Aleksander M. Derdowski

Wind Turbine optimized for

VSC-DC transmission

Master Thesis, June 2008

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Preface

This is the master thesis for Aleksander M. Derdowski and is the final step in obtaining the Masters

of Science (M.Sc.) degree at the Technical University of Denmark. The thesis has been done under

supervision of Professor Zhao Xu, Tonny Wederberg Rasmussen, Arne Hejde Nielsen and industrial

supervisor German Cláudio Tarnowski.

The project was a great opportunity to expand my knowledge within the field of wind energy, which

will be most valuable in my future work in Norway. It is also an honour to being part of a sustainable

future, by contributing to the fast and innovative development of renewable energy resources.

Finally I want to thank all people around me, for support and attention, both socially and

academically.

Technical University of Denmark

June 2008

_____________________

Aleksander M. Derdowski

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Wind Turbine optimized for VSC-DC transmission

This report was prepared by: Aleksander M. Derdowski Supervisor(s): Germán Claudio Tarnowski Zhao Xu Tonny Wedeberg Rasmussen Arne Hejde Nielsen

Department of Electrical Engineering Centre for Electric Technology (CET) Technical University of Denmark Elektrovej Building 325 2800 Kgs. Lyngby Denmark www.elektro.dtu.dk/cet Tel: (+45) 45 25 35 00 Fax: (+45) 45 88 61 11 E-mail: cet@oersted.dtu.dk

Release date:

13.06.2008

Category:

1(Public)

Edition:

1st edition

Comments:

This report is part of the requirements to achieve the master of science MSc. Degree in Wind Energy - Electrical at the Technical University of Denmark. This report represents 35 ECTS points.

Rights:

© Aleksander M. Derdowski, 2008

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Abstract

This thesis investigates the possibility of increasing power production connecting a Rotor Current

Control (RCC) generator wind turbine to a Voltage Source Converter (VSC) HVDC transmission

system. The generator is optimized in order to produce maximum power at every wind speed until

nominal power is reached. The project contains an optimization algorithm to provide the reader with

the optimum power production possible when being able to vary stator frequency, stator voltage

and rotor resistance. The power production is presented and compared with AC connected wind

turbines. The comparison of power production is performed in both steady state and in dynamic

simulations. In steady state the VSC HVDC connected generator is compared with both a variable

resistance and constant resistance AC connected wind turbine, in the dynamic simulation the

comparison of the VSC HVDC generator is only done with the constant rotor resistance AC connected

generator, as this is believed and demonstrated to be sufficient for showing the benefits of the new

topology.

The project proposes a control system for optimized power production using wind speed to generate

reference signals for stator voltage and frequency. The new control system is submitted to a tripping

of the receiving end converter (grid-side VSC) and the reaction of the new topology is shown for this

type of faults.

The project shows that there is a possibility of increasing the power production of wind turbines by

allowing the generator to work with variable stator voltage and frequency and the operational range

of the generator is presented. In Further research losses in the PWM converters have to be

investigated to show if there is increased power submitted to the Grid. Other ideas for

improvements on this field of study are also presented for future research.

Key Words: Wind power, wind turbines, induction generator, variable speed, control, power

electronics, HVDC, power systems.

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Acknowledgements

This project has been carried out at the Centre for Electric Technology at the Technical University of

Denmark.

I would like to thank my supervisors German Cláudio Tarnowski, Tonny Wedeberg Rasmussen and

Zhao Xu, where especially the help and constructive discussions with my industrial supervisor

German Cláudio Tarnowski were of great inspiration and help for this project. Further I want to

thank two fellow master degree students János Hethey for interesting point of views especially on

simulation technical issues and Xavier Le Mestre for improving my time schedule by driving me to

school during this whole period.

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PREFACE 1

ABSTRACT 5

ACKNOWLEDGEMENTS 6

1. INTRODUCTION 9

1.1 Review of related research 9

1.2 Purpose and contribution 10

2. THE MODEL REPRESENTATION 11

2.1 Mechanical Representation 11

2.2 Electrical Representation 13

2.3 Transformer representation 17

2.4 Tuning of the Steady state model 18

3. STEADY STATE OPERATION, MECHANICAL POWER AND LOSSES 21

3.1 Torque curve 21

3.2 Currents and slip 22

3.3 Mechanical Power 23

3.4 Losses 24

3.5 Generated Power 26

3.6 Chapter Evaluation 26

4. STEADY STATE OPTIMIZATION AND OPERATIONAL POINTS 27

4.1 The Optimization Algorithm 27

4.2 Operational Points AC connected Constant Rotor resistance Wind Turbine 32

4.3 Operational Points Variable Rotor Resistance AC Connected Turbine 33

4.4 Operational Points VSC HVDC Connected Wind Turbine 34

4.4.1 Necessary operational range 36

4.5 Summary of Power Production and operational points 36

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5. STEADY STATE COMPARISON OF POWER PRODUCTION 38

5.1 Comparison Strategy 38

5.2 Power Production 39

5.2.1 Evaluation 42

6. DYNAMIC MODELING 43

6.1 Wind Turbine Model 43

6.1.1 Turbine model 43

6.1.2 Pitch Controller Model 43

6.1.3 Induction Machine Model 44

6.2 PWM Model 44

6.3 HVDC Model 45

6.4 Combined Controller Model 47

7. COMPARISON BETWEEN THE STEADY STATE AND THE DYNAMIC MODEL 51

8. DYNAMIC COMPARISON BETWEEN THE AC AND DC CONNECTED WIND

TURBINE 53

8.1 The Dynamic Comparison Strategy 54

8.1.1 Evaluation 57

9. DYNAMIC RESPONSE TO SYSTEM FAULT 58

10. DISCUSSION 63

11. CONCLUSION 64

11.2 Future Research 65

APPENDIX 68

A NOMENCLATURE 68

B CALCULATIONS 69

C MATLAB OPTIMIZATION SCRIPT 72

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1. Introduction

“Wind energy installations are faced with demands from the grid operators regarding frequency and

voltage variations. Increased demands for controllability have pushed the technologies both within

the wind turbine and in the power transmission system. To meet the desired controllability, either

synchronous generators with full-scale frequency converters or doubly-fed induction generators

with slip-recovery frequency converters are favoured in the wind turbines. Meanwhile, the AC

transmission systems are reinforced with static VAr compensation and other FACTS devices. Such

topologies will remain suited for connection of wind parks to relatively strong grids, low power and

short transmission distances.

However, for particular wind parks and transmission systems, it may be advantageous to use DC

rather than AC for power transmission from a wind park to a transmission grid. In this case, HVDC

transmission using voltage-source converters (VSC) is suitable.

The HVDC-VSC is just as controllable as the wind turbine itself, and it decouples the AC transmission

system from the wind park AC grid. Hence, all the uncontrolled frequency & voltage variations

normally present at the turbine terminals disappear. As a consequence, a less complex wind turbine

may be used, for example the rotor current controlled (RCC) induction generator. One operation

mode that may be envisaged is that the High Voltage Direct Current –Voltage Source Converter

(HVDC-VSC) controls voltage and frequency to match the mean wind conditions, while each RCC

turbine thus only requires a limited speed range. The research presented will contain the RCC

induction generator turbine control and performance with variable stator frequency and voltage,

necessary RCC wind turbine speed range when connected to VSC-HVDC, Energy production

comparison between DC and AC transmission solutions and the Wind turbine response to faults in

back bone transmission grid. “(Vestas)

1.1 Review of related research

Literature considering power optimization of RCC generator connected to VSC HVCD transmission

was during this project not possible to find. Although a paper considering DFIG generators

connected to VSC HVDC transmission is found in (Erlich, 2007), where a proposed control system for

coordinated control is presented. Their main approach is to reduce the size of the grid-side

converter of the DFIG while maintaining the same operational range, this is done by using the

average wind farm slip to generate reference signal for the stator frequency and stator voltage,

where the voltage is direct assignment of the frequency (no saturation in generator). The frequency

controller is implemented without using a feed back control system. It is purposed for future

research to keep the grid-side converter constant and increase the operational range of the DFIG.

Considering loss minimization more literature was possible to find, in (Ahmed G. Abo-Khalil, 2004) a

control algorithm for minimizing losses is proposed taking into account stator and rotor iron and

copper losses. The paper shows power increase when using variable stator frequency. The project

was validated using experimental results, other research considering loss minimization of variable

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speed connected induction machines can be found in (Robert Leidhold, 2002) and (G. O. Garcia,

1994). Considering fault ride trough of wind turbines research has been found in (Watson, 2005).

The paper is based on the modification of the Irish grid code in 2004 where new stricter

requirements for wind turbines where included. The paper presents the comparison of fault ride

through of Fixed Speed active stall regulated wind turbines based on induction generators

connected to VSC HVDC and HVAC. The study shows that the fault clearing time is significantly

expanded using the VSC HVDC connection. Paper regarding modelling of VSC HVDC systems can be

found in for instance (Florin Iov, 2006).

1.2 Purpose and contribution

The main purpose of this project is to show the possible power production gain of connecting a RCC

generator to VSC HVDC transmission system for both steady state and in dynamical simulation.

Secondly propose a control system to reach this power production and check the control systems

liability when the transmission is submitted to a fault. In order to present the power production a

representative model had to be made and is therefore modelling has a significant part in this

project. The details of the contribution to research are:

• Chapter 4 presents an optimization algorithm for finding best possible operational

points for the RCC generator considering rotor and stator losses in the generator.

The optimization allows variable rotor resistance stator voltage and stator

frequency. The results are presented for steady state operation and the new

generator operational span is compared with already known AC transmission

solutions.

• Based on the operational points in Chapter 4, Chapter 5 shows a steady state

comparison of the power production from the new VSC HVDC generator topology

with AC connected variable and constant rotor resistance generators. The steady

state results show positive effect on the power production of the new topology.

• Chapter 6 presents a dynamic control system for power optimization based on the

operational points found in Chapter 4. The control system uses the wind to

generate reference signal for stator frequency and stator voltage, consisting also a

feedback loop in the control system in order to enable rate of change limiting of

frequency and voltage.

• Chapter 8 shows the gain of using the new control system in dynamic simulations

and comparing the power production to the same wind turbine connected to an

AC transmission. The dynamic simulation results supports the steady state results

and shows that there is possible gain in power production when using the

presented control system

• Finally in Chapter 9 the proposed control system is exposed to a tripping of the

receiving end converter and its response is monitored and discussed. The control

system shows relatively good coping with the problem and a proposal for

improvement is presented.

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2. The Model Representation

This chapter will in detail describe how the mechanical and electrical model was represented and

implemented. The chosen models are presented and substantiated, also numeric modifications of

former models has been performed to provide better correlation with the RCC Vestas V80 wind

turbine which is used as a base reference throughout this project.

2.1 Mechanical Representation

This section will describe the mechanical equations and explain why this representation is chosen.

The mechanical power is the power in the wind that is possible to convert to the rotor. Therefore

the static characteristics of the wind can be described by the relation between the total power in the

wind and the mechanical power of the wind turbine (Ackermann, 2005). It can be shown that an

approximation of the total power in the wind can be expressed as:

(2.1)

Where ρ is the air density 1.225 kg/m3, A is the rotor radius and vw the wind speed. However it is not

possible to extract all the kinetic power in the wind since this would result in a stand-still of the wind

on the back side of wind turbine blades, therefore the maximum theoretical utilization of the power

in the wind is given by Betz law to be 59.3% ,(Betz, 1966). This utilization of the wind is denoted Cp

and will here be described as a function of tip-speed ratio, λ and blade pitch angle, θ. At modern

wind turbines this efficiency will lay somewhere in the area of 70 -80% of the maximum given by

Betz law (Ahmed, 2006) and can be expressed as (Bose, 1983):

!"#$ (2.2)

The choice of this representation of the Cp curve is based on its simplicity together with good base

for correlation with the power curve provided in the Vestas V80 product description (Vestas, 2008).

The constants a= 0.76 and b=2.64, are self defined and chosen iteratively to give a better correlation

to the actual data. This correlation will be presented later in the project.

θ is the blade pitch angle in deg. and the tip speed ratio is [2]:

%&

(2.3)

Where R is the rotor diameter, ωr is the rotor speed and vw is the wind speed. The mechanical power

of the wind turbine will therefore be expressed as (Ackermann, 2005):

' ( ) (2.4)

12

Figure 1: Cp curves at different wind speeds as a function of Rotational Speed

When allowing variable rotational speed the optimum efficiency point can be regulated to fit a given

wind-speed as seen in Figure 1 and the mechanical power in Figure 2 can be obtained.

Figure 2: Mechanical Power at different Wind Speeds as a function of Rotational Speed, with the

maximum mechanical power for each wind-speed marked with a cross

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2.2 Electrical Representation

This section will describe the electrical representation of the Rotor Current Control generator. A RCC

is a single fed, wound rotor generator with the possibility to vary the rotor resistance. A highly

simplified sketch of this generator might be seen in Figure 3. The requirements for the electrical

model in this project is that it is able to react to changes in rotor and stator frequency, change in

external rotor resistance and change in stator voltage. The chosen model can be found in (Paul C.

Krause, 2002) and is presented in Figure 4. Al values are from now on assumed to be in p.u. if not

otherwise specified.

Figure 3: Rotor Current Control Generator

Figure 4: Arbitrary reference-frame equivalent circuits for 3-phase symmetrical induction machine

14

Since in this project it is chosen to investigate the generator losses in the desired operational points,

the necessary equations have to be deduced. The voltages can be described as:

*+ &+,*+ - %+ψ+ (2.5)

+ &+,+ - %+ψ*+ (2.6)

* &,* - %+ %ψ* (2.7)

&, - %+ %ψ (2.8)

Where the flux linkage is described as:

ψ*+ %+.+,*+ - .',*+ - ,* (2.9)

ψ+ %+.+,+ - .',+ - , (2.10)

ψ* %+.,* - .',*+ - ,* (2.11)

ψ %+., - .',+ - , (2.12)

By working in the arbitrary reference frame the number of unknowns can be reduced to improve

calculation time. Therefore the following values are set:

*+ + (2.13)

+ (2.14)

This is verified since fulfilling:

+ /*+ - + (2.15)

Further the rotor voltages can be set to zero:

* (2.16)

This is true when defining the rotor voltages over the short circuited rotor side as shown in Figure 4

and additional rotor resistance (Radd) is included in Rr. Further by combining equation(2.16), (2.5),

(2.8), (2.10) and (2.12) idr can be written as:

, !0120344 56%+.',*+7 6%.',*+7 - 6%+.',*7 - 6%+.,*7 6%.',*7 %.,*8

(2.17)

15

Since the losses are highly dependent on the currents, the currents are of great interest to single

out. This is chosen to be done with a Newton Raphson algorithm because of its history of solving this

type of initialization problems. In order to use the Newton-Raphson method to solve three implicit

unknowns, three equations has to be set to zero (William H. Press, 1992). This is done by combining

all equations from equation(2.5) until equation(2.17) obtaining:

&+,*+ - %+.+ - .',+ - !0120344 :%+.'6.',*+%+ .',*+% - %+.',* - ,*%+.

,;<%<.= ,;<%<.<>(2.18)

%+.+ - .',*+ - &+,+ %+.',* (2.19)

%+ %.',+ - & - &,* - !01 :%+ %.' - .6.',*+%+ .',*+% -

%>.=,;<-,;<%>.<,;<%>.<,;<%<.=,;<%<.< (2.20)

The deductions can be found in the Apendix.

The steady state electrical torque of the machine can be written as:

?@ %+.' &> ) +A&+& - >%+.' .++.B - %+&.++ - >&+. (2.21)

Where:

> %+ %%+

(2.22)

.++ .+ - .' (2.23)

. . - .' (2.24)

Further the stator voltage, stator frequency and air gap flux is co-dependent as shown in (Matsch,

1977) and can be presented as:

C+ DDDE+ ) F ) G+ ) H!

(2.25)

C+IJ' DDDE+IJ' ) F ) G+ ) HIJ'!

(2.26)

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Where fs is the stator frequency, Ns is the number of turns in the stator a1 is the number of the

current paths, k is the pitch factor and ψa is the air-gap flux, the subscript “nom” indicates nominal

operation. To be able to create an optimization algorithm within a reasonable time-span a

simplification had to be made, saying that the air-gap flux is constant for all stator frequencies:

H HIJ' (2.27)

Assuming this is representative and that the factors k, N and a1 remain constant the voltage

dependence on frequency will be written as:

C+ E+E+IJ'

C+IJ' (2.28)

This is valid until Vs = Vs,nom where the air-gap flux has to be reduced in order to keep the voltage

constant (Ion Boldea, 2001) a typical voltage frequency characteristic for a 220 V 50 Hz induction

machine is shown in Figure 5

Figure 5: Voltage-Frequency Characteristic for a 220 V 50Hz induction machine (Ion Boldea, 2001)

The rotor copper losses are expressed as:

KI 6, - ,* 7& - & (2.29)

The stator copper losses are expressed as:

KI+ 6,+ - ,*+ 7&+ (2.30)

Therefore the generated power of the generator is:

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L@ ' KI KI+ (2.31)

2.3 Transformer representation

This section will describe the two winding transformer connecting the wind turbine to the sending

end converter. The reason for including the transformer is to investigate the impact of the frequency

variations on the losses in the transformer. This will show if the transformer losses should be taken

into account when considering the combined control system.

Figure 6: Equivalent diagram for transformation

An equivalent diagram of a two winding transformer is seen in Figure 6 and can be found in (Matsch,

1977). Here the assumption is that the current generated by the generator is fed in to the primary

side of the transformer hence:

,+ /,+ - ,*+ (2.32)

The losses in the transformer can then be calculated:

MJ++ ,+& - ,'I0' &' - GG+@K

,+I+@K &+@K (2.33)

Where the magnetization current can be approximated:

,' ,+N&+@K - %+.+@K

N&+@K - %+.+@K - /: !0OP

- !QRSOPT! (2.34)

And im,Rm is the real part of im further:

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,+@K U+V U'WWW (2.35)

Where is and im are presented as vectors. Typical values for a transformer was provided by my

industrial supervisor and presented in Table 1

Table 1: Suitable Transformer values

Lpri Rpri Lsec Rsec Lm Rm

0.0216 p.u. 0.00225 p.u. 0.0216 p.u. 0.00225 p.u. 175 p.u. 300 p.u.

2.4 Tuning of the Steady state model

This section will present how the model was tuned in order to become representative for its

purpose.

In order to get reliable results from the further calculations it is necessary for the model to

represent a real life turbine power production as good as possible. And since some information

about the power production of the Vestas V80 is provided in (Vestas, 2008), together with electrical

parameters from a 2 MW double fed induction generator found in (Petterson, 2005) the steady state

power production could be closely represented. The model tuning was performed on the fixed stator

frequency model (AC transmission connected), and tuned to fulfill the criteria known to the author

during this project.

Figure 7: Power curve of the Vestas V80 1800 kW Wind Turbine

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From the product description (Vestas, 2008) it is known that the nominal power of the wind turbine

is reached at wind speed equal to 15 m/s. It is also known to the author that turbine works with a

fixed optimum slip, and the slip range is from 1-10% slip (Vestas, 2008). The rotor speed of the

generator is based on a combined control between the pitch angle of the blades and the control of

the additional rotor resistance (Vestas, 2008). Although for this project the optimization considering

change in pitch angle will not be considered, consequently the optimum slip was not able to follow

for all wind speeds. Therefore the chosen constrain was to reach maximum power production for all

wind turbine topologies at a slip higher than -10%. This was done by adapting the self defined a) and

b) coefficients in equation(2.2) together with choosing a step size of the additional rotor resistance,

which would provide a satisfactorily power output ( PGenerated =PMechanical-Ploss). This was performed

manually and is therefore no guaranty that this provides the optimum correlation between Figure 7

and Figure 8 , although for the scope of this project this is considered by the author to give a good

enough base for the power comparison. The original power curve is shown in Figure 7 and the power

curve obtained using equation(2.31) for all wind speeds is shown in Figure 8. The data needed to

obtain the power curve in Figure 8 is presented in Table 2 and Table 3.

Figure 8: The calculated Power Curve for the AC connected variable rotor resistance wind turbine

Table 2

Mechanical Parameters:

Rotor Diameter 80 m

Cut in wind Speed 4 m/s

Cut out wind Speed 25 m/s

Nominal wind Speed 15 m/s

Table 3

Electrical Parameters:

Rotor Resistance 0.014 p.u. *

Rotor Inductance 0.07 p.u.

Stator Resistance 0.01 p.u.

Stator Inductance 0.18 p.u.

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Magnetizing Inductance 4.4 p.u.

Nominal Power 1800 kW

Additional Rotor Resistance 10 steps of 0.02 p.u. (self defined)

*In (Petterson, 2005) the value of the Rotor resistance was 0.009 p.u. This project however is using

0.014 p.u. as minimum value of the rotor resistance and happened because of a mathematical

overlook in the optimization algorithm. This change in rotor resistance has no effect on the main

idea of the project and it was therefore considered unnecessary to recalculate all the presented

results.

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3. Steady State operation, Mechanical Power and Losses

The following chapter will provide the base for the further optimization of the VSC HVDC RCC

generator. In order to find the optimum operational points the models response to changes in the

stator voltage, stator frequency and additional rotor resistance has to be examined. This will give a

thorough examination of how torque curve, generator losses, rotational speed and internal currents

will behave when controlling the generator. The results are all based on the steady state model

derived in Chapter 2 and will now be presented. The purpose of this chapter is to show the high un-

linearity of power and losses based on operational points. This is therefore meant to substantiate

the choice of optimization algorithm.

3.1 Torque curve

The torque curve is defined by the electrical parameters and provides the base for the operational

point. The torque curve expression can be found in equation(2.21) and will now be presented for

different stator voltages/frequencies and rotor resistance.

Figure 9: Shows the Electrical torque as a function of rotational speed, for different stator frequencies

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Figure 10: Shows the torque curve as a function of rotational speed for different rotor resistances

As can be seen in Figure 9 and Figure 10 the torque characteristic is highly dependent on both the

rotor resistance and stator frequency, and therefore might be used in the tracking of the optimal

operation point of the generator.

3.2 Currents and slip

This section will show how the currents and slip will behave in the generator when adjusting stator

frequency and rotor resistance. The wind is chosen to be the constant factor.

Figure 11: Slip as a function of stator frequency and rotor resistance at wind speed = 9 m/s

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Figure 12: Rotor Current as a Function of rotor resistance and stator frequency at wind speed = 9 m/s

As can be seen from the Figure 11 and Figure 12 the slip and currents are highly un-linear when

changing the rotor resistance and stator frequency. One thing of interest is the impact of the stator

frequency on the currents in the generator. It can be seen that the Frequency variations have a

much greater impact on the currents than the change in rotor resistance when using the step sizes

and operational span of this project.

3.3 Mechanical Power

Figure 13: Mechanical Power as a function of stator frequency and rotor resistance

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In Figure 13 one can see the mechanical power possible to track by adjusting the stator frequency

and rotor resistance. As can be seen the optimization based on the rotor resistance is modest for

this wind speed compared with the optimization based on the power tracking capability enabled by

varying the stator frequency. Although by looking on the raw data, the tracking done by the rotor

resistance can be shown to be in the order of 1-6 % of the mechanical power at the given stator

frequency.

Table 4: Showing 4 operational points, the impact of the stator frequency and rotor resistance on the

mechanical power

StatorFrequency/RotorResistance 0.014 p.u. 0.214 p.u.

0.72 p.u. Pm = 0.3629 p.u. Pm = 0.3695 p.u.

1.09 p.u. Pm = 0.3596 p.u. Pm = 0.3480 p.u.

What can be noticed in Table 4 is that the optimum mechanical power is not given explicitly by the

rotor resistance or the stator frequency. This means if the stator frequency [p.u.] is above the

rotational speed [p.u.] that provides the optimum mechanical power, a low resistance will provide a

better operational point for mechanical power. Again, if the stator frequency is lower than the

rotational speed that provides the optimum mechanical power a high resistance will provide a better

operational point for mechanical power. This might be easier to visualize by studying Figure 9

together with Figure 10 and imagine how this will provide operational points in Figure 2.

3.4 Losses

Figure 14: Generator power loss as a function of stator frequency and Rotor resistance at constant wind

speed = 9 m/s

25

The loss calculations are based on stator and rotor losses showed in equation(2.29) and

equation(2.30). And will vary dependant on stator frequency and rotor resistance. In Figure 14 it can

be seen how the losses are distributed for a constant wind speed when varying the additional rotor

resistance and stator frequency. As can be seen when imagining the stator frequency constant the

losses increase with increased rotor resistance. The reader can also keep in mind that the

operational point changes with the wind speed, consequently the losses will change with the wind

speed. In Figure 15 the losses in the transformer calculated according to equation(2.33) are

presented.

Figure 15: Transformer power loss as a function of stator frequency and Rotor resistance at constant

wind speed = 9 m/s

As can be seen when comparing the Figure 14 and Figure 15 it can be seen that the losses in the

transformer are modest compared to the losses in the generator.

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3.5 Generated Power

Figure 16: Generated Power as a function of Rotor Resistance and Stator Frequency

Finally the generated power is presented in Figure 16. This figure is a result of subtracting Figure 13

from Figure 14. The optimum operational point at this wind speed is found at Stator frequency =

0.97 p.u. and with the minimum value of rotor resistance.

3.6 Chapter Evaluation

This thorough deduction was made during the progress of this project when the operational points

of the generator jet were not known. The aim was to see and understand how all possible

operational conditions to provide an optimization algorithm that was sufficient to show the possible

gain of combined controlling stator frequency, stator voltage and additional rotor resistance. But

what will be shown later is that when being able to change the stator frequency in such a wide

interval, the tracking of mechanical power can be done by the stator frequency alone, therefore the

rotor resistance can be kept at a minimum and reduce the losses in the generator, and the losses in

the transformer are not of a size which effects the operational points considering the step sizes used

in the optimization algorithm of this project. This would maybe not be the case if this project had to

consider stricter limits on the stator frequency variation, or used a higher resolution of step-size for

rotor resistance and frequencies. This was not obvious for the author until the later part of the

project. Since the losses in the transformer can be shown by the author to not impact the power

production and the operational points of the generator in a substantial way, this is from now on

neglected in the project.

On the other hand, the rotor resistance has a positive effect on the AC connected model, where

there is no possibility of power tracking by using variable stator frequency. Therefore the rotor

resistance is used in the power optimization of the AC connected model.

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4. Steady state optimization and Operational Points

This chapter will describe the optimization algorithm and how it was implemented in MatLab. As

mentioned earlier the objective is to find the maximum power output for the generator, as shown in

Figure 17 within the operational boundaries shown in later presented in

Table 8. The variables in the algorithm are the rotational speed, the stator frequency, stator voltage

and rotor resistance.

Figure 17: Energy yield for the RCC generator model

When examining the steady state model presented in Chapter 2.2 and the mechanical equations in

Chapter 2.1 it is seen that we are operating with highly un-linear co-dependent equations. The main

thought for the algorithm was for it to be robust and thorough, and to be able to check a wide range

of operational conditions to show the picture of how the power output of the generators where

responding according to these changes. The algorithm then stores the necessary data for validation,

power production, losses, currents and optimum operational points for each wind speed. The

simplified diagram of the algorithm is presented in Figure 19. The further presentation might require

some knowledge about MatLab, but is tried to be presented in a more general way.

4.1 The Optimization Algorithm

The algorithm is based on “for loops “and is starting by setting wind speed equal to 4 m/s and

proceeds to the setting of the stator frequency equal to 0.5 p.u. (30 Hz) where the stator voltage is

calculated according to equation(2.28). The rotor resistance is then set to 0.014 p.u.. Based on this

information the rotor speed can be calculated by using equation(2.4) and equation(2.21) resulting in:

'% % ) ?@ (4.1)

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This is solved by plotting the two graphs on a common axis of rotational speed (step size of 1/33

p.u.) where the operational points in between the steps is approximated using a “spline”

interpolation. The intersection point is found using the Matlab function “fsolve”, which is a solver for

implicit unknowns similar to a Newton-Raphson algorithm. The benefit of using this function instead

of a self defined algorithm is its possibility of providing an “exitflag” which tells something about the

accuracy of the solution. This “exitflag” is therefore used as the first constrain to ensure that the

intersection of the two graphs is of desired mathematical accuracy (converging to a real root). Other

constrains at this point in the algorithm is that the stabile operational point marked with a circle in

Figure 18 and that the difference in between the two curves illustrated by the arrows in Figure 18 is

sufficient. And since the generator is not supposed to work at slips lower than -10% and the rotor

speed should be less than 1.5 p.u. these are included as constrains.

Figure 18: Mechanical Power and Electrical Power Torque

If all constrains are not fulfilled a bypass is initiated, setting rotational speed, currents, powers and

slip equal to zero before storing the data with label. Than the rotor resistance is increased and the

same calculations and verification is initialized. When all constrains are fulfilled the rotor speed is

used for current calculations and mechanical power calculations. The currents are calculated using a

Newton-Raphson method on equation(2.18) to equation(2.20) and finally finding the last current by

solving equation(2.17) with the currents already obtained. The choice of using the Newton-Raphson

method was based on its history of solving this type of mathematical problems. Since all operational

points here already where filtered earlier, the problem with the algorithm converging towards

unrealistic solutions is not an issue at this point in the algorithm. More information about Newton-

Raphson can be found in (William H. Press, 1992).The mechanical power is calculated according to

equation(2.4)

Based on the currents the losses in the stator and rotor are calculated according to equation(2.29)

and equation(2.30). The generated power is calculated according to equation(2.31). The data is

29

stored in a 3D matrix where the data is stored with its respective label of [ωs,Rr, Vw]. ωs is the stator

frequency, Rr is the rotor resistance and Vw is the wind speed.

When Rr reaches Rr,max , ωs is increased with one step, and the same procedure is initiated. When ωs

reaches ωs,max, Vw is increased by one step until Vw,max is reached. This ensures that all operational

conditions are checked, verified and stored.

30

Figure 19: Simplified diagram of the optimization algorithm

31

When all data is stored a second algorithm is initiated finding the maximum power production and

at which operational conditions these are obtained, these operational conditions together with their

power production are saved and used as the base for the rest of the project.

This same algorithm is used for all three turbine topologies, only difference is the amount of

variables chosen to be constant. The VSC HVDC topology is the one described above, the AC

connected variable resistance is keeping the stator frequency constant, and the AC connected

constant rotor resistance topology additionally keeps the rotor resistance constant. A summary is

seen in Table 5.

Table 5: Operational range and step size for the different generator topologies

Limits/topology VSC HVDC AC Variable

Resistance

AC Constant

Resistance

Rotor resistance 0.014 0.014 0.075

Resistance step size 1/50 p.u. 1/200 p.u. -

Stator frequency 0.5 -1.5 p.u. 1 p.u. 1 p.u.

Frequency step size 1/33 p.u. - -

The values chosen in Table 5 where changed and modified continuously during this project, the final

solution however ended with the presented values.

This because:

• The VSC HVDC topology prefers to find operational points by changing stator frequency and

keeping the rotor resistance minimum, in order to minimize losses. Therefore the resistance

could in general be set to constant, but for illustrative reasons it is changing to verify this

statement.

• Since the AC variable resistance is more dependent on the rotor resistance a higher

resolution on the step size was needed in order to provide smooth operational points.

• The AC connected topology with constant resistance, has rotor resistance = 0.075 since this

is the value the “Variable Resistance” topology uses at wind speed = 15 m/s and reaches the

power production of 1 p.u. (can be seen in Figure 21)

If the reader should be interested in reading the MatLab script appended the constrains are

presented more orderly in Table 6.

Table 6: List of Constrains

MatLab

Denotation

Mathematical Denotation

1st Constrain Choosing the correct intersection

2nd Constrain Solution has to be a real root

3rd Constrain Sufficient difference between electrical power torque and mechanical power

4th Constrain Slip > -10%

5th Constrain Rotational speed ≤ 1.5

32

4.2 Operational Points AC connected Constant Rotor resistance

Wind Turbine

For comparison of the three types of wind turbine generator topologies it was chosen to show how

the slip changes of the AC connected constant speed generator. This generator will have no

possibility of performing any kind of power tracking, and the rotational speed will only be affected

by the change in wind speed. The slip characteristic is shown in Figure 20

Figure 20: Slip of generator with Constant Rotor Resistance

33

4.3 Operational Points Variable Rotor Resistance AC Connected

Turbine

Since the Vestas V80 OptiSlip has a very advanced optimization system the author had to simplify

this in order to be able to provide a model within the time scope of this project. In reality the

OptiSlip is combined with an OptiTip system, which provides coordinated control of the slip of the

generator and the slip will further be used as the active power reference. This system has many

advantages like increased power quality, less flicker, and less harmonic disturbance together with

higher power output(Vestas, 2008). Since this project mainly focus on the investigation of power

production, the author has simplified the control system and will use only the rotor resistance to

provide a maximum power production tracking. This simplification distance the model from the real

V80 since an optimum slip function is not possible to obtain with this control system. On the other

hand, the VSC HVDC model is also not considering the OptiTip function, so a slight justification of the

comparison of the models can be drawn. Although this has to be kept in mind when the final results

of this project is presented. If this report should be basis for a new control strategy of wind-turbines

it is highly recommended to include possible gain of using the OptiTip function.

Figure 21: Optimum Rotor Resistance for AC connected Wind Turbine

As can be seen in Figure 21 the rotor resistance is kept low when the wind is low to minimize losses

and a good mechanical power tracking is performed. When the wind speed increase additional rotor

resistance is provided so the rotational speed increase to perform a better tracking of the

mechanical power, while maintaining low losses. When the wind reaches 13 m/s the mechanical

power is high so the most profitable solution is to reduce the losses by reducing the rotor resistance.

How these operational points effect the slip is shown in Figure 22.

34

Figure 22: Slip of the AC connected Wind Turbine

4.4 Operational Points VSC HVDC Connected Wind Turbine

In order to create the dynamic control system the optimum operational points of the new topology

has to be singled out. This section will provide the reader with the operational points and the

approximation used in between operational points for use in the dynamical simulation program

DIgSILENT.

Figure 23: Optimum Stator Frequency as a function of Wind Speed

35

Figure 24: Optimum Rotational Speed as a Function of Wind Speed

Figure 25: Optimum Slip as a Function of Wind Speed

In Figure 23, Figure 24 and Figure 25 the optimum operational points are published. Al figures are

presented until wind speed equal to 13 m/s since here the nominal power production is reached. In

Figure 23 and Figure 24 the spline approximation of the operational points is also presented. This

line shows how the operational points later implemented in the dynamic simulation in DIgSILENT

Power Factory. In the VSC HVDC connected generator the optimum operational points always

resulted in the minimum value of the rotor resistance, this since all the power tracking is performed

by the converter regulating the stator frequency.

36

4.4.1 Necessary operational range

From the operational points presented the necessary operational range can be derived. To be able

to follow the tracking method presented it will be necessary for the rotor to rotate in the interval

showed in Table 7.

Table 7: Operational range needed for Induction machine

Min rotational speed Max Rotational Speed

0.5 p.u. 1.3985 p.u.

30 Hz 83.9 Hz

4.5 Summary of Power Production and operational points

As is shown, the effect of controllability of the generators has a significant impact on the power

production performed by the generators. The more controllable the generator topology, the more

power can be withdrawn from the wind, while limiting the losses so the generated power will

maximize. To present a final steady state overview, the author has chosen to present the operational

points of the three wind turbine topologies on common mechanical power curves. This is meant to

illustrate and describe for the reader, the essence of the Power Optimization Theory.

Figure 26: Operational Points of the three generators on the power curve provided by a wind of 6 m/s

37

Figure 27: Operational Points of the three generators on the power curve generated by a wind of 13 m/s

As is seen in Figure 26 and Figure 27 the operational points of the three generators are different for

the different wind speeds. The controllability of the VSC HVDC generator provides it with the benefit

of keeping the rotor resistance low, and use the stator frequency to decide the rotational speed. This

is an optimal solution since this provides minimum losses and maximum mechanical power for all

wind speeds. The generator with variable rotor resistance has the ability to consider if the extra

rotational speed obtained with the increase of extra rotor resistance will provides a favourable

relation between losses and mechanical power. When comparing Figure 21 and Figure 27 one can

see that the impact of decreasing the rotor resistance is favourable for this wind speed. This because

the mechanical power curve here is relatively flat, so the gain in mechanical power of adding extra

rotor resistance will be lost with interest when considering the losses. The normal AC connected

generator has no chance of optimization and will therefore changing rotational speed based on the

wind only.

38

5. Steady State Comparison of power production

This chapter will use the models derived earlier to show the difference of the energy production

from the AC transmission connected RCC generator and the VSC HVDC connected RCC generator.

The chosen way of performing this analysis is described and how the results can be interpreted

5.1 Comparison Strategy

In order to provide reasonable results in the power production comparison between the generators

connected to the VSC HVDC and the generator connected to regular AC transmission, the general

details has to be defined. By checking the product description (Vestas, 2008), the operational

interval of the generator is spanning from 1-10 % slip. Therefore it is assumed that this same

constrain applies to the new wind turbine design with variable stator frequency. The three turbines

investigated in steady-state are the same turbines presented in Chapter 4, and the power curves

derived from the presented operational points.

As mentioned, for simplicity the pitch angle is not used in the optimization algorithm. The pitch

angle is firstly used when the generators reach nominal power, and is then used to reduce the

mechanical power. When the nominal power is reached, the optimization is terminated and both

generators keep producing nominal power until the cut out wind speed of 25 m/s is reached. Both

turbines are assumed to have the same cut in wind speed, 4 m/s.

The yearly energy production will be based on weibull distributed wind speeds multiplied with the

respective power curves, assuming that both generators run without stop for the whole year. A

more tidy presentation of the physical constrains is listed in Table 8 .

Table 8: Constrains for the two generators

Generator connected to AC

Transmission

Generator connected to VSC

HVDC Transmission

Nominal Power 1800 kW 1800 kW

Nominal Wind Speed 15 m/s Unknown (13 m/s)

Operational stator frequency 50 Hz 25 to 75 Hz

Operational rotor frequency 50 to 55 Hz 25 to 75 Hz

Maximum Slip 10% 10%

Cut in wind speed 4 m/s 4 m/s

Cut out wind speed 25 m/s 25 m/s

39

5.2 Power Production

This section will present the results of the two different wind turbine connections. Using the

optimization algorithm the power curves for the different wind turbines was obtained. To make the

power calculations valid the chosen wind speeds where taken from Sklinna Fyr in Norway

(Norwegian Metrological Institue, 2008). Based on 1 hour’s values from May 2007 until May 2008 a

weibull distribution was generated in MatLab. The yearly energy production can be described as:

X YZ [ \]<^<C]_

`) ab(cEC]cC] (5.1)

Where Vw is the wind speed and Wblpdf represents the Weibull distribution, the reader can refer to

(Johnson, 1994) for the weibull calculation technique. 8760 is the number of hours in a year

considering no operational problems for the wind turbine. For Illustrative purposes also the energy

production at given wind speeds is presented and can be calculated like:

XC] \]<^<C] ) ab(cEC] (5.2)

The power curves together with the weibull distribution for the wind is presented in Figure 28 and

the power production generated at different wind speeds is shown in Figure 29.

Figure 28: Power Curves for the two wind turbines together with the weibull distribution of the wind

40

Figure 29: Power Production distributed at different wind Speeds

Figure 30: The Power Curves of the three wind-turbine topologies

Based on the yearly energy production equation it is now possible to compare the production of the

generators and the results are presented in Table 9.

Table 9

Turbine Type Power Production

AC Connected Constant Resistance 6.359 MWh

AC Connected Variable Resistance 6.464 MWh

VSC HVDC Connected 6.819 MWh

41

Increase compared to: Power Production Increase of the VSC

HVDC Connected Wind Turbine

AC Connected Constant Resistance 7.25 %

AC Connected Variable Resistance 5.49 %

Further a dynamic computation of energy will be performed with high resolution wind data in

Chapter 8. The wind data used is described more in detail in chapter 8.1. Therefore firstly a steady

state computation of the energy gain will be presented, for later comparison with the dynamic

model. In order to save some time in the dynamic modelling, the choice was to compare the VSC-

HVDC topology with the AC connected constant rotor resistance topology. This was considered by

the author to be sufficient to illustrate the possible gain of the new topology. This assumption was

made by analysing Figure 30, seeing that the difference between the constant and variable

resistance AC connected wind turbine is very small compared with the difference up to the VSC

HVDC topology. This simplification saved the author potential time used for implementing and

tuning a rotor resistance control system.

Figure 31: Power Production distributed on different wind speeds

Figure 31 shows the weibull distribution of the high resolution wind together with the energy

production of the two wind turbine topologies calculated according to equation(5.2). The Power

Production calculated according to equation (5.1) is presented in Table 10.

Table 10

Turbine Type

AC Connected Constant Resistance 6.975 MWh

VSC HVDC Connected 7.224 MWh

Increased Power Production 3.56%

42

5.2.1 Evaluation

It can be seen from Figure 28 that if the shape of the wind will have a higher density in the area

above 10 m/s the difference will be more favorable for the DC connected wind turbine. On the other

hand if the density of the wind is compressed more in the below 10 m/s region the difference in

power production might decrease. The favorability of this new concept using VSC HVDC transmission

will therefore be strongly dependant on the wind speed. The author also find it reasonable to

assume that the shape of the wind will have a strong impact on the favorability since the power

production of the VSC HVDC connected and the variable rotor resistance wind turbine has a very

comparable power production around the average wind area of 7-10 m/s. Therefore the favorability

of the new VSC HVDC concept has to be determined sight specific in order to see the possible gain

when looking at power production. What has not been considered in these calculations and will

most likely determine if there will be any benefit, are the losses in the PWM converters compared to

normal HVAC transmission systems. A general loss description of the most common transmission

systems was found in (Watson, 2005) and is presented in Table 11.

Table 11: Losses in typical transmission systems, losses expressed in percent of nominal power

Alternative Convention Losses Line Losses Total Losses

AC 0 % 1.2 % 1.2 %

Conventional HVDC 1.4 % 0.5 % 1.9 %

VSC HVDC 5 % 1.5 % 6.5 %

As seen from Table 11 the total losses for VSC HVDC transmission is 5.3 % higher at nominal power

in the VSC HVDC transmission compared with the AC transmission. More detailed data considering

the two topologies is provided in (Watson, 2005). The details of how the state of the art PWM

converter losses are compared to the ones presented here, and how the losses are represented in all

operational conditions was beyond the scope of this project. Including these losses is highly

recommended by the author in future research. Conventional HVDC is tyristor/diode based HVDC

solutions and will not be discussed in detail in this project. If the reader wants more information on

this topology a detailed description can be found in (Kundur, 1993).

43

6. Dynamic Modeling

For further investigation of the stability of the new topology, a dynamic model has to be developed.

The chosen program is DIgSILENT Power Factory because of its history of being used for representing

wind turbine models. The model has to contain all components important for dynamic investigations

and to show dynamic power optimization. Further the VSC HVDC transmission receiving end station

control system need to be represented realistic, and a control system for the sending end converter

has to be implemented. The generator and PWM converters are chosen from the Power Factory

component library and modified to fit better with the wind turbine investigated.

6.1 Wind Turbine Model

The wind turbine model implemented in Power Factory was based on the model from the steady

state analysis. This was done to obtain correlation and therefore a dynamic validation of the steady

state results. Since the steady state analysis was performed without the considering the effect of the

shaft, this was also neglected in the dynamic model. Therefore the whole inertia was moved to the

generator. The wind turbine generator is therefore represented by a pitch controller, turbine model

and the wound rotor single cage induction generator.

6.1.1 Turbine model

The turbine model is implemented using the equations derived in Chapter 2.1. The Cp curve is

provided by a spline approximation between values of and θ from equation(2.2). The lambda and

mechanical power is computed as in equation(2.3) and equation(2.4).

6.1.2 Pitch Controller Model

The pitch angle controller used is a slightly modified version of the pitch controller used in the Power

Factory DFIG example. This is implemented by comparing the rotational speed of the generator to a

given reference. This signal is fed to a PI controller with limits of the maximum and minimum pitch

angle. This generates the reference signal for the pitch angle, which is compared with the actual

pitch angle and providing the desired pitch angle. In order to smoothen the pitching a rate limiter on

the rate of change of the pitch angle is implemented. If the reader wants to use a similar approach

to reconstruct this model, remember that the reference speed has to be changed in order to cope

with the variations desired in the rotational speed of this project. For more detailed information

about the pitch controller the reader can referee to (Anca D. Hansen, 2003).

44

6.1.3 Induction Machine Model

The induction machine used in the dynamic model is the Type ASM Asynchronous Machine. The

electrical equations of the machine are derived in the same way as the ones presented in chapter

2.2. The voltage equations are described as:

+ <+,+ - cH+%J'cd - e %+

%J'H+ (6.1)

cH%J'cd - e %+ %

%J'H (6.2)

Removing the derivatives from equation (6.1) and (6.2) then comparing with equation(2.5) to

equation(2.8) the equation are identical. DIgSILENT state in the technical reference that stator

transients are neglected in this model, so ωref is fixed to global or local reference .The author has not

found which reference is used in the presented model, but based on the results presented in

Chapter 7 it is clearly shown that the change in stator frequency affects the generator model. One

theory is that the local reference frame chosen in Power Factory is linked to the output frequency of

the PWM converter of the sending end of the HVDC, and therefore identical with the stator-

frequency. The very good correlation between the steady state model and the dynamic model

presented in Chapter 7 was only obtainable when considering change in generator reactance based

on stator frequency in the steady state model! All other approaches ended in larger deviation for the

two models (for instance setting the ωs = ωn in the steady state model and running the dynamic

model in an EMT simulation (DIgSILENT2, 2007)). A more detailed description of the induction

machine can be found in the DIgSILENT technical reference for induction machines (DIgSILENT2,

2007).

6.2 PWM Model

The PWM Model chosen for this project is also one of the standard models from the DIgSILENT

Power Factory component library. The model represents a self commutated voltage source AC/DC

converter. The model is based on the fundamental frequency approach and supports sinusoidal and

rectangular modulation. Further the model supports a number of different control conditions

(DIgSILENT, 2007):

• Vac-phi: Specifies magnitude and phase of AC-terminal. Typical control modes for motor-side

converters in variable speed drive applications.

• Vdc-Phi: Specifies the DC-voltage and the AC-voltage phase. No typical application.

• PWM-phi: Load-flow setup without control. The pulse-width modulation factor is directly

set in magnitude and phace.

• Vdc-Q: Specifies DC-voltage and reactive power. Typical applications: STATCOM, shunt-

converter of UPFC, grid side converte of doubly fed induction machine and VSC-HVDC

applications

45

• Vdc-P: Specifies AC-voltage magnitude and active power. This is equivalent to a “PV”

characteristic of conventional synchronous machines. Typical applications: Grid side

converter of converter driven synchronous machines, VSC HVDC.

• P-Q: Specifies P and Q at the AC-side. This control mode is equivalent to a “PQ”-

characteristic of synchronous machine. Typical applications: Same as “Vac-P”

For more detailed information the reader can refer to the DIgSILENT technical reference (DIgSILENT,

2007).

6.3 HVDC Model

This section will determine the choice of VSC HVDC model and present a stepwise modeling solution

for use in Power Factory. During the history of VSC based HVDC models, many control strategies for

the converter has been proposed. This project will not consider them all but concentrate at some

solutions fitted for this project.

Firstly, even though the two converters topologies are equal, the main functions of the two

converters are different, for the receiving end converter the important property is to feed the active

power transmitted by the sending end converter and maintaining the DC voltage at a desirable level

and controlling the reactive power (Erlich, 2007) ,(Florin Iov, 2006). The sending end converter will

transmit the power from the wind farm and control/maintain the frequency and voltage on the wind

turbine AC side (Erlich, 2007).

Based on this information the implemented control system for the receiving end converter was

made to control the reactive power and DC – link voltage, the system is chosen to operate in the d-q

reference frame and is highly inspired by the control system presented in (Florin Iov, 2006). The

sketch of the control system is shown in Figure 32 and the supported control condition chosen from

Chapter 6.2 is the Vdc – Q control.

The DC link voltage is controlled using a PI controller where the measured voltage from the DC-Link

is compared with a fixed reference. The fixed reference is set to be the nominal voltage of the DC-

link, 150 kV (1 p.u. of connected bus-bar). The output from the DC-Voltage controller is used as the

reference for the current in the d-axis.

Referring to (Florin Iov, 2006) the set point of the reactive power can be set to zero, it should be

noticed that the system operator can demand a different set point. Although for simplicity this is

used in this project and therefore the output of the reference output (current reference in q axis) of

the Q-controller is set to zero.

46

Figure 32: Control system for the receiving end Converter

Now when the reference parameters are determined the measured currents have to be transformed

into the d-q axis. This is done by implying a Phase-Locked-Loop to calculate the currents from real

and imaginary to the d-q axis as shown in equation (6.3) and (6.4)

, , fghi - ,hjk i (6.3)

,* , hjki - ,fgh i (6.4)

Where id is the d-axis current, iq is the q-axis current, ir is the real part of grid current and ii is the

imaginary part of the grid current. The Phase-Locked-Loop measures sin(ϕ) and cos(ϕ). These

currents are now fed into a first order filter before compared to the reference signals. The offset is

than fed to the current controller and generate the control signals Pmd and Pmq of the PWM

converter.

47

The relation between the AC and DC voltage is then described as(DIgSILENT, 2007):

Clm CnmIo '

(6.5)

pnmI* o '*plm (6.6)

Where UAC,d is the d-axis grid voltage, UAC,q is the q axis grid voltage, UDC is the DC voltage and K0 is

based on the type of modulation, in this project a sinusoidal modulation is chosen and therefore K0 is

defined as(DIgSILENT, 2007) :

o qrq (6.7)

6.4 Combined Controller Model

The combined controller model is designed to increase the maximum power production of the wind

turbine generator. By using the sending end converter the voltage and frequency of the wind turbine

stator side can easily be controlled. Of the supported control conditions in Power Factory described

in Chapter 6.2 the sending end controller will utilize the VAC –phi system. This system easily provides

the possibility of quick and accurate settings of voltage and frequency. The sketch of the complete

control system can be seen in Figure 33

48

Figure 33: Control system for the sending end controller

The input signals of the converter in this model will be Pm_in and f0. And since the receiving end

converter now controls the DC – Link voltage, and this can be assumed to be constant at 150 kV in

normal operation, the voltage of the AC side in kV is:

nm qr ) FCq 's (6.8)

The input signal f0 is directly proportional to the AC frequency in p.u. The generator stator frequency

will therefore be described as:

E+ tu ) E (6.9)

For simplicity the grid voltage was considered proportional to Pm_in. This is true for normal operation,

but as shown in Chapter 9 is not representative when the system is submitted to a fault, although it

is considered by the author to be a sufficient approximation in this project.

49

In (Anca D. Hansen, 2003) it is stated that the use of a moving average is a realistic way of filtering

wind data in power optimization. The method is based on a fixed window, moving in time, averaging

the values within, and can be expressed mathematically as:

vwL xd - xd y xd zz - (6.10)

Where t is the instant time, and t – 1 is the time with one step delay (e.g. one second ago). x is the

value at the given time.

The averaging time is determined experimentally by trying with different numbers until an averaging

time providing favorable power production was obtained. The averaging time is chosen to be 1 sec

averages.

To impose a control system that changes the stator voltage and frequency it is very important to

take the inertia of the generator into account. If the stator frequency is increased to fast the

rotational speed of the generator might not be able to follow, and the generator can be forced into

under magnetization and in worst case resulting in a change in operational mode. If the stator

frequency exceeds the rotational frequency, the generator turns to motor and starts consuming

power from the grid. And if the stator frequency is lowered to fast, the slip might increase, providing

unfavorable high currents (Kundur, 1993). This problem has been solved by implementing a rate

limiter on the output of the frequency and voltage controller. One way of implementing a rate

limiter in Power Factory is shown in Figure 34

50

Figure 34: PI controller with rate limiter in Power Factory

As can be seen the error is multiplied with a constant to represent the actual frequency. Than the

rate limits can be set by setting a maximum and a minimum on the raw error signal. When

integrating between these boundaries the actual frequency is obtained, with the desired rate of

change limits. The rate of change limits where set manually at the rate that provided smooth

variations of the stator frequency and power production. The same rate of change was imposed on

the voltage controller to ensure correspondence between the voltage and frequency.

The reference signals of the frequency and voltage is generated as a function of wind speed, based

on the operational points presented in Figure 23 where the voltage is limited on 1 p.u. This is

implemented in Power Factory using a lookup table where a spline interpolation provides the values

in-between the operational points.

51

7. Comparison between the Steady State and the Dynamic model

This chapter will provide a systematic comparison between the steady state model and the dynamic

model presented earlier. The purpose of this comparison is to increase the reliability of the results

presented, and provide an estimate of how the results should be interpreted. The method used in

this project was to force the dynamic model to work in steady state

To force the dynamic model into steady state is done by fixing a number of parameters that result in

a steady output. In this project the author chose to fix the rotor resistance, stator frequency, stator

voltage and wind speed. The outputs chosen for comparison was P-mechanical, P-generated and

rotational speed, ωr. The reason for the choices is as followed:

P-mechanical:

The mechanical Power is used to see how accurate the mechanical representation of the model in

Power Factory was achieved. This contains the accuracy of the Cp and λ representation although it

has to be noticed that this is affected by the operational point ωr. One of the important issues of this

comparison was to find a suitable resolution of the Cp-curve. The Cp curve in Power Factory was

implemented as a spline approximation matrix, where the inputs where the tip speed ratio (λ) and

pitch angle (θ). This was made on a try and fail basis until a sufficient correspondence was obtained.

Rotational Speed:

The rotational speed was a good measure to see the influence of the rotor resistance on the

operational point. This shows the intersection between the electrical and mechanical torque curves.

This also increased the demands on the number of operational points in the optimization algorithm.

It was shown that to provide a sufficient accuracy there was needed a step size of 1/33 p.u. of the

stator frequency and rotor frequency. Also it showed that the suitable approximation in the

optimization algorithm was to use a linear approximation between the operational points, when this

gave the lowest error between the steady-state and dynamic model.

P-generated:

The generated power is used to see the accuracy of the operational point. Since the optimization

algorithm is considering losses as an optimization factor it is important to see if the dynamic model

has comparable losses at the same operational points.

The chosen operational points for comparison are the optimum operational points from the

optimization algorithm. This is because these values are already calculated and presented and is of

interest for the project. The correspondence between the steady-state and dynamic model is shown

in Table 12

52

Table 12: Comparison of the Steady-state and Dynamic model at optimum operational points

As can be seen from the table the two models correspond with a relative good accuracy. The error in

rotational speed can be shown by the author to be removed if the step-size of the rotor and stator

frequency in the optimization algorithm is decreased, tough this is too time consuming to perform

for all wind speeds. The less obvious error is the Pgenerated which seems to be consequently deviating

from the steady-state model. Although this error is small in p.u. it can be seen in Table 13 that this

have an effect in percent, especially at the lower wind speeds. The author was not able to find the

reason for this difference during this project, but consider the results to be corresponding

adequately to perform further analysis.

Table 13: Comparing the steady-state and dynamic model losses

Wind Speed Steady-State model Dynamic Model Error [p.u.]

Losses in p.u. Losses in % of Pm Losses in p.u. Losses in % of Pm DM-SSM

4 0.0007 2.24 0.0012 3.83 0.0005

5 0.0012 1.89 0.0016 2.52 0.0004

6 0.0016 1.46 0.0022 2.01 0.0006

7 0.0027 1.55 0.0034 1.95 0.0007

8 0.0043 1.66 0.0049 1.89 0.0006

9 0.0068 1.84 0.0073 1.97 0.0005

10 0.0117 2.31 0.0119 2.35 0.0002

11 0.02 2.96 0.02 2.96 0

12 0.035 4.00 0.0338 3.86 -0.0012

13 0.0607 5.45 0.0572 5.13 -0.0035

Wind

Speed

Operational Conditions Steady – State Model Dynamic Model

fs

[p.u.]

Vs

[p.u.]

Rr

[p.u.]

ωr Pmechanical PGenerated ωr Pmechanical PGenerated

4 0.5 0.5 0.029 0.5019 0.0313 0.0306 0.52 0.0313 0.0301

5 0.5313 0.5313 0.029 0.5349 0.0634 0.0622 0.535 0.0634 0.0618

6 0.6563 0.6563 0.029 0.6614 0.1095 0.1079 0.6615 0.1095 0.1073

7 0.75 0.75 0.029 0.7573 0.1739 0.1712 0.7572 0.174 0.1706

8 0.875 0.875 0.029 0.8844 0.2595 0.2552 0.8842 0.2595 0.2546

9 0.9688 0.9688 0.029 0.9809 0.3697 0.3629 0.9807 0.3697 0.3624

10 1.0625 1 0.029 1.0798 0.5071 0.4954 1.0794 0.5072 0.4953

11 1.1563 1 0.029 1.1813 0.6748 0.6548 1.1811 0.6749 0.6549

12 1.25 1 0.029 1.2868 0.876 0.841 1.2862 0.8762 0.8424

13 1.3438 1 0.029 1.3985 1.114 1.0533 1.3774 1.1141 1.0569

53

8. Dynamic Comparison between the AC and DC Connected Wind

Turbine

This chapter is meant to show a more realistic power production where the wind is fluctuating and

the inertia of the generator is taken into account. Since rotational speed cannot be changed

instantaneous because of the inertia of the generator the optimum tracking of the generator is

provided with a delay. This is why the proposed control system is of great interest to analyse in

dynamic simulation. The full system implemented in Power Factory is shown in Figure 35

Figure 35: Full system for Dynamic Simulation, implemented in Power Factory

The system shown is by the author divided in to three areas, shown in Figure 35 to be area A, B and

C. The idea of this is to keep are A in both simulations, The HVDC connected simulation and the AC

connected simulation. The External grids in B and C are identical. The HVDC model contains 2 x 150

kV 75.2 µF (Du, 2007) capacitors to decrease fluctuations of the DC-Link. The cable is modeled as a

150 kV DC cable with resistance of 0.0608 Ω/km and reactance of 0.4187 Ω/km the DC voltage is

150 kV and the external grid is connected at 30 kV. In area A the generator with nominal voltage of

690 V is connected to a transformer with the ratio of 0.69/30 kV (Bonus Energy a/s, 2001). The

inertia of the generator is determined by acceleration time constant and is set to 2 s. When

simulating the AC connected system area B is disconnected and put out of operation, so only system

C is connected to system A. When the HVDC connected system is simulated, system C is

disconnected and put out of operation. Here only system A and system B is in operation.

54

8.1 The Dynamic Comparison Strategy

Since the aim of this project was to create a control system for optimum operation of the generator

designed to increase the yearly production, a reasonable comparison strategy was needed. The high

resolution wind data available for the author during this project was based on a wind generator

algorithm provided by the industrial supervisor. This data is divided in a step-size of 0.05 s and have

duration of 10 min. Further details of the wind data are presented in Table 14 and its weibull

distribution was earlier presented in Figure 31.

Table 14: Characteristic of the Wind data used in the dynamic model

Mean Wind Speed 9.65 m/s

Min Wind Speed 7.11 m/s

Max Wind Speed 13.07 m/s

The two system combinations presented in Figure 35 are now exposed to the same wind data, and

the power production data is stored and exported to MatLab.

Figure 36: The wind used for dynamic comparison

55

Figure 37: AC connected generator Power Production presented in p.u.

Figure 38: DC Connected generator Power Production presented in p.u.

56

Figure 37 and Figure 38 shows the energy production of the two generator topologies based on the

wind in Figure 36. The reason for the negative scale of the y-axis of the power production graphs is

because the machine is in generator mode. The data stored in these graphs could easily be exported

to MatLab using the Power Factory “export to windows clipboard” feature, and the results where

integrated and divided by time.

IwL > [ cd

|

(6.11)

Where Ppu is the power in p.u. stored in the graphs of Figure 37 and Figure 38, and 600s is the

simulation time. The answer obtained is then the average Power Production in p.u. Assuming the

wind data is representative for a year, the yearly energy production can be written as:

X@ IwL ) Y~a ) YZ (6.12)

Therefore the increase of power production based on the dynamic model can be written as:

X s\zzdcsIwL s\zzdcsIwL

(6.13)

The results of the dynamic simulation are shown in Table 15

Table 15: Average Power Production [p.u.] for the two wind turbine topologies

AC connected wind turbine 0.4473 p.u.

VSC HVDC connected wind turbine 0.4590 p.u.

Increased Power Production = 2.62%

57

8.1.1 Evaluation

The results so far show the possibility of increasing the power production of a wind turbine both in

steady state and in dynamic models by varying the stator frequency and stator voltage. Although it

has to be remembered that in order to compare this strategy with a state of the art wind turbine like

the Vestas V80, a more accurate model has to be developed. Also the possibility of regulating the

slip by actively using the pitch angle should be included in the optimization algorithm and this

control might be needed to implement in the dynamic simulation. Lastly the power submitted to the

grid will include losses in the DC transmission and cables. This should be considered in future

research.

58

9. Dynamic Response to system fault

As the last part of this project it was desired to check how the model with control system reacts to

tripping of the one of the converters. The base of the dynamic study is to see how the active and

reactive power, stator voltage rotational speed and the receiving end controller react to the tripping

of the converter. The system study investigated will be tripping of the receiving end controller. This

can represent a malfunction of the converter, or a fault on the external grid which forces the

converter to disconnect. The system is modelled as shown in Figure 35 and contains system A and B.

The analysis of the tripping of the grid side converter is performed by opening the switch connected

to T DC shown in Figure 35. The opening of the switch occurs at time = 10 s and the closing of the

switch at time = 10.3 s. The parameters for investigation are DC Voltage, stator voltage, stator

frequency, rotor speed, receiving end current controller, active power and reactive power.

Figure 39: Active power during Converter tripping

59

Figure 40: Reactive Power during Converter Tripping

150

Figure 41: Rotational speed during Converter tripping

60

Figure 42: Stator side Frequency during Converter Tripping

Figure 43: Receiving End Current controller during Converter Tripping

61

Figure 44: DC -Link Voltage during Converter tripping

Figure 45: Generator Side AC Voltage during Converter Tripping

62

Figure 46: Rotor speed during converter tripping, with constant wind = 12 m/s

As can be seen from the presented results the VSC HVDC system reacts properly to a situation where

the receiving end converter needs to trip. The generator will continue energy production during the

fault and charge the DC capacitors. This result in an increase of the DC – Link voltage as seen in

Figure 44 and the receiving end control system will try to retain the DC voltage, Figure 43. The DC

voltage will then oscillate until the PI controller has managed to stabilize the voltage. Since the

Sending end converter is implemented on the assumption of a constant DC Voltage, the oscillation of

the DC voltage is directly proportional to the generator AC side voltage as seen in Figure 45, the

reason the AC voltage stabilizes around 33 kV is because the stator frequency here is over 1 p.u.,

earlier showed in Figure 5. The increase of the rotor speed is not a sign of instability thus only a

result of continues controlling of the stator frequency based on the wind- speed as seen when

comparing Figure 41 with Figure 42, for illustrative purposes it was also decided to show the

rotational speed for the same fault with wind speed constant equal to 12 m/s as shown in Figure 46.

The measurement of the voltage of the sending end converter was for simplicity in this project

assumed to be proportional with the Pm_in signal to the converter. This is true when the system

operates in normal operation. Although as was seen in the simulations presented this assumption

provides direct proportionality between the voltage variations in the DC-Link to the generator side

AC terminal. In order to reduce this influence of DC voltage variations it might be interesting to use

an external voltage measurement on the station1/PCC in Figure 35 instead. This might give lower

dependency between the AC and DC voltage, although this hypotheses was not tested by the author

during this project.

63

10. Discussion

This project has investigated the possible gain of using a VSC HVDC transmission system for

increasing the power production. The results presented show a positive effect of the new control

possibilities when connecting such a generator to HVDC transmission system. Although the reader

should have in mind the assumptions and simplifications this project has taken to account. Firstly

wind farms connected to a VSC HVDC system will most likely not be represented by a single wind

turbine, but rather by a significant number, as an example Horns Rev is represented by 80 wind

turbines. When working with these types of wind farms there is no guaranty that the wind speed is

equal for all the turbines, therefore common optimal operational point might be hard to find and

the efficiency and power production presented in this project hard to obtain. The shaft system is

neglected during the whole project and will influence the dynamic analysis and also possible change

the operational points, although for the power production comparison it is assumed by the author

the comparison is representative since none of the turbines in the project has this model included.

This project has also not considered the effect of saturation in the generator which might be found

in (Maria Imecs, 2001). The assumption of neglecting the saturation was based on the simplicity

obtained. If saturation should be included the whole optimization algorithm would have to be

reconstructed and the difficulty level increased since the co-dependency of currents and flux no

longer would allow the setting of rotational speed in the way presented earlier. The model

presented is also based on picking parts (data) from different wind turbines where references have

been found. In order to provide a product for use in real life these parameters have to be more

exact, so the models can be customized in a way that produces the best possible correlation with

real life. The losses in the transformer was initially included in the model, but removed during the

project because of its modest impact considering losses and operational points. What should be

included in future research is the losses in the PWM converter, as these are believed by the author

to be determinant for the new topology feasibility considering increased power production

compared to standard AC transmission.

64

11. Conclusion

The project shows, based on its assumptions and simplifications, that there is a possible gain in

power production by implementing coordinated control between VSC and the RCC generator,

although it prevail uncertainty if this gain will compensate for the losses in the PWM converters. The

power production increase is shown in both steady state and in dynamic simulations. Basing the

conclusion on the wind data used in the dynamic simulation the steady state model shows an

increase in power production of 3.56 % compared to the same generator working as a constant rotor

resistance AC-transmission connected wind turbine. The dynamic control system is based on using

wind speed to provide a reference signal for stator frequency and stator voltage, and a rate limiter

implemented so the variations should fit better with the inertia of the generator. The gain of using

the control system based on the same wind speed as the one used in the comparison in steady state

is shown to be 2.62%. The difference from steady state to dynamic simulation is because of the

inertia of the generator that withstands sudden changes in frequency. But if losses in the PWM are

of the size shown in Table 11 the VSC-HVDC generator will provide a lower power output to the grid

than a conventional AC transmission at this wind distribution. Therefore it is believed by the author

that the new control system presented will have a future in the situations where PWM converter

transmission is the only topology that fulfils the requirements set by the transmission system

operators. In these cases the optimization performed in this project will be pure gain.

The models presented in this project are believed by the author to be representative to show the

idea of the new topology. Although to initiate product development of such a system more exact

data is needed, and considerations of state of the art OptiPitch control and HV PWM converters

should be investigated and possibly included.

65

11.2 Future Research

The following topics are recommended as future research in this field and recommended to do in

the following order:

• Including losses of state of the art PWM converters in order to determine power

output to the grid compared with the alternative AC transmission

• Use wind data from existing wind parks to see possibilities for coordinated control of a

whole wind park, alternatively see how many sections a wind park should be divided

into in order to get sufficient controllability.

• Include saturation and iron losses in the generator model

66

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[2] Ahmed G. Abo-Khalil, H.-G. K.-C.-K. (2004). Maximum Output Power Control of

Wing Generation Systems Considering Loss Minimization of Machines. Busan: IEEE.

[3] Ahmed, N. &. (Aug 2006). Stand-Alone Hybrid Generation System Combining Solar Photovoltaic and Wind Turbine with Simple Maximum Power Point Tracking

Control. IEEE Power Electronics and Motion Control Conference , s. 1-7.

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[10] Du, C. (2007). VSC-HVDC for Industrial Power Systems. Göteborg:

CHALMERS UNIVERSITY OF TECHNOLOGY.

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Farms Connected to the Grid through VSC-HVDC Link. Duisburg: IEEE.

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for Active Stall Wind Farms to Grid. Aalborg/Risø: IEEJ Trans.

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Engineers. Englewood Clifs: Prentice-Hall, Inc.

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Systems. Wiley.

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Induction Generators for Wind Turbines. Department of Energy and Environment,

Division of Electric Power Engineering. Göteborg: CHALMERS UNIVERSITY OF

TECHNOLOGY.

67

[22] Robert Leidhold, G. G. (2002). Field-Oriented Controlled Induction

Generator With Loss Minimization. IEEE.

[23] Vestas. (2008).

[24] Vestas. (u.d.). Project Description.

[25] Watson, A. R. (2005). Comparison of VSC based HVDC and HVAC

Interconnections to a Large Offshore Wind Farm. IEEE.

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68

Appendix

A Nomenclature

Symbol Description

A Rotor Radius

a1 Number of Current Paths

Cp Power Coeficient

f Frequency

f0 Frequency modulation

ϕ Angele between Current and Voltage

ψ Flux

i Current

k Pitch Factor

k0 q

q

L Inductance

λ Tip Speed Ratio

N Number of Turns

ρair Air Density

Pc Copper Loss

Pgen Generated Power

Pm Mechanical Power

Pmd Modulation d axis

Pmq Modulation q axis

R Resistance

s Slip

Te Electrical Torque

θ Blade Pitch Angle

v Voltage

Vw Wind Speed

Subscripts: Description

a Air Gap

avg Average

AC Alternating Current

DC Direct Current

d d axis

q q axis

r Rotor

s Stator

69

m Magnetizing

nom Nominlal

max Maximum

min Minimum

pu Per Unit

B Calculations

Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(w-

wr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-(w-wr*(Lm+Lr)),Rr]]);

> Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(w-

wr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]);

> A:=Vector([iqs,ids,iqr,idr]);

>B:=Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(w-

wr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]);

>

Rs w Ls + Lm( ) 0 w Lm

-w Ls + Lm( ) Rs -w Lm 0

0 w - wr( ) Lm Rr w - wr( ) Lm + Lr( )

- w - wr( ) Lm 0 -w + wr Lm + Lr( ) Rr

éêêêêêêêë

ùúúúúúúúû

Rs w Ls + Lm( ) 0 w Lm

-w Ls + Lm( ) Rs -w Lm 0

0 w - wr( ) Lm Rr w - wr( ) Lm + Lr( )

- w - wr( ) Lm 0 - w - wr( ) Lm + Lr( ) Rr

éêêêêêêêë

ùúúúúúúúû

A :=

iqs

ids

iqr

idr

éêêêêêêêë

ùúúúúúúúû

B :=

Rs w Ls + Lm( ) 0 w Lm

-w Ls + Lm( ) Rs -w Lm 0

0 w - wr( ) Lm Rr w - wr( ) Lm + Lr( )

- w - wr( ) Lm 0 - w - wr( ) Lm + Lr( ) Rr

éêêêêêêêë

ùúúúúúúúû

70

> B.A;

>

>

> idr:=solve(-(w-wr)*Lm*iqs-((w-wr)*(Lm+Lr)*iqr)+(Rr*idr),idr);

> subs(idr=solve(-(w-wr)*Lm*iqs-((w-

wr)*(Lm+Lr)*iqr)+(Rr*idr),idr),B.A);

> B:=Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(w-

wr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]);

> B;

> A:=Vector([iqs,ids,iqr,idr]);

Rs iqs + w Ls + Lm( ) ids + w Lm idr

-w Ls + Lm( ) iqs + Rs ids - w Lm iqr

w - wr( ) Lm ids + Rr iqr + w - wr( ) Lm + Lr( ) idr

- w - wr( ) Lm iqs - w - wr( ) Lm + Lr( ) iqr + Rr idr

éêêêêêêêë

ùúúúúúúúû

idr := Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr

Rr

B . A

B :=

Rs w Ls + Lm( ) 0 w Lm

-w Ls + Lm( ) Rs -w Lm 0

0 w - wr( ) Lm Rr w - wr( ) Lm + Lr( )

- w - wr( ) Lm 0 - w - wr( ) Lm + Lr( ) Rr

éêêêêêêêë

ùúúúúúúúû

Rs w Ls + Lm( ) 0 w Lm

-w Ls + Lm( ) Rs -w Lm 0

0 w - wr( ) Lm Rr w - wr( ) Lm + Lr( )

- w - wr( ) Lm 0 - w - wr( ) Lm + Lr( ) Rr

éêêêêêêêë

ùúúúúúúúû

A :=

iqs

ids

iqr

Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr

Rr

éêêêêêêêêêë

ùúúúúúúúúúû

71

> B.A;

éêë

Rs iqs + w Ls + Lm( ) ids + w Lm Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr( )

Rr

éêë

ùúû,

-w Ls + Lm( ) iqs + Rs ids - w Lm iqr[ ], éêë

w - wr( ) Lm ids + Rr iqr

+ w - wr( ) Lm + Lr( ) Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr( )

Rr

ùúû, [

- w - wr( ) Lm iqs - w - wr( ) Lm + Lr( ) iqr + Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr

]ùúû

72

C Matlab Optimization Script % This M-File Contains the Newton Raphson initialization of Generator % Currents, rotor speed dependancy on rotor resistance, and optimization % regarding the maximum power output clear all close all clc syms iqs ids iqr Rr ws wr vs %Electrical Constants Ls=0.18; %Stator Inductance Lr=0.07; %Rotor Inductance Lm=4.4; %Manetizing Inductance Rs=0.01;%Stator Resistance Lss=Lm+Ls; Lrr=Lm+Lr; %Mechanical Constants teta=0; % Blade Pitch Angle beta=0; % Blade Pitc Angle rho=1.225; % Air Density Awt=40; %Radius of turbine Blade Wmin=4; %Minimum Wind Speed Wmax=4; %Maximum Wind Speed Omega_s=33; %Inverse stepsize factor for stator frequency Omega=33; %Inverse stepsize factor for rotor frequency %Base Values wb=1; % Base value for Frequencies, could be removed Pnom=1.8*10^6; %Nominal Power Of Wind Turbine %Self defined factors for computation R_factor=200; %Rotor resistance step size lfac=2.64; % Lambba factor prop=0.355; %Proporsionality Factor %Trafo Parameters Kundur 2-vinding transformer Rprim=0.00225; %Primary Resistance Lprim=0.0216; %Primary Inductance Rsec=0.00225; %Secondary Resistance Lsec=0.0216; %Secondary Inductance Lmt=175; %Magnetizing Inductance Rmt=300; %Magnetizing Resistance LV=690; %Low Voltage HV=33000; %High Voltage % Electrical Equations equal to Zero F1=(-vs)+(Rs*iqs)+(ws*(Ls+Lm)*ids)+(ws*Lm*((Lm*iqs*ws)-(Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr)))/Rr; F2=(-ws*(Ls+Lm)*iqs)+(Rs*ids)-(ws*Lm*iqr); F3=((ws-wr)*Lm*ids)+(Rr*iqr)+((ws-wr)*(Lm+Lr)*((Lm*iqs*ws)-(Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr)))/Rr; %Creating Jackobian Matrix

73

G1=[diff(F1,iqs) diff(F1,ids) diff(F1,iqr)]; G2=[diff(F2,iqs) diff(F2,ids) diff(F2,iqr)]; G3=[diff(F3,iqs) diff(F3,ids) diff(F3,iqr)]; %Defining Matrixes for Newton - Raphson J=[G1;G2;G3]; F=[F1;F2;F3]; %Unknown currents, Initial guess iqs(1)=0.005; ids(1)=0.005; iqr(1)=0.005; x=zeros(3,4); x(1,1)=subs(iqs); x(2,1)=subs(ids); x(3,1)=subs(iqr); for Vw =Wmin : Wmax; for Ws = 1 :Omega_s; ws=(0.5-(1/(Omega_s-1)))+(Ws/(Omega_s-1)); %Calculation of stator frequency if ws <= 1 ; vs=ws %Calculation of stator voltage else vs=1; %Calculation of stator voltage end for rr=1:10 Rr=0.009+(rr/R_factor); %Calculation of Rotor resistance %Initiazing Step for Finding and veryfing opperational point for Ws_in = 1 :Omega_s; ws_in=(0.5-(1/(Omega_s-1)))+(Ws_in/(Omega_s-1)); if ws_in<=1 vs_in=ws_in; else vs_in=1; end for Wr_in = 1:Omega wr_in=(0.5-(1/(Omega-1)))+(Wr_in/(Omega-1)); %Equations seen from the Electrical Side s(Wr_in,Ws_in)=(ws_in-wr_in)/ws_in; topp(Wr_in,Ws_in)= (ws_in/wb)*(Lm^2)*Rr*s(Wr_in,Ws_in)*vs_in^2; bottom(Wr_in,Ws_in)=(Rs*Rr+s(Wr_in,Ws_in)*((ws_in/wb)^2)*(Lm^2-Lss*Lrr))^2+((ws_in/wb)^2)*(Rr*Lss+s(Wr_in,Ws_in)*Rs*Lrr)^2; Te(Wr_in,Ws_in)=topp(Wr_in,Ws_in)/bottom(Wr_in,Ws_in); PmEl(Wr_in,Ws_in)=(-1)*wr_in*Te(Wr_in,Ws_in); %Equations seen from the mechanical side lambda_in(Wr_in)=(Awt*wr_in*lfac)/Vw; Cp_in(Wr_in)=prop*(lambda_in(Wr_in)-0.022*(teta^2)-5.6)*exp(-0.17*lambda_in(Wr_in));

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Pm_in(Wr_in)=((rho/2)*pi*Awt^2*Cp_in(Wr_in)*Vw^3)/Pnom; end %Finding the rotor rotational speed in the opperation Point wr_plot=linspace(0.5,1.5,Omega)'; ff= @(A) interp1(wr_plot,Pm_in,A,'linear')-interp1(wr_plot,PmEl(:,Ws_in),A,'linear'); %First Constrain [wr_vector(Ws_in,rr),feval(Ws_in,rr),exitflag(Ws_in,rr)]=fsolve(ff,(ws_in)); wr_vector1(Ws_in,rr,Vw)=(wr_vector(Ws_in,rr)); % Calculating third constrain a=(PmEl(:,Ws)==max(PmEl(:,Ws))); aa=ind2sub(size(PmEl),find(a==1)); % Initializing second, third and fourth constrain if exitflag(Ws,rr) == 1 & PmEl(aa,Ws)>Pm_in(aa) & wr_vector(Ws,rr)/ws <= 1.1 wr=wr_vector(Ws,rr); slip(Ws,rr,Vw)=(ws-wr_vector(Ws,rr))/ws; % Calculating Curents using Newton-Raphson for i=2:4 x(:,i)=subs([x(1,i-1);x(2,i-1);x(3,i-1)]-subs(J)\subs(F)); iqs=x(1,i); ids=x(2,i); iqr=x(3,i); idr=(((Lm*iqs*ws)-(Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr))/Rr); iqs1(Ws,rr,Vw)=x(1,i); ids1(Ws,rr,Vw)=x(2,i); iqr1(Ws,rr,Vw)=x(3,i); idr1(Ws,rr,Vw)=(((Lm*iqs1(Ws,rr,Vw)*ws)-(Lm*iqs1(Ws,rr,Vw)*wr)+(ws*Lm*iqr1(Ws,rr,Vw))+(iqr1(Ws,rr,Vw)*ws*Lr)-(iqr1(Ws,rr,Vw)*wr*Lm)-(iqr1(Ws,rr,Vw)*wr*Lr))/Rr); flux_qs(Ws,rr,Vw)=Ls*iqs1(Ws,rr,Vw)+Lm*(iqs1(Ws,rr,Vw)+iqr1(Ws,rr,Vw)); flux_ds(Ws,rr,Vw)=Ls*ids1(Ws,rr,Vw)+Lm*(ids1(Ws,rr,Vw)+idr1(Ws,rr,Vw)); flux_qr(Ws,rr,Vw)=Lr*iqr1(Ws,rr,Vw)+Lm*(iqs1(Ws,rr,Vw)+iqr1(Ws,rr,Vw)); flux_dr(Ws,rr,Vw)=Lr*idr1(Ws,rr,Vw)+Lm*(ids1(Ws,rr,Vw)+idr1(Ws,rr,Vw)); flux_s(Ws,rr,Vw)=sqrt(flux_qs(Ws,rr,Vw)^2+flux_ds(Ws,rr,Vw)^2); flux_r(Ws,rr,Vw)=sqrt(flux_qr(Ws,rr,Vw)^2+flux_dr(Ws,rr,Vw)^2); flux_m(Ws,rr,Vw)=sqrt(flux_r(Ws,rr,Vw)+(Lr^2).*(sqrt(iqr1(Ws,rr,Vw)^2+idr1(Ws,rr,Vw)^2)));

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ir(Ws,rr,Vw)=sqrt(iqr1(Ws,rr,Vw)^2+idr1(Ws,rr,Vw)^2); is(Ws,rr,Vw)=sqrt(iqs1(Ws,rr,Vw)^2+idr1(Ws,rr,Vw)^2); end else wr_vector(Ws,rr)=0; wr=0; slip(Ws,rr,Vw)=0; ids=0; idr=0; iqr=0; iqs=0; ir(Ws,rr,Vw)=0; is(Ws,rr,Vw)=0; end end %Calculating losses in the transformator Zprim=sqrt((ws*Lprim)^2+Rprim^2); Zsec=sqrt((ws*Lsec)^2+Rsec^2); Zmt=((1/((ws*Lmt)^2))+(1/Rmt))^-1; PrimLoss=(iqs^2+ids^2)*Rprim; Imt=(iqs^2+ids^2)*(Zprim/(Zmt+Zprim)); MagLoss=Imt*((ws*Lmt)/(Rmt*(ws*Lmt)))*Rmt; SecLoss=((iqs^2+ids^2)-Imt)*Rsec*(LV/HV); Tloss(Ws,rr,Vw)=PrimLoss+MagLoss+SecLoss; lambda=(wr*Awt*lfac)/(Vw); Cp=prop*(lambda-0.022*(teta^2)-5.6)*exp(-0.17*lambda); Ploss(Ws,rr,Vw)=((((iqr^2+idr^2))*Rr)+((iqs^2+ids^2)*Rs)); Pm(Ws,rr,Vw)=((rho/2)*pi*Awt^2*Cp*Vw^3)/Pnom; Pgen(Ws,rr,Vw)=Pm(Ws,rr,Vw)-Ploss(Ws,rr,Vw); P(Ws,rr,Vw)=Pgen(Ws,rr,Vw)-Tloss(Ws,rr,Vw); Ps(Ws,rr,Vw)=vds*ids+vs*iqs; Qs(Ws,rr,Vw)=vs*ids+vds*iqs; %initializing 5th Constrain if ws >= wr | wr > 1.5 Pm(Ws,rr,Vw)=0; Pgen(Ws,rr,Vw)=0; Ploss(Ws,rr,Vw)=0; P(Ws,rr,Vw)=0; Ps(Ws,rr,Vw)=0; Qs(Ws,rr,Vw)=0; slip(Ws,rr,Vw)=0; ir(Ws,rr,Vw)=0; is(Ws,rr,Vw)=0; end end end Vw end

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% Finding Maximal Power Production and Corresponding operational points for Vw=Wmin:Wmax g=(P==max(max(P(:,:,Vw)))); [I1(Vw),I2(Vw),I3(Vw)] = ind2sub(size(P(:,:,Vw)),find(g==1)); PowerCurve(Vw)=P(I1(Vw),I2(Vw),Vw); RotorSpeed(Vw)=wr_vector1(I1(Vw),I2(Vw),I3(Vw)); end %Operational Points omega_s=((0.5-(1/(Omega_s-1)))+(I1./(Omega_s-1)))*50 Rotor_resistance=0.009+(I2./R_factor)

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