐴𝑥

ωBv_Mid : speed of the mid-speed gear of the bevel gear connected to auxiliary turbine.

θBv_Mid : angle of the mid-speed gear of the bevel gear connected to auxiliary turbine.

Jrg_Ax : inertia of the rigid part of the blades plus the hub of the auxiliary turbine.

152

KAxt_Bvl : stiffness of the shaft between the bevel gear and hub in auxiliary turbine.

DAxt_Bvl : damping of the shaft between the bevel gear and hub in auxiliary turbine.

The dynamic response of the three-shaft bevel gear is given by (6.13) for the medium-

speed gear, by (6.14) for low-speed gear, and by (6.15) for the high-speed disk.

𝐵𝑣_𝑀𝑖𝑑 = 𝜔𝐵𝑣_𝑀𝑖𝑑

(6.13)

𝐵𝑣_𝑙𝑤 = 𝜔𝐵𝑣_𝑙𝑤

(6.14)

𝐵𝑣_𝐻𝑔ℎ = 𝜔𝐵𝑣_𝐻𝑔ℎ

(6.15)

where

ωBv_Hgh : is the rotational speed of the high-speed gear of the bevel gear.

θBv_Hgh : is the angle of the high-speed gear of the bevel gear.

ωBv_lw : is the rotational speed of the low-speed gear of the bevel gear.

θBv_lw : is the angle of the low-speed gear of the bevel gear.

ωGen : is the rotational speed of the generator.

]..[.]..[.

)()(1

______

______

_______

_

HghBvHghBvMidBvMiBvHghMdMiBv

HghBvHghBvMidBvMiBvHghMdMiBv

AxrgMidBvBvlAxtAxrgMidBvBvlAxtMidBv

MidBv

RRDRRRKR

DKJ

ωω

θθ

ωωθθω

+−

+−

−−−−=

]]..[.

]..[.

)()([1

_____

_____

_______

_

HghBvlHghBvlwBvllwBvHghlwlwBv

HghBvlHghBvlwBvllwBvHghlwlwBv

MtrglwBvlMtBvlMtrglwBvlMtBvllwBv

lwBv

RRDRRRKR

DKJ

−

−

+−

+−

−−−−=

ωω

θθ

ωωθθω

]]..[.

]..[.

]..[.

]..[.

)()([1

_____

_____

_____

_____

_____

lwBvlwBvHghBvHghBvhilwHghBv

lwBvlwBvHghBvHghBvhilwHghBv

HghBvHghBvMidBvMiBvhimidHghBv

HghBvHghBvMidBvMiBvhimidHghBv

GenHghBvGenBvlGenHghBvGenBvlHghBv

HghBv

RRDRRRKRRRDRRRKR

DKJ

ωω

θθ

ωω

θθ

ωωθθω

+−

+−

+−

+−

−−−−=

−

−

−

−

−

153

θGen : is the angle of the generator.

ωrg_Mt : is the rotational speed of the rigid part of the main turbine.

θrg_Mt : is the angle of the rigid part of the main turbine.

JBv_lw : is the momentum inertia of the low-speed disk of the bevel gear.

JBv_Hgh : is the momentum inertia of the high-speed disk of the bevel gear.

JBv_Mid : is the momentum inertia of the medium-speed disk of the bevel gear.

Klw_Hgh : the stiffness between the low-speed and high-speed disks.

Dlw_Hgh : the mutual damping factor between the low-speed and high-speed disks.

KMd_Hgh : the stiffness between the medium-speed and high-speed disks.

DMd_Hgh : the mutual damping factor between the medium-speed and high-speed disks.

RBv_lw : radius of the low-speed disk of the bevel gear

RBv_Hgh : radius of the high-speed disk of the bevel gear

RBv_Mid : radius of the medium-speed disk of the bevel gear

The dynamic response of the rigid portion of the blade plus the hub of the turbine is

given by (6.16).

𝑟𝑔_𝑀𝑡 =1

𝐽𝑟𝑔_𝑀𝑡[−𝐾𝐵𝑙𝑀𝑡𝜃𝑟𝑔_𝑀𝑡 − 𝜃𝑓𝑙𝑥_𝑀𝑡 − 𝐷𝐵𝑙_𝑀𝑡𝜔𝑟𝑔_𝑀𝑡 − 𝜔𝑓𝑙𝑥_𝑀𝑡

−𝐾𝐵𝑣_𝑀𝑡𝜃𝑟𝑔_𝑀𝑡 − 𝜃𝐵𝑣_𝑙𝑤 − 𝐷𝐵𝑣_𝑀𝑡𝜔𝑟𝑔_𝑀𝑡 − 𝜔𝐵𝑣_𝑙𝑤]

(6.16)

𝑟𝑔_𝑀𝑡 = 𝜔𝑟𝑔_𝑀𝑡

where

ωflx_Mt: is the rotational speed of the flexible part of the main turbine.

θflx_Mt : is the angle of the flexible part of the main turbine.

Jrg_Mt : momentum inertia of the rigid part of the main blades plus the inertia of the hub.

KBv_Mt : stiffness of the interface shaft between bevel gear and the main hub.

DBv_Mt : damping factor of the interface shaft between bevel gear and the main hub.

The dynamic response of the flexible portion of the main blade in the DRWT is given in

(6.17):

154

𝑓𝑙𝑥𝑀𝑡 =1

𝐽𝑓𝑙𝑥_𝑀𝑡[𝑇𝑀𝑡 − 𝐾𝐵𝑙_𝑀𝑡 . 𝜃𝑓𝑙𝑥_𝑀𝑡 − 𝜃𝑟𝑔_𝑀𝑡

−𝐷𝐵𝑙_𝑀𝑡. 𝜔𝑓𝑙𝑥_𝑀𝑡 − 𝜔𝑟𝑔_𝑀𝑡]

𝑓𝑙𝑥_𝑀𝑡 = 𝜔𝑓𝑙𝑥_𝑀𝑡

(6.17)

where

TMt : aerodynamic torque captured by main turbine in the DRWT

Jflx_Mt: momentum inertia of the flexible portion of the main blade.

The dynamic response of the flexible portion of the main blade in the DRWT is given in

(6.18):

𝐺𝑒𝑛 =1𝐽𝐺𝑒𝑛

[𝑇𝑒 − 𝐾𝐵𝑣𝐺𝑛(𝜃𝐺𝑒𝑛 − 𝜃𝐵𝑣𝐻𝑔ℎ) − 𝐷𝐵𝑣𝐺𝑛(𝜔𝐺𝑒𝑛 − 𝜔𝐵𝑣𝐻𝑔ℎ)] (6.18)

where

Te : electromagnetic torque

KBv_Gn: stiffness of the shaft between the high-speed disk and the generator.

DBv_Gn: mutual damping of the shaft between the high-speed disk and the generator.

The state variable vector for SRWT is given below by (6.19):

(6.19)

The overall ‘A’ matrix of the DRWT is presented by (6.20):

(6.20)

[ AxrgAxflxGenMtflxMtrgHghBvlwBvMidBvAxrgAxflx _________ θθωωωωωωωω

]GenMtflxMtrgHghBvlwBvMidBv θθθθθθ _____

=DRWTA

155

where,

,

,

,

, , ,

,

, , ,

, , ,

10000000000000000100000000000000001000000000000000010000000000000000100000000000000001000000000000000010000000000000000100000000

00000000000000000000000000000000000

00000000000000000000000000000000000000000000000000

16_813_88885

15_714_77776

15_614_6676664

16_513_512_511_558555453

14_413_412_4464544

13_311_310_3353332

11_210_229232221

10_1191211

AAAAAAAAAAAAA

AAAAAAAAAAAAAA

AAAAAAAAAAAA

AAAA

Axflx

AxBl

JD

A_

_11 −=

Axflx

AxBl

JD

A_

_12 =

Axflx

AxBl

JK

A_

_19 −=

Axflx

AxBl

JK

A_

_110 =

Axrg

AxBl

JD

A_

_21 −=

Axrg

bvlAxAxBl

JDD

A_

__22

+−=

Axrg

bvlAx

JD

A_

_23 =

Axrg

AxBl

JK

A_

_29 =

Axrg

bvlAxAxBl

JKK

A_

__10_2

+−=

Axrg

bvlAx

JK

A_

_11_2 =

MidBv

bvlAx

JD

A_

_32 =

MidBv

HghMdMiBvbvlAx

JDRD

A_

_2

__33

.+−=

MidBv

HghMdHghBvMiBv

JDRR

A_

___35

..−=

MidBv

bvlAx

JK

A_

_10_3 =

MidBv

HghMdMiBvbvlAx

JKRK

A_

_2

__11_3

.+−=

MidBv

HghMdHghBvMiBv

JKRR

A_

___13_3

..−=

156

, , ,

, ,

,

,

, ,

,

,

, ,

,

,

,

, ,

,

,

,

lwBv

HghlwlwBvMtBvl

JDRD

A_

_2

__44

.+−=

lwBv

HghlwHghBvlwBv

JDRR

A_

___45

..−=

lwBv

MtBvl

JD

A_

_46 =

lwBv

HghlwlwBvMtbvl

JKRK

A_

_2

__12_4

.+−=

lwBv

HghlwHghBvlwBv

JKRR

A_

___13_4

..−=

lwBv

MtBvl

JK

A_

_14_4 =

HghBv

himidHghBvMiBv

JDRR

A_

___53

..−=

HghBv

hilwHghBvlwBv

JDRR

A_

___54

..−=

HghBv

hilwHghBvhimidHghBvGenBvl

JDRDRD

A_

_2

__2

__55

.. ++−=

HghBv

GenBvl

JD

A_

_8_5 =

lwBv

himidHghBvMidBv

JKRR

A_

___11_5

..−=

lwBv

hilwHghBvlwBv

JKRR

A_

___12_5

..−=

HghBv

hilwHghBvhimidHghBvGenBvl

JKRKRK

A_

_2

__2

__13_5

.. ++−=

HghBv

GenBvl

JK

A_

_16_5 =

Mtrg

MtBv

JD

A_

_4_6 =

Mtrg

MtBvMtBl

JDD

A_

__6_6

+−=

Mtrg

MtBl

JD

A_

_7_6 =

Mtrg

MtBvMtBl

JKK

A_

__14_6

+−=

Mtrg

MtBv

JK

A_

_12_6 =

Mtrg

MtBl

JK

A_

_15_6 =

Mtflx

MtBl

JD

A_

_6_7 =

Mtflx

MtBl

JD

A_

_7_7 −=

Mtflx

MtBl

JK

A_

_15_7 −=

Mtflx

MtBl

JK

A_

_15_7 −=

Gen

MtBv

JD

A _5_8 =

Gen

MtBv

JD

A _8_8 −=

Gen

MtBv

JK

A _13_8 =

Gen

MtBv

JK

A _16_8 −=

157

6.3 Proposed Method to Reduce the Risk of SSR

According to section 2.4.4, all the approaches proposed to damp the oscillations due to

the SSR are mainly focused on the electrical side of this phenomenon. However, it is

possible to reduce the risk of SSR at the design phase of the mechanical components of

the wind turbine. In this chapter, a method is introduced to reduce the risk of the SSR.

The principle of this approach is to avoid any coincidence of the complementary of the

natural frequency of the grid and the torsional modes of the shaft system. In section

2.3.3.5, the types of the SSR have been discussed in more detail. It can be seen that only

the IGE is not influenced by the number of torsional frequencies. On the other hand, the

risk of other types of SSR, such as TI, TA and TICU is influenced by the number of the

torsional frequencies. The higher the number of torsional frequencies, the higher the risk

of SSR occurrence can be imagined.

Theoretically, the level of the series compensation varies from zero to 100% of the line

impedance. Consequently, according to (2.1), the complementary of the natural

frequency can sit between 0 to 50Hz. Nonetheless, based on [11], the normal range of

the compensation level is between 20% and 70% of the series reactance of the

transmission line. This matter is also followed by the ABB Company, which is one of

the pioneering and internationally well-recognized companies in the electrical industry

[110]. In the document produced by ABB, the degree of compensation is between 25%

and 50%, which is more conservative. It can be concluded that, in practice, the range

that the natural frequency complementary may be placed in, is less than the full range.

Through using (2.1), the practical range is calculated to be almost between 22 to 42Hz.

Since the number of the rotating components is fixed, it is not possible to decrease the

number of the torsional frequencies for the DRWT. It is feasible to diminish the risk of

SSR through delimiting the mechanical torsional frequencies from the range with

possessing a high risk of the placement of the complementary of the network natural

frequency. The torsional mechanical frequencies of the shaft system are mostly

dominated by the momentum inertia of the elements and the stiffness of the interface

shafts. The target is to delimit the torsional frequencies from the range of 22 to 42Hz,

through optimizing the values of momentum inertia and stiffness. In this way, there is

no chance for the natural mechanical and electrical frequencies to approach each other

when the level of series compensation is between 20% and 70%. It is really hard to

reach the target through adjusting the mechanical parameters manually. This trial and

158

error approach is time-consuming. Genetic algorithm (GA) can be a good solution to

optimize the parameters quickly, while including some constraints. There are some

limitations on selection of the mechanical parameters, such as the total weight of the

components at the top of the tower or other mechanical characteristics that should not be

violated.

6.4 Genetic Algorithm Settings

GA is an all-inclusive search method based on the similarity with biology, in which a

group of solutions evolves through a process of natural selection and the survival of the

fittest. Each control variable in GA is called a gene. When all control variables are put

together as a vector, they form a chromosome. Each chromosome shows a solution for

the problem. The population is comprised of several chromosomes depending on its

size. Applying the evolutionary rules to the current population generation, results in

new population, i.e. next generation. To reach this stage, three operators are needed:

elite selection, crossover and mutation.

GA needs essential settings to function appropriately. These settings include: problem

formulation, initialization, chromosome fitness, selection pater, mutation factor,

crossover factor, population size and stopping criteria. Each of the above-mentioned

factors will be explained below.

6.4.1 Problem Formulation

The target is to define two boundaries with a lower boundary of 22 Hz and an upper

boundary of 42Hz. To reach this stage, the torsional frequencies which fall in the high

risk range are divided into two groups. The first group contains the frequencies which

are closer to lower boundary (22Hz) and the second group includes the torsional

frequencies which are closer to upper boundary (42 Hz). The former and latter

frequency groups are, in this study, called bottom frequency group (BFG) and upper

frequency group (UFG) for more convenience. The GA function is in charge of pushing

down the BFG to be less than 22Hz and pushing up the UFG to be placed above the

42Hz. To achieve this target, two frequencies are selected as targets. One of them is

12Hz, and the distance between the frequencies of BFG and this target should be

minimised. The frequency of the other target is 52Hz and the distance between the

frequencies in UFG with this frequency should be minimised.

To do this, an array must be formed containing the natural frequencies of the wind 159

turbines. The natural frequencies which are not placed in the range between 22Hz and

42Hz should be left out and the rest should be grouped into UFG and BFG. The general

form of the objective function described in (6.21) must be minimized by GA.

(6.21)

where

fBFC : are the frequencies in BFG

N : is the number of the frequencies in BFG.

fUFC : are the frequencies in UFG

M : is the number of the frequencies in UFG.

In design procedure, designers always face practical boundaries for selecting

mechanical parameters. To make this research more realistic and practical, it is better to

include these boundaries into our study. Since momentum inertia and stiffness of

mechanical equipment dominate the value of the natural frequencies, these parameters

are considered to be the control variables. Limitations are applied at the lower and upper

limits of the mechanical parameters, such as stiffness and momentum inertia of the

rotating equipment in the simulation. Stiffness of the components is not only function of

its material. It is also influenced by dimension of the component. For example shaft

stiffness keys to Young's modulus of the material, shaft cross section and length. The

stiffness of a shaft is calculated as follows: K=10-3*A*E/Ls. where A, Ls and E are

respectively the cross section, length and Young module of the shaft. To it is possible to

have a specific stiffness through choosing appropriate dimension and material.

On the other hand, the free variables of the optimization are limited to fairly narrow

boundaries by the designed GA (upper and bottom limits are respectively 0.5 and 1.5

times of the original standard value of each free variable).

6.4.2 Initialization

The population initialization is an important step towards getting the nearest value to

the optimal solution. In this chapter, the random initialization method is adopted.

Generally, this method has been used mostly for population initialization. The

simulation results must be applicable for the manufacturers. So for generating the

chromosomes, narrow boundaries are selected for the control variables to be chosen.

2_

21_

2_

21_ )52(...)52()12(...)12( MUFGUFGNBFGBFGTorsional ffffFit −++−+−++−=

160

The data related to the first chromosome is mentioned in Table 6.1. The lower and

upper boundaries for each control variable are, respectively, 0.5 and 1.5 times of its

associated variable in the first chromosome.

6.4.3 Chromosome Fitness

Fitness is a quantity which shows the ability of chromosome for optimizing the

problem. This is a satisfactory criterion for comparing the chromosomes. After

calculating the fitness, the constraints must be checked for determining any violation.

Although some of the chromosomes meet the fitness limit, they may violate the

constraints which are defined during the programming. To solve this problem, penalty

terms must be added to the obtained fitness’s for increasing the amount of it in case of

any violation. We have defined the fitness for ith chromosome as:

𝐹𝐹𝑖𝑛𝑎𝑙 = 𝐹𝑖𝑡𝑇𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 + 𝑃𝑒𝑛𝑧

𝑁𝐶

1

(6.22)

where,

FitTorsional : is the fitness function defined by (6.21).

Penz : is the penalty term due to the violation by the zth constraints.

NC : is the number of constraints.

The penalty is calculated by (6.23), which must be applied to the fitness evaluation

function in (6.21).

Table 6.1 Initial Chromosome

Generator Moment of Inertia 297kg.m2

Rotor Inertia Momentum Main=1.8*104kg.m2 Aux.=6*103

kg.m2

Flexible portion of Blades Inertia Momentum Main=1.12*105kg.m2 Aux.=4*103kg.m2

Inertia Momentum of wheel 75 kg.m2

Inertia Momentum of pinion 1 650kg.m2

Inertia Momentum of pinion 2 1127*103 kg.m2

Effective Main Blade Stiffness 3*106 N.m/rad Effective Auxiliary Blade

Stiffness 5.4*106 N.m/rad

Shaft Stiffness between main blade and bevel gear 4*106 N.m/rad

161

Shaft Stiffness between auxiliary blade and bevel gear 5.2*106 N.m.s/rad

Shaft Stiffness between bevel gear and Generator 1.7*106 N.m.s/rad

Effective stiffness between wheel and pinion 1 3.8*106 kg.m2


𝑃𝑒𝑛𝑧 = 𝑝𝑒 . |𝑘𝑖 − 𝑘𝑙𝑧| 𝑖𝑓 𝑘𝑖 < 𝑘𝑙𝑧

𝑝𝑒 . |𝑘𝑖 − 𝑘ℎ𝑧| 𝑖𝑓 𝑘𝑖 > 𝑘ℎ𝑧 (6.23)

Where pe is the penalty coefficient, klz and khz are the lower and upper limits of the zth

constraint. As long as ki is between klz and khz then pe is equal to zero. In addition, it is

advisable not to let the penalty coefficients increase too much [111].

6.4.4 Selection

Improvement of the average fitness of the population is achieved through selection of

individuals as parents from the completed population. The selection is performed in

such a way that chromosomes having higher fitness are more likely to be selected as

parents. The selection function used in this study is Stochastic Uniform.

6.4.5 Crossover

After the selection, the GA chooses a pair of selected chromosomes to create two new

chromosomes. We employed MATLAB function @crossovertwopoint as a tool.

6.4.6 Mutation

Mutation is applied to expand the population diversity. Gaussian mutation has been

chosen as the mutation function. This function adds a random number taken from a

Gaussian distribution with mean of zero to each element of the parent vector. The

standard deviation of this distribution is determined by the values of the scale and

shrink parameters. The scale parameter assigns the standard deviation at the first

generation. The shrink parameter controls how the standard deviation shrinks as

generations go by. The scale and shrink are set to 0.5 and 0.75, respectively.

162

6.4.7 Crossover Fraction

The crossover fraction specifies the fraction of each population, other than elite

children, that are made up of crossover children. Crossover fractions of 1 means all

children are crossover children, while a crossover fraction of zero means all children are

mutation children. The crossover fraction is set to 0.6.

6.4.8 Population Size

The population size should be large enough to create sufficient diversity to cover, the

possible solution space. Obviously, a more complex problem needs a higher population

size, since there are higher numbers of feasible mixtures of variables. In the problem

under study, the population size is set to 128.

6.4.9 Stopping Criteria

Stopping criteria determine what causes the algorithm to terminate. In this chapter the

employed GA is set to have fixed fitness limit. The algorithm stops if the best fitness

value is less than or equal to 15Hz.

6.5 The Constraints of the Proposed Method

In this study, two main constraints are included in the optimization procedure. For the

first constraint, whenever the fitness function by (6.21) meets the stopping criteria, the

high-risk range should be checked and none of the torsional frequencies should exist in

this range. This matter will be explained in more detail in section (6.5.1). The second

constraint is to avoid any combining of two torsional frequencies. In other word none of

the torsional frequencies should be identical after optimization. The details are given in

section 6.5.2.

6.5.1 No Torsional Frequency in High Risk Range

As the GA tries to force the torsional frequencies already existing in the high-risk area,

to leave this area, the other torsional frequencies that are already outside of this range

must be supervised so as to not enter the high-risk area after optimization. Therefore, if

any chromosome meets the fitness function, the high-risk area must be checked to be

devoid of any torsional frequency. The mathematical version of the above-mentioned

statement is given in:

163

𝑓1 < 22 𝑜𝑟 𝑓1 > 42

𝑓2 < 22 𝑜𝑟 𝑓2 > 42

⋮

𝑓𝑘 < 22 𝑜𝑟 𝑓𝑘 > 42

(6.24)

where ‘k’ is the total number of the torsional frequencies.

6.5.2 Torsional Frequency Combination

The GA is only in charge to empty the high-risk area of the torsional frequencies.

Consequently, some of the torsional frequencies outside the high-risk range may be

combined together or in other word approach each other. If two eigenvalues approach

each other, there might be an oscillatory response with poor damping for the whole

system. This is due to the superposition effect of the torsional frequencies which adjoin

each other. This issue is discussed mathematically below. Assume that a system has two

pairs of eigenvalues as indicated by (6.25):

𝐹𝑢𝑛(𝑠) = (6.25)

It can be expanded into four first-order partial fractions as presented by (6.26), or two

second order fractions as shown by (6.27):

𝐹𝑢𝑛(𝑠) = (6.26)

𝐹𝑢𝑛(𝑠) = (6.27)

For more convenience, the format by (6.27) is chosen for the rest of the calculations in

this section. The time domain form of (6.27) is achievable by taking the inverse Laplace

shown by (6.28):

𝐹𝑢𝑛(𝑡) = 𝐶1. 𝑒−𝛼1𝑡 cos(𝜔1𝑡 + 𝜃1) + 𝐶2. 𝑒−𝛼2𝑡 cos(𝜔2𝑡 + 𝜃2)

+𝐶3. 𝑒−𝛼1𝑡 cos(𝜔1𝑡 + 𝜃1) + 𝐶4. 𝑒−𝛼1𝑡 cos(𝜔1𝑡 + 𝜃1) (6.28)

).2)(.2( 2222

22111

201

ωωξωωξ +++++

ssssasa

)()()()( 22

*

2211

*

11 dddd jsB

jsB

jsA

jsA

ωσωσωσωσ +++

−++

+++

−+

))(())(( 22

22

0121

21

01

ωαωα +++

+++

+s

BsBs

AsA

164

If the two eigenvalues approach each other, for any reason, then (6.25) can be rewritten,

as presented by (6.29):

(6.29)

The time domain form of the above equation is achievable by taking its inverse Laplace:

(6.30)

From (6.30) it can be seen that the magnitude of the components is a function of the

time and the exponential components. On the one hand, the magnitude tends to increase

due to the time factor (t) and, on the other hand, the exponential factor with negative

power (e-αt) has a weakening impact on the magnitude. Through comparing the time

domain response of (6.28) and (6.30), it is obvious that the response of the system

becomes more oscillatory and the damping factor is poorer when two eigenvalues

approach each other. This matter reduces the equivalent damping torque of the DRWT

and consequently its dynamic stability margin should be decreased. The lifetime of the

mechanical components is also reduced because of the high rate of fatigue. As an

example, the system responses by (6.28) and (6.30) are overlayed in Fig. 6.3 to show

the impact of the superposition effect of the eigenvalues. The superposition effect is also

studied in [112] and it confirms that when a system with two non-similar real poles

turns to a system with two identical real poles, then the damping characteristic of a

system shifts from overdamped to critically damped. It signifies that the damping factor

becomes poorer when two poles approach each other.

Fig. 6.3. Response of the system with identical eigenvalues versus normal system

22111

201

).2( ωωξ +++

ssasa

)cos(..)sin(...)( 11211111 θωθω αα +++= −− teKtteKttA tt

10 12 14 16 18 20 22-6

-4

-2

0

2

4

6

A(t)

time (sec)

Normal SystemSuper-positioned System

165

So to decrease the rate of fatigue, whenever the fitness function is met by a

chromosome, the value of the eigenvalues should be checked by the GA to avoid any

undesirable matching of the two torsional frequencies. The chromosomes that meet the

fitness function but make the eigenvalues approach each other must be ignored. Based

on the explained constraint in this section, the distance between any pair of the torsional

frequencies should not be less than 1.5Hz.


The objective of this study is to investigate the ability of the GA for selecting the

mechanical parameters of the DRWT in such a way as to reduce the risk of SSR in this

system. In order to initially reach this stage, the state space model of the DRWT and

SRWT has already been formed in section 6.2.1 and section 6.2.2, respectively. The

parameters of the mechanical systems are listed in Table 6.1. Based on the latter, the

torsional frequencies of the SRWT and the DRWT have been calculated and are given

in Table 6.2.

Table 6.2. Torsional frequencies of the DRWT and the SRWT with data from initial chromosome

f0 f1 f2 f3 f4 f5 f6 f7

DRWT 2.71 10.24 23.41 27.91 37.56 45.73 61.84 150.34

SRWT 2.57 8.84 33.75 41.37 58.67 - - -

According to the results given in Table 6.2, three torsional frequencies of the DRWT are

placed in the high-risk range. Thus, based on the fitness function by (6.21), the

frequencies of 23.41Hz and 27.91Hz are considered as BFG and should be attracted to

the target frequency of 12 Hz and the frequency of 37.56Hz is UFG and should be

pushed to the target frequency of 52Hz.

The progression of the GA, as it searched in the solution space for the best fitness,

generation by generation, is shown in Fig. 6.4. In this figure, for each population, its

average fitness and the best fitness in that population are presented. Finally, the best

fitness met the stopping criteria at the 88th generation. The validity of this chromosome

is approved by the constraints. Based on the data processing procedure provided by GA,

which is described in section 6.4, and the constraints in section 6.5, the designed

flowchart is presented in Fig. 6.5.

Freq.(Hz) System

166

Fig. 6.4. Average and best fitness of each population

167

Fig. 6.5. The optimization flowchart for pushing away the torsional frequencies of the high-risk range

The best chromosome delivered by the GA is given by Table 6.3.

Start

Initialization Randomly assign initial values to momentum inertias and stiffness's

Fitness For each chromosome: - Calculate the objective function - Check the constraints for violation, impose penalties when necessary - Calculate the fitness

GA Procedure: - Selection - Elitism - Crossover - Mutation

Natural Frequency For each chromosome calculate natural Frequencies of the dual rotor.

Fitness Limit

Met

End Yes

No

FitnessCalculate Fitness for each chromosome

168

Table 6.3. The best chromosomes selected by GA

Generator Moment of Inertia 339kg.m2

Rotor Inertia Momentum Main=2.7*104kg.m2 Aux.=5.4*103 kg.m2

Flexible portion of Blades Inertia Momentum Main=1.28*105kg.m2 Aux.=3.24*103kg.m2

Inertia Momentum of wheel 105 kg.m2

Inertia Momentum of pinion 1 865kg.m2

Inertia Momentum of pinion 2 0.84*103 kg.m2

Effective Main Blade Stiffness 2.38*106 N.m/rad

Effective Auxiliary Blade Stiffness 6.49*106 N.m/rad

Shaft Stiffness between main blade and bevel gear 3.71*106 N.m/rad

Shaft Stiffness between auxiliary blade and bevel gear 2.78*106 N.m.s/rad

Shaft Stiffness between bevel gear and Generator 1.67*106 N.m.s/rad



The parameters of the state space model of the DRWT in (6.20) were updated based on

the data yielded by the best chromosome. The torsional frequencies of the updated ‘A’

matrix in (6.20) were recalculated and the results are given in Table 6.4.

Table 6.4. Torsional frequencies of the DRWT with data from the best chromosome

f0 f1 f2 f3 f4 f5 f6 f7

DRWT 4.52 8.67 13.41 18.26 43.56 54.39 67.65 162.73

From Table 6.4, it can be seen that the high-risk area of the frequency spectrum (22Hz <

f <42Hz) is empty of the torsional frequencies while there is enough margin between

the torsional frequencies. The frequency difference between each individual torsional

frequency of the DRWT is more than 1.5Hz. It means that the DRWT based on the new

parameters introduces less risk of SSR in comparison to the DRWT with original

System Freq.(Hz)

169

parameters without lessening the overall damping factor of this type of wind turbine. To

check the damping factor of the dynamic response of the DRWT with the updated

parameters, a load switching was performed to force the generator to oscillate. The

technology for this test is FSIG. The test set up is illustrated in Fig. 6.6.

Fig. 6.6 Test set up to check the damping of the system

The load switching was carried out for speed and torque controlling modes. In Fig. 6.7a,

the oscillations of the generator speed are presented. The total mechanical torque

provided by the main and auxiliary turbine is depicted in Fig. 6.7b. It shows that, in

speed control mode, the fluctuations of the quantities of the DRWT are well damped

after updating the parameters.

a) Generator speed

b) Mechanical torque coming from turbines

Fig. 6.7. Dynamic response of DRWT after updating the parameters in speed control mode

Generator : Graphs

280 290 300 310 320 330 340 350 360 370

0.9900 0.9950 1.0000 1.0050 1.0100 1.0150 1.0200 1.0250 1.0300 1.0350

y (p

u)

Gen Speed DRWT

Untitled 1 : Graphs

280 290 300 310 320 330 340 350 360 370

-0.700

-0.650

-0.600

-0.550

-0.500

-0.450

-0.400

y

DRWT Tm

170

The damping characteristic of the DRWT with parameters from the best chromosome

was tested in torque control mode. The results are given in Fig. 6.8. It can be seen that

DRWT also presents accepted damping in torque control mode.

a) Generator speed

b) Mechanical torque coming from turbines

Fig. 6.8. Dynamic response of DRWT after updating the parameters in torque control mode

It is worth mentioning that there is no aging factor for the shaft stiffness and momentum

inertia of the components. In other words, stiffness and the inertia do not change over

the lifetime of the mechanical system. So it can be predicted that the location of the

torsional frequencies stay at the same spot over the life time of the wind turbine.

6.7 Conclusion

The impact of the DRWT on the angle, voltage and frequency transient stabilities was

discussed in previous chapters. In this study, the investigation of the impact of the

DRWT on the risk of SSR was also included as a part of the power system transient.

This chapter showed that the risk of TI and TA was higher in the DRWT, due to the

higher number of shaft system torsional frequencies. Some additional torsional

frequencies were imposed by extra rotating components of the auxiliary turbine. To

overcome this drawback of the DRWT, it was proposed to delimit the torsional

frequencies from the high-risk range (22Hz < f < 42Hz). In this way, even though the

number of torsional frequencies remained the same, it was possible to avoid the

interaction between the torsional frequencies and the complementary of the grid natural

Generator : Graphs

280 290 300 310 320 330 340 350 360 370

0.9950

1.0000

1.0050

1.0100

1.0150

1.0200

1.0250

1.0300

1.0350

y (p

u)

Gen Speed DRWT

Untitled 1 : Graphs

290 300 310 320 330 340 350 360 370

-0.900 -0.850 -0.800 -0.750 -0.700 -0.650 -0.600 -0.550 -0.500

y

DRWT_Tm

171

frequency. Thus, the risk of the TI-SSR and TA-SSR was reduced considerably.

To achieve the target, a genetic algorithm (GA) was designed to optimize the

mechanical parameters of the DRWT to make the high-risk frequency range empty of

the torsional frequencies. By updating the mechanical parameters of the DRWT

according to the best chromosome given by the GA, it was feasible to reduce the risk of

adjoining of the mechanical and electrical natural frequencies of the system. Two

constraints were checked for each chromosome that met the fitness function. Firstly, the

distances between each pair of torsional frequencies should be sufficient. Secondly, the

high-risk area should be checked to be empty of torsional frequencies.

The damping characteristic of the DRWT with the parameters from the best

chromosome was assessed through a numerical simulation result and it was seen that the

updating of the mechanical parameters of the DRWT did not introduce any negative

impact on the dynamic response of the wind turbine. The upper and lower boundaries

for the parameters were selected to be narrow to avoid any unreasonable results.

172

Conclusion and Future Works Chapter 7

The dual-rotor wind turbine (DRWT) was introduced into the market to enhance the

aerodynamic efficiency over that provided by the single-rotor wind turbine (SRWT).

The relative efficiency of the DRWT is almost 9% more than the SRWT under the same

conditions. Therefore, according to this reported advantage, the DRWT has a good

potential for commercialization in the near future. However, before making any

decision, the DRWT should be competitive in terms of the conventional SRWT in a

number of different aspects. This thesis is dedicated to creating the initial basis on

which to analyse the impact of the DRWT on the transient stability margin of the local

power system. This research mainly investigated four aspects of transient stability,

consisting of: transient angle stability; transient frequency stability; transient voltage

stability; and, sub-synchronous resonance.

Firstly, the effect of the DRWT-based wind farm on the transient angle stability

margin was evaluated against that of the SRWT. To facilitate this assessment, the

dynamic model of each component of the mechanical drive of the DRWT was

developed, and these were linked to each other through the multi-objective method.

Eigenvalue analysis was used to compare the natural damping of the DRWT with that

of the SRWT. It was seen that, after adding the state equations of the auxiliary turbine

to the state space model of the main turbine and the generator combination (the control

system state equations were excluded), the real part of some of the eigenvalues shifted

towards the left, which suggests that the natural damping of the DRWT is higher than

that of the SRWT. To check the respective margins of transient angle stability of the

DRWT and SRWT, the most common approach – called the ‘critical rotor speed’ – was

adopted. The main criterion of this method is the minimum distance between the stable

and unstable operating points during the fault and post-fault period; the less the

distance, the less the margin is considered for the induction generator. So, the

calculation of the acceleration rate was considered to be a key factor in identifying

whether the DRWT or SRWT is more susceptible to transient angle instability. It was

found that under the same conditions, because of the extra momentum inertia by the

auxiliary turbine in the DRWT, the acceleration rate of the SRWT was greater than for

the DRWT. This suggests that the transient angle stability margin of the DRWT is

higher than that of the SRWT when the pitch angle is regulated by a PI controller.

Application of the droop loops is able to improve the damping factor of the wind 173

turbines. From the investigation above, it can be concluded that the integration of the

droop loop into the pitch control system introduces a higher damping degree to the

DRWT than it does to the SRWT, due to the functioning of the droop loop of the

auxiliary turbine.

Secondly, the influence of the DRWT-based wind farm on the transient frequency

stability margin was assessed and compared with that of the SRWT. Based on the wind

speed, the wind turbines operated under one of the de-loading modes to give the

frequency control ability of the wind farms. The investigation was accomplished for

three de-loading modes, including: pitch control de-loading mode; sub-optimal curve

de-loading mode; and a combination of these modes, termed, in this study, the

combination mode. In previous reports, the KE has been identified as the only factor

dominating the inertial response of the wind turbines. However, sensitivity analysis

demonstrated that, during the transient deviation of the generator speed, the

aerodynamic energy was dominated by the operating point excursion along the

aerodynamic curve. Thus, the view was formed that the route of the excursion is a

function of the variations of both the blade angle and the generator rotational speed.

Therefore, from this study, the conclusion has been drawn that, during the rotational

speed transient deviation, the energy released by the wind-generating unit as the inertial

response is determined by both the kinetic energy (KE) and the transient variation of the

aerodynamic energy coming from the blades. Additionally, depending on the de-loading

mode, the transient variations of the aerodynamic energy have a weakening or boosting

effect on the inertial response capability of the unit. In pitch control de-loading mode,

the excursion of the operating point happens in the under-speed area in the case of any

drop in grid frequency; conversely, in sub-optimal de-loading mode, the excursion

occurs in the over-speed area during the frequency fall. So, based on the aerodynamic

characteristic curve of the wind turbine, it has been concluded that the transient

variation of the aerodynamic energy has a decreasing impact on the inertial response

strength, while this impact is increasing for the sub-optimal de-loading mode. To

investigate the transient frequency control capability of the DRWT against that of the

SRWT, their characteristics were compared with respect to both the relative KE-

releasing potential and relative degree of impact of the transient aerodynamic change. It

was found that the SRWT releases slightly more KE than does the DRWT in the same

situation; thus, it can be assumed that they are almost the same in this aspect. So, the

174

transient variation of the aerodynamic energy is recognized as the integral factor in

determining which type of wind turbine is superior in performance. The transient

frequency support capability of the DRWT was seen to be better than that of the SRWT

in pitch control de-loading mode, and consequently, the assertion is made that this is

due to the higher speed drop of the SRWT rotational speed in the under-speed, which

leads to a higher weakening effect of the operating point excursion of the SRWT in

comparison to that of the DRWT. For the sub-optimal de-loading mode, the SRWT was

found to be more successful in arresting the network transient frequency nadir in

comparison to the DRWT. It is thus claimed that, because of the higher speed drop of

the SRWT rotational speed in the over-speed region, the operating point excursion of

SRWT results in a higher boosting impact than that of the DRWT. In the combination

mode, it has been confirmed that, through suitable selection of the droop factor for the

pitch control system of the auxiliary turbine, the DRWT is more effective in limiting the

transient frequency deviation. The main turbine in the DRWT and the turbine in the

SRWT have the same droop system.

Thirdly, the effect of the DRWT-based wind farm on the transient voltage stability

margin of the local network was evaluated against that of the SRWT. The current most-

common method for assessing the transient voltage stability margin of the IG-based

generating units has been critiqued above. It was concluded that, although this method

is quite accurate for the prediction of the transient angle stability margin of the

induction generators, it does not cover all influential factors in the transient voltage

stability margin. A method was proposed as the tool of comparison and its validity was

checked for three energy conversion scenarios, including FSIG, DFIG in nominal

condition, and DFIG when its supplied reactive power hits the capacitive limit. The

criterion of this method is the peak of the transient apparent power generated by the

wind turbines during the transient period; the higher the apparent power delivered by

the generating unit, the less the transient voltage stability margin can be predicted for

the local grid. For all three scenarios, the wind turbine was working in the maximum

power point tracking (MPPT) mode and the disturbance was chosen to be a big load

switching which lead to generator deceleration. For the first scenario with FSIG as the

energy conversion system, it was uncovered that the peak of the transient apparent

power delivered by DRWT, as a response to the load switching, is greater than that of

the SRWT in MPPT mode. Consequently, for this scenario, the transient voltage

175

stability margin of the SRWT was greater than that of the DRWT, which confirms the

expectation of the above-mentioned method. For the second scenario, the energy

conversion technology was DFIG. In reality, due to the reactive power losses of the

electrical components, like cables, transformers and transmission lines as interfaces

between the wind turbines and grid, the capacitive area of the capability curve is

reduced dramatically when the wind farm is operating close to the nominal ratings. In

other words, the whole capability curve of the wind farm is shifted towards the lagging

area. Therefore, to make the study more practical, it was suggested to use the capability

characteristic curve of the grid connection bus of the wind farm to identify the reactive

power limits, rather than to use the algebraic summation of the capability curves of the

individual wind turbines recommended by the literature. It was revealed that, as long as

the wind farm matches the required reactive power, the transient voltage support

performances of both wind turbines are quite similar. However, in cases where the

reactive power provided by the wind farm reaches the reactive power capacity limit of

the grid connection bus during the transient period, the SRWT keeps its advantage over

the DRWT for the same reason as in the first scenario. This indicates that the wind

turbines with the greater potential for transient apparent power generation present less

voltage stability margin when the DFIG is saturated. This phenomenon agrees with the

rationality of the proposed method for assessing the transient voltage stability margin of

induction-based wind farms.

Lastly, the risk of sub-synchronous resonance (SSR) in DRWT-based wind farms was

assessed against that of SRWT-based wind farms. Based on the state matrices, the

numbers of torsional frequencies were eight and five for, respectively, the DRWT and

SRWT. This confirmed the expectation of the higher risk of the torsional interaction

SSR (TI-SSR) and torsional amplification SSR (TA-SSR) for the DRWT, in

comparison to the SRWT. In this thesis, the main target was the reduction of the SSR

risk at the design stage of the prime mover of the DRWT. To reduce the risk facing the

DRWT, it was proposed to optimize the mechanical parameters of the DRWT in order

to delimit the torsional frequencies from the high-risk frequency range. In this thesis,

the high-risk range was defined as the range that the complementary of the natural

frequency of the grid may be placed in due to the series capacitor variation. It was

recognized as being between 22Hz and 42Hz. Therefore, by delimiting the torsional

frequencies, it was possible to decrease the risk of any matching between the torsional

176

frequencies and the complementary of the natural frequency of the grid. Then, a GA

was designed as the optimization tool. In this way, it was possible to reduce the risk of

the coincidence of the torsional frequencies and the grid natural frequency. To do so,

12Hz and 52Hz were defined as the two target frequencies outside of the high-risk range

and the GA was in charge to make the torsional frequencies, which were already inside

the high-risk area, as close as possible to the target frequencies. Two constraints were

applied to every chromosome that met the stopping criteria. If two torsional frequencies

approach each other, then the damping characteristics of the dynamic response is

degraded. So, one of the constraints was allocated to check that the margin between the

adjacent torsional frequencies was not closer than a specified value. Another constraint

was put in place to check whether there was any torsional frequency in the high-risk

area. The designed GA was run and it was successful in removing the torsional

frequencies from the high-risk area without damaging the damping characteristic of the

dynamic response of the DRWT. Thus, it can be concluded that the risk of sub-

synchronous resonance is decreased dramatically through the proper selection of

mechanical parameters, thereby allowing the DRWT to achieve superior commercial

competitiveness.

7.1 Future Works

There are some subjects related to the DRWT that are worth investigation in the future.

These subjects will be discussed in more detail below.

7.1.1 Gearless DRWT

The transient characteristic of the DRWT, normally called the T-gear DRWT, was

studied in comparison with the SRWT. The term T-gear DRWT is used is because of

the bevel configuration that is quite similar to the letter ‘T’. There is also another brand

of the DRWT in which the interface gearbox is omitted [113]. It can be called ‘gearless

DRWT’ here. In the gearless DRWT, the efficiency of the mechanical drive is higher

and less down time can be imagined for it. This is due to the omission of the gearbox.

Its configuration is given in Fig. 7.1.

Among all the components of the gearless DRWT, the structure of the generator is

changed significantly compared to the T-gear DRWT version. The configuration of the

generator employed in the DRWT is given in Fig. 7.2. The main and auxiliary turbines

are respectively connected to the stator and rotor of the generator without using any

177

gearbox for speed adoption. It can be predicted that the dynamic model of the gearless

DRWT should be considerably different from the T-gear DRWT. The main reasons are

the removal of the bevel gear and the stator of the generator, which rotates and delivers

the power to the grid through slip rings. While in the T-gear DRWT, the bevel serves as

an interface between the turbines and the generator to transfer the power and, on the

other hand, the stator of the generator is stationary and does not rotate.

Fig. 7.1. Gearless DRWT [114]

Fig. 7.2. Configuration of the employed generator in the DRWT [114]

It is worth studying the impact of the gearless DRWT on the transient stability margin

of the power system.

178

7.1.2 Impact of DRWT on the Transient Voltage Stability Margin in Sub-optimal

mode

In Chapter 5 a method was proposed to evaluate the impact of the DRWT on the short-

term voltage stability of the network versus that of the SRWT. The validity of the

method was checked for MPPT mode of operation. In this mode the steady state

operating point is located at the peak of the aerodynamic curve and the wind turbine is

de-loaded by the pitch control system. The interpretation of the test results have been

given in 5.6. However, sub-optimal mode is another de-loading mode that is preferred

over the pitch control de-loading mode for the high-speed winds. There is potential to

investigate the effect of the DRWT on the short-term voltage stability margin of the grid

when both the DRWT and SRWT are de-loaded by operating on sub-optimal curves.

Based on the theoretical arguments in Chapter 5 it can be predicted that the DRWT

presents better transient voltage support performance in sub-optimal mode. The reason

could be a higher boosting effect of the operating point excursion of the SRWT on its

energy generation strength during the transient period versus that of the DRWT in over-

speed area. So the SRWT is able to generate more power during the transient period in

sub-optimal mode which, according to the proposed method, leads to less short-term

voltage stability margin. Though, this claim should be approved by the simulation

results.

7.1.3 Impact of the DRWT on the Risk of the IGE-SSR

In Chapter 6 the impact the DRWT system on the risks of the torsional interaction (TI-

SSR) and torsional amplification (TA-SSR) have been assessed. Another subcategory of

the SSR is the induction generator effect (IGE-SSR) which didn’t study in this chapter.

Therefore, there is enough potential to analyse the risk of the IGE-SSR for the DRWT

systems. The IGE is explored more in details in section 2.3.3.5.1. It was seen that, as the

slip corresponding to the grid natural frequency become smaller, the risk of the IGE

goes up. This is due to the higher negative values of the equivalent resistance attained

by the rotor equivalent resistance. As the wind speed drops down or the compensation

level grows up the equivalent resistance become more negative. Regarding the influence

of the compensation level the both SRWT and DRWT should be the same. Regarding

the influence of the sudden drop of the wind speed, it can be predicted that the risk of

the IGE-SSR should be higher in the SRWT system. The ground for this prediction is

the higher rate of speed reduction of the SRWT. Thus, at the same amount of wind 179

speed drop, the minimum speed reached by the SRWT during the transient period is less

than that of the DRWT. Therefore, smaller slips can be reached by the SRWT during

the transient period in comparison to the DRWT. Consequently, the induction

generators in the SRTW system experience higher negative values of the rotor

equivalent resistance which means higher risk of IGE. However, the prediction in this

section should be confirmed through simulation results.

7.1.4 Using GA to Reduce the Risk of SSR in SRWT

Just like Chapter 6 it is possible to design a GA in order to optimize the mechanical

parameters of the SRWT to delimit the torsional frequencies of the SRWT from the

high-risk frequency range. So in this way it is also possible to lower the risk of SSR for

SRWT system.

7.1.5 Down Time Evaluation of DRWT in Comparison to SRWT

Although the DRWT introduces positive impacts on the aerodynamic efficiency and

transient stability margin of the wind turbines, however the amount of downtime for this

technology should be higher than that of the SRWT. This is due to the higher number of

components in the DRWT in comparison to the SRWT which possesses higher risk of

component failure in the DRWT system more than the SRWT. The matter can be a draw

back for the DRWT in the view of the owners of the wind farms. The higher the amount

of the downtime, the less the energy can be delivered by the wind farm which results in

less profit for the companies. Therefore, an investigation is required to explore and

compare the failure rate of the DRWT with that of the SRWT through employing the

Markova Chain method.

180

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192

APPENDIX A

Table A.1 Electrical Parameters of Induction Generator

Rated Voltage 0.69 kV

Rated Power 1.5 MVA

Moment of Inertia 3 sec

Frequency 50 Hz

Machine Damping 0.3 p.u.

Stator Resistance 0.066 p.u.

Stator Leakage Reactance 0.1 p.u.

Rotor Resistance 0.05 p.u.

Rotor Reactance 0.2p.u.

Unsaturated Magnetizing Reactance 2.5 p.u.

Table A.2

Parameters of the network system Transformer ratio 0.69/63 kV

Base MVA 2 MVA

Positive sequence reactance 0.3 p.u

Line length 100 km

Line resistance 0.1781E-3 Ω/m

Line inductive reactance 0.514E-3 Ω/m

Line capacitive reactance 27354.48 M Ω*m

Table A.3 Mechanical Parameters of Gear Box and Turbine

Spur base circle radius r1=0.1m r2=1m

Bevel base circle radius rav1=0.1m rav2=0.5m rav3=1m

SRWT blade diameter 51m

DRWT blade diameters Main=51m Aux.=26.4m

Rotor Damping Factor Single=Main=3 Aux.=1.5 p.u.

Blade Damping Factor Single=Main=2 Aux.=1.0 p.u.

Rotor Inertia Momentum Single=Main=0.28e5 Aux.=0.8e4 kg.m2

Blade Inertia Momentum Single=Main=0.1E5 Aux.=0.16E4kg.m2

Effective Blade Stiffness 0.21e6

All Shaft Stiffness’s 2.5e5

All Shaft Damping Factors 0.6p.u.

193

APPENDIX B

Fig.B.1. Voltage versus slip in induction machine

APPENDIX C

Fig.C.1. Bevel gear parameters

194

Fig.C.2. Pressure angle

Fig.C.3. Spur Gear parameters

195

APPENDIX D

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196

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