Power Electronic Generator Systems

Power Electronic Generator Systems for Wind Turbines

Script of a Study Course Up to the chapter 6 end / 28.11.2011 Chapter 5 revised/28.11.2011

Prof. Dr.-Ing. Friedrich W. Fuchs Christian-Albrechts-University of Kiel

Kiel, 2010/2011 Revision 2011/12

II

Introduction

Preface Wind power stations are becoming a growing source of energy, with high growing rate in the past, up to 25 % per year, and also assumed for the future. Wind power stations enable a sustainable energy supply with moderate energy costs. These costs are only moderately higher than that of conventional power stations. On the other hand there is an experience of more than 20 years with industrial systems in the field, so it can be called a proven technology. Wind power stations are relative complex systems. Engineering fields are grounding, tower, nacelle, rotor blades, drive train including gear or not, generator and power electronics, transformer, compliance with the standards concerning safety, noise, electrical performance at the mains. Several engineering disciplines are engaged in design and development of wind power stations. There is still immense work in this field. The goals are for example optimizing the system, increasing reliability, creating stations with higher power. Nevertheless, the growing feed in of electrical power from wind turbines in the mains, 8 % as mean value in Germany in2009, 40 % in northern Germany, necessitates stronger standards concerning wind turbine‟s contribution to the grid stabilization. Former wind power stations have been equipped with generators directly feeding into the mains. The next step of evolution have been the dual peed generators (Danish system), with better energy output. However, these systems have strong drawbacks as poor grind performance and strong dynamic mechanical loading. Actually produced higher power wind turbines are equipped with variable speed generators, connected to the mains via power frequency converters. A variety of power electronic generator systems is actually being installed from the manufacturers, while every manufacturer has its own specialty. Here a deep introduction into the power electronic generator systems and grid feed in is given. The content is restricted to the frequency converter fed converter systems, whereupon the given information enables to analyze also directly grid connected systems. Starting with the basics of electrical engineering as three phase systems, the mains components of the PEGS are covered. These are the transformer, the various generator systems as induction machine with short circuit rotor, doubly fed induction machine, synchronous machine with field winding and permanent magnet synchronous machine. The steady state performance of the machines is presented. After that, the power electronic frequency converters are introduced. The dynamic performance of the different systems and exemplary suitable control concepts are shown. Some introduction to the grid feed in and the grid regulations for this are given. Finally the operating control and plant management are figured. The content is intended to be a basic introduction for these readers, who have no knowledge in this field or for those who want to renew their knowledge and bring it up to date, concerning newest technology in wind turbines, or for those, who want to have collected information on the systems to look up if necessary. It is also intended to be readable also for those, who are not very familiar with electrical engineering. Thanks to the members of our institutes team who helped to write down the hand written and bring the manuscript in a good shape. Actually this script is in its first typed version. As it is inevitable, there will be mistakes. The readers are encouraged to inform the author of these mistakes, please via email to [email protected].

Introduction

III

Contents Preface ................................................................................................................................................ II Contents............................................................................................................................................ III List of Symbols and Abbreviations ............................................................................................... VIII 1

Introduction ............................................................................................................................ 14

2

Power Generation by Wind Turbines..................................................................................... 15

3

2.1

Wind Flow ................................................................................................................... 15

2.2

Conversion of Wind Power to Mechanical Power....................................................... 20

2.3

Setup of a Wind Turbine and a Generator System ...................................................... 26

Basic Mathematics and Requirements ................................................................................... 29 3.1

Time dependent representation of alternating values .................................................. 29

3.2

Complex numbers, Phasors ......................................................................................... 29

3.3

Three-Phase-Systems ................................................................................................... 30

3.4

Power ........................................................................................................................... 31

3.5

4

3.4.1

Power in a single phase system .................................................................... 31

3.4.2

Complex construction of the power ............................................................. 33

3.4.3

Power in an arbitrary three phase system ..................................................... 33

3.4.4

Power in a symmetrical three phase system ................................................. 33

Space Vectors .............................................................................................................. 34 3.5.1

abc-to-αβ-Transformation ............................................................................ 34

3.5.2

αβ-to-dq-Transformation .............................................................................. 36

Electrical Machines and Transformers .................................................................................. 39 4.1

Transformer ................................................................................................................. 39 4.1.1

4.1.2

Single-Phase Transformer ............................................................................ 39 4.1.1.1

Basic performance and voltage equation.................................... 39

4.1.1.2

Equivalent circuit diagram, phasor diagram and magnetization 40

4.1.1.3

Losses ......................................................................................... 42

4.1.1.4

No Load and Short-Circuit Operation ........................................ 43

Three-Phase Transformer ............................................................................. 46

IV

Introduction 4.1.3

Exercises Transformer .................................................................................. 48 4.1.3.1

Exercise 1 Voltage transformer .................................................. 48

4.1.3.2

Exercise 2 Transformer; No load- / Short-circuit - Operation.... 48

4.2

The Rotating Field in Electrical Machines .................................................................. 50

4.3

Topologies of Electrical Machines .............................................................................. 51 4.3.1

4.3.2

4.4

Induction Machine (IM) ............................................................................... 51 4.3.1.1

Inductions Machine with Short Circuit Rotor (IM-SC).............. 51

4.3.1.2

Induction Machine with Slip Ring Rotor (e.g. DFIG (IM-DF)) . 52

Synchronous Machine (SyM) ....................................................................... 53 4.3.2.1

Salient Pole Machine .................................................................. 53

4.3.2.2

Non-Salient Pole Machine.......................................................... 54

4.3.2.3

Permanent Magnetic Synchronous Machine (PMSM) ............... 54

Induction Machine (IM-SC) ........................................................................................ 55 4.4.1

General Function .......................................................................................... 55

4.4.2

Voltage Equations ........................................................................................ 56

4.4.3

Equivalent Circuit Diagram and Voltage and Current Phasor Diagrams ..... 57

4.4.4

Current Locus Diagram, Heyland Circle ...................................................... 59

4.4.5

Power ............................................................................................................ 64

4.4.6

Torque .......................................................................................................... 66 Formula of Kloss .......................................................................................... 67 Speed-Torque-Diagram ................................................................................ 67

4.4.7

Speed-Control............................................................................................... 68 Introduction .................................................................................................. 68

4.5

4.4.8

Realization and Application ......................................................................... 74

4.4.9

Exercises....................................................................................................... 75 4.4.9.1

Exercise 1 ................................................................................... 75

4.4.9.2

Exercise 2 ................................................................................... 75

4.4.9.3

Exercise 3 ................................................................................... 76

Synchronous Machine (SyM) ...................................................................................... 78 4.5.1


4.5.2

Mathematical Equations ............................................................................... 80

Introduction

4.6

V

4.5.3

Equivalent Circuit Diagram ......................................................................... 81

4.5.4

Phasor Diagram ............................................................................................ 81

4.5.5

Stator Current Locus Diagram ..................................................................... 84

4.5.6

Power............................................................................................................ 87

4.5.7

Torque .......................................................................................................... 88

4.5.8

Speed control ................................................................................................ 89

4.5.9


4.5.10

Exercises ...................................................................................................... 91

Doubly-Fed-Induction Machine (IM-DF) ................................................................... 92 4.6.1


4.6.2

Equivalent Circuit ........................................................................................ 94

4.6.3

Mathematical Equations ............................................................................... 94

4.6.4

Voltage and Current Phasor Diagram .......................................................... 95

4.6.5

Power and torque.......................................................................................... 96

Torque 97

5

4.6.6

Speed-Control .............................................................................................. 98

4.6.7


4.6.8

Exercise (Wind turbines with doubly-fed induction machine.).................... 99

Three Phase AC to Three Phase AC Frequency Converters for Generators ........................ 105 5.1

Introduction ............................................................................................................... 105

5.2

Voltage Control by means of switching – DC/DC-Converter ................................... 105 5.2.1

Buck-Converter (Step-Down-Converter) ................................................... 106

5.3

Basic DC/AC Converter Functions – Phase Leg ....................................................... 112

5.4

Three Phase Voltage Fed DC/AC-Converters ........................................................... 114 5.4.1

Basic Circuit ............................................................................................... 114

5.4.2

Pulse Width Modulation of the output voltage .......................................... 115 5.4.2.1

Mean Voltage Equivalence Method ......................................... 115

5.4.2.2

Sine-Triangle Modulation ........................................................ 115

Exercise (sine-triangle control) .................................................................. 117 5.4.2.3

Space Vector Modulation ......................................................... 117

VI

Introduction

5.4.3 5.5

5.4.2.4

General Remarks ...................................................................... 123

5.4.2.5

Realization and Implementation ............................................... 124

5.4.2.6

Output voltage of the pwm converter with diode bridge feeding 125

5.4.2.7

Exercise space vector control ................................................... 126

Composed Circuits ..................................................................................... 126

Power Semiconductors .............................................................................................. 126 5.5.1.1

Rectangle-Triangle Modulation................................................ 128

Variants of Pulse Width Modulation ...... Fehler! Textmarke nicht definiert. 6

7

Variable Speed Generator Systems (Stationary Performance) ............................................ 133 6.1

Speed control of induction machines ( repetition) ..................................................... 133

6.2

Operation range of the frequency converter .............................................................. 136

6.3

Drive with DC link converter and induction machine ............................................... 137

6.4

Example ..................................................................................................................... 137

6.5

Exercise...................................................................................................................... 140

Control of Generators in Wind Turbines ............................................................................. 140 7.1

Introduction................................................................................................................ 141

7.2

Basics of Automatic Control...................................................................................... 142

7.3

Electrical and Mechanical Control (Overview) ......................................................... 143

7.4

Closed-Loop-Control of the Induction Machine (IM-SC) ......................................... 144

7.5

7.6

7.7

7.4.1

Equivalent Circuit....................................................................................... 145

7.4.2

Derivation of the dynamic Model............................................................... 146

7.4.3

Block Diagram ........................................................................................... 152

Closed-Loop-Control of the Doubly-Fed-Induction Machine (IM-DF) .................... 154 7.5.1


7.5.2


7.5.3

Block Diagram ........................................................................................... 156

Closed-Loop-Control of the Permanent Magnetic Synchronous Machine (PMSM) . 157 7.6.1


7.6.2


7.6.3

Block Diagram ........................................................................................... 158

Closed-Loop-Control of Line Side Converters .......................................................... 159

Introduction

8

9

VII

Supervisory Control in Wind Turbines ................................................................................ 162 8.1

Operation Modes of Wind Turbines .......................................................................... 162

8.2

Control Strategy of Wind Turbines during Power Optimization Mode .................... 163

8.3

Control Strategy of Wind Turbines during Power Limitation Mode ......................... 165

8.4

Start Up ...................................................................................................................... 166

Requirements and Realization to Wind Turbines Concerning Power Quality..................... 167 9.1

Grid Connection of Wind Turbines ........................................................................... 168

9.2

Grid Codes ................................................................................................................. 168

9.3

Auxiliary Services ..................................................................................................... 168

10

Usedand Recommended Literature ...................................................................................... 169

11

Appendix .............................................................................................................................. 171

VIII

Introduction

List of Symbols and Abbreviations Register of notations and symbols used Symbol

Meaning

German meaning

A

area

Fläche

a

numberof parallel conductors

Anzahl paralleler Leiter

a

three-phase-operator

Drehstromoperator

B

magnetic induction

magnetische Induktion

D

diameter

Durchmesser

D

drilling diameter

Bohrungsdurchmesser

E

electrical field strength

elektrische Feldstärke

F

force

Kraft

f

frequency

Frequenz

H

magnetic field strength

magnetische Feldstärke

I1

stator current at s = infinite

primärer Strom bei s = unendlich

I10

stator open circuitcurrent

primärer Leerlaufstrom

IA

armature current

Ankerstrom

primary related secondary current

primär bezogener sekundärer Strom

imaginary part of a complex value

Imaginärteil einer komplexen Größe

conjugate-complex current

konjugiert komplexer Strom

If

excitation current

Erregerstrom

If´

rotor field current, related to stator

Feldstrom, bezogen auf den Stator

IFe

current, because of iron losses

Strom, der durch RFe fließt

Ig

current of negative phase-sequence system

Strom des Gegensystems

Im

current of positive- sequence system

Strom des Mitsystems

I0

current of zero-sequence system

Strom des Nullsystems

Iμ

magnetizing current

Magnetisierungsstrom

i

peak value of current

Scheitelwert des Stromes

~ I

root mean square value of current

Effektivwert des Stromes

k

magnetic coefficient of coupling

magn. Kopplungsfaktor

Im {...}

Introduction

IX

k1, k2

machine constants of DC machine

Maschinenkonstanten der Gleichstrommaschinen

L

inductance

Induktivität

l

length

Länge

l

axial length of iron

axiale Eisenlänge

L1h

primary main-inductance

primäre Hauptinduktivität

L1ζ

Primary stray inductance

primäre Streuinduktivität

L2ζ´

primary related secondary stray inductance

bezogene sekundäre Streuinduktivität

M

torque (engl. T)

Drehmoment

M

mutual inductance

Gegeninduktivität

m

number of phases

Phasenzahl

N

number of slots

Nutenzahl

n

speed

Drehzahl

P

active power

Wirkleistung

p

number of pole pairs

Polpaarzahl

PD

power of rotating field

Drehfeldleistung

Pi

internal power

innere Leistung

Q

reactive power

Blindleistung

q

number of slots per zone

Anzahl der Nuten je Zone

R

resistance

Wirkwiderstand

R2´

primary related secondary resistance

Primär bezogener sekundärer Widerstand

RA

armature resistance

Ankerwiderstand

Re {...}

real part of complex value

Realteil einer komplexen Größe

Rf

excitation resistance

Erregerwiderstand

RFe

resistance to represent iron losses, primary side

Widerstand, der die Eisenverluste repräsentiert (Primärseite)

Rm

magnetic resistance

magnetischer Widerstand

S

apparent power

Scheinleistung

s

distance

Strecke

X

Introduction

s

slip

Schlupf

ü

transformation ratio

Übersetzungsverhältnis

u, u(t)

instantaneous value (example: voltage)

Augenblickswert, (exemplarisch: Spannung))

arithmetic mean value (of voltage)

arithmetischer Mittelwert

u

ac component of voltage

Wechselanteil der Spannung

û

peak value of voltage

Spitzenwert der Spannung

u

DC-component

Gleichanteil der Spannung

root-mean-square (RMS) value

Effektivwert

ν-th harmonic (RMS-value)

ν-te Harmonische (Effektivwert)

vector

Vektor

u, u

space vector

Raumzeiger

U, UV, ,UL

phase-to-phase voltage

Außenleiterspannung, verkettete Spannung

u2´

Primary related secondary voltage

bezogene sekundäre Spannung

UA

armature voltage

Ankerspannung

Uf

excitation voltage

Erregerspannung

Up

rotor voltage (synchronous machine)

Polradspannung

USt, Uy

phase voltage, star voltage

Strangspannung

û12

voltage peak value between terminal 1 and 2

Spannungsscheitelwert zwischen den Klemmen 1 und 2

w

number of turns

Windungszahl

W

energy, work

Energie

X

Reactance, reactive part of impedance

Blindwiderstand

Xd, X1, Xq

synchronous reactance, transversal reactance (direct, quadrature)

synchrone Reaktanz, Querreaktanz

Ψ

magnetic flux linkage

magn. Fluß(-verkettung)

Z

impedance

komplexer Scheinwiderstand

z

total number of conductors

Anzahl der Leiter

α

pole-covering-factor (conductor in exciting field) Polbedeckungsfaktor (Leiter im

Introduction

XI Erregerfeld)

ε

angle

Winkel

γ

mismatch of two windings

Wicklungsversatz zweier Wicklungen

ε

angle

Winkel

β

electrical winding mismatch

el. Wicklungsversatz

δ

length of airgap

Luftspaltlänge

η

efficiency

Wirkungsgrad

Φ

magnetic flux

magnetische Durchflutung, magnetische Quellspannung

Φ

angle

Winkel

rotor angle

Polradwinkel

μ0

magnetic field constant, permeability (in vacuum)

magnetische Feldkonstante

μr

relative permeability

Relative Permeabilität

ν

ordinal number of harmonic

Ordnungszahl der Harmonischen

ξ

total winding factor

Gesamtwicklungsfaktor

ξZ

zone winding factor

Sehnungswicklungsfaktor

ζ

total stray coefficient

Gesamtstreuziffer

ζ1

primary stray coefficient

primäre Streuziffer

ζ2

secondary stray coefficient

sekundäre Streuziffer

η

time constant

Zeitkonstante

ηN

splitting of slots

Nutteilung

ηp

splitting of poles

Polteilung

cos 

power factor

Leistungsfaktor



phase difference angle

Phasenverschiebungswinkel

Φ

magnetic flux

magnetischer Bündelfluß

ψ

magnetic flux linkage

magnetische Flussverkettung

ω

angular frequency

Kreisfrequenz

XII

Introduction

Register of indices used Index

meaning

1, 2, 3, M, S special points in a circuit M = center, midpoint, S = neutral point, star point

German translation spezielle Punkte einer Schaltung z.B.: M = Mittelpunkt, S = Sternpunkt

0

open circuit

Leerlauf

1

primary

primär

2

secondary

sekundär

Ab, dissip

dissipated value e.g.: Power

abgeführte Größe, z.B Leistung

Cu

copper

Kupfer

D

phase sequence

Drehfeld

el

electrical

elektrisch

Fe

iron

Eisen

GS, DC

direct current (IM brake)

Gleichstrom (ASM Bremse)

h

main-, major-, lead- value, e.g.: magnetizing reactance X1h

Haupt z.B. Hauptreaktanz X1h

Hy

hysteresis

Hysterese

i

internal (e.g. Power)

innere (z.B. Leistung)

K, SC

short circuit

Kurzschluß

Kipp, BO

breakover point

Kippunkt

mech

mechanical

mechanisch

opt

optimal, e.g. optimal operating point

optimaler Betriebspunkt

R, S, T

conductor marking (Phases) in three-wire system

Leiterbezeichnung (Phasen) im Dreileiternetz

res

resulting

resultierend

RS, ST, TR

value between phases R and S, S and T etc.

Wert zwischen den Phasen R und S usw.

ζ

leakage, strain

Streuung

Str,

phase value

Strangwerte

syn

synchronous

synchron

V, loss

losses

Verluste

V, L

conductor value, interlinked value, e.g. phase- Leiterwerte, verkettete Werte

Introduction

XIII

to-phase voltage W

eddy-current

Wirbelstrom

Zu, in

feed value e.g. power

zugeführte Größe z.B Leistung

14

1 Introduction

1 Introduction Electrical energy from wind is a fast growing resource for today‟s energy consumption. In actual wind turbines, variable speed generators are used and thus the power electronic generator system is a key component. This script gives an introduction into this field. The generators used in wind turbines and their associated power electronics are presented for stationary performance. Dynamic performance and control of this electric system are also covered.

2 Power Generation by Wind Turbines

15

2 Power Generation by Wind Turbines 2.1

Wind Flow

Origination of the Wind The sun shines on the earth and warms the air masses. Large-area differences in the radiation, due to the spherical shape of the earth, earth's rotation and tilt, and also clouds, lead to pressure and temperature differences. This causes an air circulation – the winds from the high to the low pressure area and from the cold to the warm temperature areas.

a) Coriolis force pressure Fig. 2.1:

b) Windflow at high and low

Windflow near the ground (northern hemisphere) Source: Gehrtsen (left); Kleemann (right)

Due to the Coriolis force, depending on the rotation of the earth, the countervailing winds in the northern hemisphere are deflected to the right (southern hemisphere to the left). In an extreme case they flow in parallel to the isobars (geostrophic wind instead of gradient wind). Coriolis force: apparent force; wind with a velocity part in the direction rectangular to the earth rotation; to the north flowing, the earth has a lower rotational speed there, i.e., the wind seems to have a higher speed in the direction of the earth rotation.

16


Fig. 2.2: Simplified global circulation of the earth, Source: Kleemann, according to WMO 1981

bright arrows: high winds dark arrows:

winds near to ground

These various factors lead in the latitude range of  30° to the Hadley circulation. Due to strong sun radiation, warm air ascends at the equator, flows toward the poles as southwest high wind and cools down at about  30° latitude. It sinks and flows back as the northeast trade winds, due to the Coriolis force. (south: equivalent, south-west wind) In the remaining region of the earth, the Rossby circulation takes place, predominantly westerly winds that are deflected by the pressure conditions. They are caused by the temperature differences between the 30° and 70° north and south latitude. The Rosby circulation has a higher energy potential than the Hadley circulation. (Rosby: Jetstream) Locally there are many other factors, such as the difference in temperature behavior between sea and land by day and night – wind flowing to and from the sea, water and land regions, mountains, etc., influencing the wind flow.


17

Windflow The wind strength is classified according to Beaufort: Wind strength according to Beaufort

Wind speed m/s

0 1 2 3 4 5 6 7 8 9 10 11 12

0.0 0.3 1.6 3.4 5.5 8.0 10.8 13.9 17.2 20.8 24.5 28.5 32.7

Calm Light air Light breeze Gentle breeze Moderate breeze Fresh breeze Strong breeze High wind Fresh gale Strong gale Storm Violent storm Hurricane force

… … … … … … … … … … … … …

Impact on the environment knots 0.2 1.5 3.3 5.4 7.9 10.7 13.8 17.1 20.7 24.4 28.4 32.6 36.9

0 1 4 8 12 16 22 28 34 41 48 56 64

… … … … … … … … … … … … …

1 3 7 11 15 21 27 33 40 47 55 63 71

Smoke rises vertically Wind motion visible in smoke Wind felt on exposed skin Leaves and smaller twigs in constant motion Small branches begin to move Branches of a moderate size move Whistling of the wind Whole trees in motion Progress on foot is seriously impeded Roof tiles fall down Trees broken off or uprooted Widespread damage to vegetation Windows may break, heavy destruction

Tab. 2.1: Wind strength and wind speed, Source: Kleemann

According to the explanations of the origination of the wind, the medium wind strength varies a lot depending on the place of location as shown in Fig. 2.3

a) World

b) Germany

Fig. 2.3: Average annual wind speed, Source: Kleemann (Abb. 10.4/10.5)

To run wind turbines economically, today a medium wind speed of 4 to 5 m/s is a precondition. This can be found in Germany predominantly in the coast regions, but also in some regions inside the country, i.e. on hills and mountains. (D: Sylt = 7.13 m/s; Passau = 1.86 m/s) Regarding the wind flow worldwide, there is to be stated, that at nearly every coast the necessary wind speed is given but also within some distance far from the coast.

18


The instantaneous wind speed varies considerably during the course of a year. The presentation of the probability takes place over the wind speed with the Rayleigh or Weibull distribution, or summed with the sum frequency distribution, see Fig. 2.4

a) Measurement

b) Rayleigh (Weibull)-distribution

Fig. 2.4: Probability distribution of the wind speed, Source: a) Heier, b) Quaschning Generally the wind speed increases from the ground to the height because of lower friction. Thus, high wind turbine towers can be interesting. The power being contained in the wind, here taken as the power density pW, related to the flown through area A, is proportional to the 3rd power of the wind speed.

wind speed specific weight of the air (

at 25°C)

This results in high power differences at small speed differences as can be seen in tab. 2.2. There is also given the relative energy e per area to be harvested in one year. It can also be seen, that high


19

wind speeds (v1h) generate a high power of the system and thus to the energy, also if they appear only a short period of the year. [m/s] 4 5 6 7 30

pW [W/m2] 64 125 239 343 2700

E = E/A [kWh/(m2. a)] 608 1188 2078 3300

Tab. 2.2: Influence of the wind speed on the wind power pW an the energy per m2 and year

The wind turbine has to be economically designed according to its nominal power at maximum speed in relation to its power at medium speed; and it has to be safety relevant adapted in regard to robustness against destruction at high wind speeds.

20

2.2


Conversion of Wind Power to Mechanical Power

Ideal wind energy converter In an idealized view the wind energy converter is flown through by evenly distributed air. This air drives the rotor and loses in speed. Due to the continuity condition the mass flow is equal at the inlet and outlet of the converter, the pressure as well, but the area flown through is unequal.

Fig. 2.5: Wind flow in an ideal wind turbine in principal

The mass flow in the air in general and in special is given by

specific weight of the air (

at 25°C)

area flown through The power of the air flow, the energy that is flowing through per time, results to:


21

Power (A = πR2)

Power per area

Ideal wind energy converter Taking the pressure of the air to be constant in front and behind the turbine, what is realistic, as the Figure above shows, the power taken out of the wind is equal to: with

The total power contained in the wind cannot be extracted, since the speed of the air behind the converter would be equal to zero and there could be no mass flow. The harvested power is defined in the related form by the power coefficient cp. Reference value is the wind power at the rotor (A2) with unbridled speed (v1). This derivation has been done by Betz (Betz-law).

with Thus, 59.3% of the power in the wind can be harvested under ideal conditions as a maximum. This value is not achieved by real turbines.

22


Wind energy converter with resistance rotor The analysis is done in a simplified example.

Fig. 2.6: Wind flow in an ideal wind turbine in principal

Here the wind force acts on the blown area A (according to Kleemann or Gasch). The resistance force FW takes effect, which contains the face velocity as the difference of the wind and mechanical rotor velocity:

drag coefficient wind speed rotor velocity This results in a dependency of the power coefficient on the ratio of mechanical rotor speed and the wind speed as well as the drag coefficient. The maximum is: with The largest with the resistance rotor achievable power coefficient is 0.34 for a C-profile, which has the largest resistance drag coefficient of cw,max = 2.3. Compared to other types of rotors, this value is very low. The relationship with the value for a maximum in the range of λ = 1.

at the wing tip is called tip speed ratio λ and is always at


23

Wind energy converter with lifting style airfoil Rotor designs according to the resistance principle are not favorable because of their low energy capturing. Instead of this, rotors according to the principle of sailing are used. Better power coefficients can be achieved using lifting style airfoils as rotor.

a) principle

b) oncoming flow

c) forces

Fig. 2.7: Lifting style airfoil, Source: Quaschning, 1998

A rotor in the form of a modern airplane wing is used. The incident air flow of the rotor blade is given by the vector sum of the wind speed vW and the rotor velocity u. The rotor blade is blown with the velocity vA. The velocities of the blade and the incident air flow are larger than the wind speed, similar to a sailboat, which runs diagonally to the wind. In the circulation of the rotor blade with a suitable profile, in particular a slight curve in the direction of flow, similar to airplane wings, it comes in addition to smaller flow velocities on the bottom. This leads to high pressure on the bottom side. Accordingly higher flow velocities cause low pressure on the top side. This generates a buoyancy force FA and also a drag force FW. The resultant force is denoted by FR. The drag force is considerably smaller than the buoyancy force (1% … 0.25%), i.e. normally is FR = FA. The resulting force can be decomposed into a tangential component FRT, which turns the rotor blades, and in an axial component FRN, which is transmitted to the shaft and must be absorbed by the tower. The angle at which the wind hits the rotor must be chosen to achieve the largest possible tangential component. One possibility to maintain a constant angle is to vary the rotor velocity. Since the circumferential speed considerably varies over the blade length, but the wind speed is equal, a differential approach is necessary. Therefore a suitable basic adjustment of the blade angle of every differential blade segment must be found, which thus varies over the blade length.

24


Power coefficients of different rotor types Different rotor types yield to various power coefficients cp. It is declared as a function of the tip speed ratio

circumferential speed at the blade tip wind speed with a value for a resistance rotor in the range of 1.

Fig. 2.8: Power coefficients of different rotor types, Source: Hau,1996

The today commonly used three bladed rotors offer the best power coefficient of the types presented. Nevertheless, four or five blade rotors – not shown here - seem to show even greater coefficients.


25

Control of the power extraction Wind direction tracing To achieve maximum power extraction, the rotor shaft has to be moved parallel to the wind (for horizontal-axis wind turbines). Therefore the nacelle can be turned using a crown gear with a servo motor.

Pitch angle control and variable speed for power optimizing To achieve maximum power extraction out of the wind, the pitch of the blades has to be optimal. This implies that the relation between circumferential velocity (rotor speed) and wind speed has to be optimal. There are three typical methods of control: 





Control of the pitch and speed In order to optimize the wind turbine for a large wind speed interval, the pitch of the blades and the speed must be controllable. Therefore a system to adjust the pitch of the blades and a converter fed machine shall be provided. This type of system becomes more popular nowadays. Control of the pitch To run wind turbines with widely fixed speed as optimal as possible, i.e. directly grid coupled machines, the pitch of the blades needs to be controllable. However it is not possible to maximize the power output. Rotor without adjustment of the pitch (stall-operating) The power output of wind turbines with fixed rotor blades is even smaller, because the mean deviation from the optimal point is much bigger.

Power limiting Power limiting is required in the case of excessive wind. This can be done by turning the rotor shaft out of the wind, by using the blade adjustment (pitch) or happens because of the stall-effect of the fixed bladed rotor.

Power speed diagram of a rotor system In order to design a wind turbine the power P, the torque M and the thrust load FRA are of particular importance. They depend for a given blade specification on wind and circumferential speed. For the rotor the number and shape of the blades, the length, the width and the basic blade angle over the blade length are important. The calculation, e.g. according to Schmitz, is complex (aerodynamics). The general description of an airfoil is given by a dimensionless power coefficient cp as a function of the tip speed ratio  and for the design of a real wind turbine the power P as a function of the rotor speed N. For fixed bladed turbines the switch off coefficient aus dim is used for dimensioning the turbine.

26


a) dimensionless diagram (1: Vestas V44-600) Fig. 2.9:

b) dimensionful diagram

Power-speed-diagrams of a wind turbine without blade adjustment dimensioned tip speed ratio  = 7.5, Source: Quaschning

The trajectories can be approximated with suitable fitted equations.

2.3

Fig. 2.10:

Setup of a Wind Turbine and a Generator System

Wind power station with most necessary components (nascelle contains converter and sometimes , not shown here;


27

Fig. 2.11: Setup of a wind turbine and its basic components- U.S. Department of Energy, EERE

Fig. 2.12: Mechanic setup with gear box in 3D view

(Nacelle additionally equipped with converter; transformer often located on the bottom)

28


Drive Train Fig. 2.13 shows the setup and the components of the drive rain: rotor, gearbox if included, generator, transformer, and filter. Additional contactors and fuses are both shown here.

Fig. 2.13: Setup of a wind turbine and its basic components U.S. Department of Energy, EERE

In general, three different basic turbine and drive train concepts can be found in the megawatt range. These differ from low, average and fast generator rotor speed. Example

Rotor speed

Gearbox

Generator speed

Weight of nacelle (offshore)

Standard

Fast

2-ary

Normal (1500 min-1), different types

450 t

Multibrid

Average

1-ary

Enercon

slow

none

Slow (approx. 750

min-1)

Very slow (30 min-1)

Tab. 2.3: Overview of different wind turbine concepts

350 t 550 t

3 Basic Mathematics and Requirements

29

3 Basic Mathematics and Requirements 3.1

Time dependent representation of alternating values

Electrical power systems contain sinusoidal alternating voltages u(t) and currents i(t) as for example shown in the equations. Here and are peak values. Steady state performance is assumed, so the angular frequency and the magnitude are constant.

They can be shown as for example in Fig. 3.1.

Fig. 3.1: Time dependent representation of voltage and current (

3.2

Complex numbers, Phasors

Complex numbers or phasors are often used to calculate in electrical energy or communication systems. This representation is valid only in steady state. A complex number or phasor U, marked by underlining, consists of a real part URe and an imaginary part UIm, where j represents the imaginary prefix:

The time dependent representation of for example the voltage can be transformed into the complex representation for a given angular frequency ω by taking the imaginary part in the following way where the equation is used.

For the complex number or the phasor from inside the brackets the root mean square value of the voltage U and the phase angle are taken, not the angular frequency. This must be known

30


separately. Thus the phasor of the voltage is written as below. The same applies for the current or other electrical values.

Thus, the time dependent representation is equal to the representation with phasors or complex numbers, when for the latter the angular frequency is known. This representation is only valid in steady state.

Fig. 3.2: Voltage and current in the complex plain (

3.3

Three-Phase-Systems

For each electrical system with higher power (> 5 kW), a three phase system is used. It is a symmetrical sinusoidal system where the three phases have the same magnitude and a phase shift of 120° to each other.

Fig. 3.3: Three phase symmetrical system

A three phase system has three phases, here called a, b and c, and is described by the following equations, here for example for the voltages:


The voltage represents that of the first phase, for phase three. The magnitude is given by included.

31

describes it for phase two and , the phase shift equal to 120° or

is

These voltages or the currents in the same way can be represented as phasors, according to the previous chapter. Therefore the phase shift between the phases can be written as

where a is called three phase operator.

3.4

Power

3.4.1

Power in a single phase system

The momentary value of the power s(t), being dependent on the time, is defined by the following:

Inserting a sinusoidal voltage and current the time dependent function can be formed.

This equation shows a constant component with double the angular frequency ω, with following figure.

and a component alternating with . This performance is shown in the

Fig. 3.4: Picture of the time dependent power s(t)

Further decomposition leads to the following:

32


The components with are denominated time dependent real power p(t). One subpart is a constant part (multiplication with 1 in the brackets). There is also a part oscillating cosinusoidal with double angular frequency around the former constant value. The components with are called time dependent reactive power q(t). They alternate sinusoidally with double angular frequency around zero. An example is shown in the next figure.

Fig. 3.5: Instantaneous power s(t), p(t) and q(t)

The power which is able to execute work is the active or real power p(t). Its mean value is usually unequal to zero. Thus, the active power, not time dependent, is called P and defined as the mean value of the instantaneous power s(t):

This mean value can be taken from the equation s(t) above. With the definition real power P can be written as:

The amplitude of the oscillating component with the factor Q:

The instantaneous power, time depending, can thus be written as:

, this

is called reactive power

3 Basic Mathematics and Requirements 3.4.2

33

Complex construction of the power

The different power definitions can also be calculated by means of phasors of voltage and current. With the phasors and . Here means the conjugate complex value of , charateristic is the minus-sign in front of the imaginary part. Thus it can be written: Real power:

Reactive power:

Complex apparent power:

Apparent power, absolute value:

3.4.3

Power in an arbitrary three phase system

The total power of an arbitrary three phase system is the sum of the powers in the three phases.

Systems in “star connection” Systems in “delta connection” For a star system this is written as:

3.4.4

Power in a symmetrical three phase system

A symmetrical power source and drain is precondition. With the equation of the complex apparent power, at the period of time with the sum of the total transferred power at symmetric load is:

34


+

The symmetrical three phase system always contains always constant real power and is therefore optimal, as there are no power fluctuations. The reactive power of all three phases added becomes zero, as seen from the equation, nevertheless, in each phase the reactive power is still there.

3.5

Space Vectors

3.5.1

abc-to-αβ-Transformation

Electrical values of three phase systems without a zero component, which can be assumed in most fault free cases, can be transformed into a two phase system for easier analysis. Space vectors are used to present electrical values especially for dynamic or control purposes. They are generated via the following formula, where x stand for voltage u or current I or other values, where the underlining marks the space vector. Other markings are for example an arrow underlining.

The scaling factor of gives the space vector the same size, the phase value in time dependent representation actually has. Transformation procedure:

Fig. 3.6: Transformation from three to two phase system

The origin of space vectors is the analysis of electrical machines and their fields. It can be demonstrated with the figure above. A three phase winding system a,b,c in an electrical machine is


35

shown on the left, producing a magnetic field in the plain (real: in the volume). This field could also be generated by means of the two phase alpha/beta winding system on the right. The currents of the left system have therefore to be transformed to the right one in the right way. This procedure can be applied to all electrical values. Given the voltages of the three phase system:

The transfer law to the alpha/beta system in real and imaginary part, taking the star voltages, against the midpoint (Index M), is:

The precondition is, that there is no zero system, which can be defined as:

This is equal to the matrix representation:

This transformation law is valid not only for voltages, but for all electrical values. It is called Clark-transformation. It can be developed out of the basic equation in the beginning of this chapter:

This space vector is a complex, time varying vector. It can be separated in its two components, a real part and an imaginary part. with: An example for this space vector in the alpha/beta coordinate system can be given. If there is a symmetrical sinusoidal voltage, the space vector is given by:

36


This means a pointer of constant length rotating with constant angular frequency. This is shown in the next figure. It looks comparable to the field in an electrical machine.

Fig. 3.7:

Space vector of a symmetrical sinusoidal voltage (left); decomposition to the phase voltages (right)

The one pointer that gets created has the two components maximum of the linked three phase system.

3.5.2

and

and it always points to the

αβ-to-dq-Transformation

For purpose of analysis and control it is often useful to transform the values from the fixed αβ coordinate system to a rotating coordinate system, here called dq. One specialty is, that with suitable selection of the coordinate system, the values in the dq system can be dc values in steady state, easy for control. Therefore, the coordinate system is positioned to lay in the direction of space vector x, the transformation angle has to be chosen in this way. For a transformation from the -system into the dq-system, the following transformation equation is to be used, here x is the representative for any electrical value:

and reverse:

In order to get straight from the abc-system to the dq-system this transformation can be used:

This transformation matrix is the so called Park-Transformation. It was developed in 1920 by R.H.Park. It is also used for the inverse transformation:


37

The following figure shows the transformation from the αβ and to the dq coordinate system

Fig.3.8: Transformation into a rotating reference frame

By superscripts it can be indicated, in which coordinate system the value is analyzed. For example means that the value x is represented in the dq coordinate system. This coordinate transformation

Typically, dq-frame and

can also be writen as:

have the same rotational speed

So, have only DC components (in steady state) and if there is only a real component of the space vector in the dq system. Space vectors are especially used for dynamic operation, where they are fully valid.

38


For later use:

Fig. 3.9: Space Vectors

The mathematical implementation of the dq-transformation is done by the multiplication of the signal vector with the alternating values of the abc-system and the Park-transformation matrix. If the alternating values are given with their derivation a combination of d and q can be useful. This leads to the following equation.

4 Electrical Machines and Transformers

39

4 Electrical Machines and Transformers 4.1

Transformer

Transformers are used to transform a voltage from a higher to a lower value or vice versa, to galvanically isolate to electrical circuits or to do both. In energy generation, transmission and distribution as well as in electrical devices for industrial and private use transformers are a standard component. The same applies for wind power stations: Usually the power of the wind power station is generated at lower voltage and via transformer transformed to a higher voltage of the grid. 4.1.1

Single-Phase Transformer

4.1.1.1

Basic performance and voltage equation

A transformer consists of an iron core with primary and secondary windings laid around it. The core itself consists of the vertical bars and the horizontal cover. The iron core, its cross section area and length, as well as the windings, their turn number and cross section area, define the transformation behavior of the transformer. The currents in the windings generate magnetic flux. The flux is to the right circulating the current. There are parts of the flux that flow through primary and secondary winding (main or mutual flux Φ12, Φ21) and parts that are connected only to one winding (stray flux Φ1ζ, Φ2ζ).

Fig. 4.1:

Two magnetic linked choke Fig. 4.2: coils with iron core

General and communications engineering equivalent circuit diagram

The voltages in primary and secondary winding can be determined by the derivations of the fluxes via time.

The ohmic voltage drop, small losses because of the electrical resistances R in the stator and rotor windings, has to be added:

40


To make technical calculations, it makes sense to have u1 and u2 in the same dimension. For most transformers there is a voltage step up or down function of the transformer (e.g.: u1 >> u2 or u1 0, overexcited

There are four typical operation modes, that can be well characterized in the phasor diagram. These are the no load, generator and motor as well as the phase shift operation. They are shown in a combined figure. Another important point is the kind of reactive power at the stator of the machine and how it is generated. There is operation possible in over excitation and under excitation. The excitation is


83

done via the rotor current and its resulting inductor voltage Up. If the projection of the inductor voltage on the stator voltage is bigger than it, this is called over excited, in the other case under excited. This operation is shown in the diagram as well as for motor as for generator operation.

Fig. 4.42:

Phasor diagrams of the synchronous machine with field winding, R1 neglected; Top left: no load operation, I1=0; top right: phase shift operation, θ = 900; medium left: generator operation, > 00, underexcited; medium right: generator operation, > 00,

84

4 Electrical Machines and Transformers overexcited; Bottom left: motor operation, operation, < 00, overexcited

< 00, under excited: bottom right: motor

In no load operation, the generator does not generate any torque, the pole displacement angle is zero. The stator current I1 is zero, thus the stator U1 and inductor voltage UP have to be equal, otherwise a current would flow via the synchronoues reactance. The resultant diagram is shown. In phase shift operation, no active power but only reactive power is generated or taken up. Thus, the stator current lies in the imaginary axis. The voltage drop at the stator reactance is rectangular to the current and thus in direction of the stator voltage with . Stator voltage and inductor voltage thus have the same direction but different amplitudes. This operation shows well the ability of the machine to generate or take up reactive power, which is also the case in motor in generator application. In generator operation, a power and relevant torque has to be applied mechanically to the rotor shaft. The mechanical power is transformed to electrical power and fed out of the machine via the stator. The current is in the lower half of the complex plain, thus for consumer representation a generator operation is given here. Because of the now flowing stator current I1, a voltage arises at the synchronous reactance. The voltage difference between these two sources leads to a phase shift of the rotor voltage to the stator voltage of the grid, described by the pole displacement angle , which is positive in generator operation. On the other hand this angle is necessary to have a phase shift between stator and rotor field to be able to generate a torque. In generator operation the rotor leads the stator field. When the stator current lies in the right complex half plane, this indicates a consummation of inductive reactive power. In motor operation, the machine takes up power via the stator, converts this to mechanical power and feeds it into the rotating shaft. Again a pole phase displacement occurs, now in the opposite direction as in generator mode, the value is negative. The rotor runs behind the stator field. There is a voltage drop at the stator inductance, so inductor and stator voltage have a phase shift and maybe different amplitudes. When the current lies in the right complex half plane, this indicates a consummation of inductive reactive power.

4.5.5

Stator Current Locus Diagram

The current locus diagram shows the current phasor for different operating points in the complex plane, similar to the Heyland circle for the IM-SC. The stator voltage and frequency is assumed to be constant. The stator current can be determined by the complex difference of the stator and inductor voltage with the parameter stator reactance .

From this equation the stator current locus diagram can be drawn. The first component in the equation is constant and positioned in the negative imaginary direction as shown in the diagram. The second component is variable, in amplitude and phase. So for constant amplitude and varying phase the current locus is concentric circles around the top of the phasor of the first component with different radiuses. Three circles are drawn for example.


85

Fig. 4.43: Current locus diagram of the synchronous machine with field winding

The addition of both components yields the stator current. As seen in the diagram, the phasor of the stator current begins in the origin. Dependent on the amplitude and phase of the inductor voltage the machine operates in capacitive – over excited or inductive – under excited mode. For [projection of UP on U1 U1] the machine is under excited, in the other case over excited. The pole displacement angle varies with the position of the inductor voltage UP. For all machines it is necessary to have a look at the allowable operations. These operations are depending on the maximum voltage and current and the stability. These limitations are e.g.: 

stability limit ①



maximum allowable UP ②



maximum stator current ③



maximum driving power (would cause a limitation on the Re-Axis (vertical line))

Allowable operation:

86

4 Electrical Machines and Transformers Limits: 1. stability reasons 600 lines 2.

p

p,max

voltage limit Circle around – with

p,max

3. Circle around origin with

(rated

current)

Fig. 4.44:

Allowable operating range (shaded) for the synchronous machine with field winding in the stator current locus diagram

4 Electrical Machines and Transformers 4.5.6

87

Power

The active power and reactive of the synchronous machine are derived from the equations of the stator power. Here m denominates the phase number, usually this is 3. Stator and iron losses are neglected, so this is equal to the mechanical power.

These equations are modified according to interrelations in the phasor diagram as shown for the current locus diagram, given in the next figure.

Fig. 4.45: Special phasor diagram of a synchronous machine

These interrelations are:

Where the minus sign in the b equation results from the directions and algebraic signs of the angles. These expressions set into the base equations give:

The active stator power, equivalent to the mechanical power, and reactive power can now be written based on the basic values of the synchronous machine, which are the stator voltage , the inductor voltage and the pole displacement angle .

88 4.5.7

4 Electrical Machines and Transformers Torque

The torque can be developed from the mechanical power torque equation combined with the stator power equation from the last section: for

kipp

kipp

Similar to the power, the torque is dependent on and . The torque depends sinusoidally from the pole displacement angle. The maximum torque, the break over torque, is generated at an angle of . The figure shows the torque characteristic. The stability for stationary operation is according to this figure given up/down to an angle of +/-900. For lower angles, the torque decreases for increasing load angle equal to increasing loading, the machine falls out of synchronism. The same applies for higher load angles. In practice, the limit value is 600 or lower.

Fig. 4.46: Torque versus pole displacement angle diagram of the synchronous machine

Speed-torque-characteristic The synchronous machine supplied from the mains can be operated only at one speed with variable torque. This is shown in the following diagram.


89

Fig. 4.47: Torque speed characteristic of the synchronous machine with fixed stator frequency

4.5.8

Speed control

The machine is fed with variable stator voltage and frequency by means of a converter. The machine is fed as derived for the IM-SC with constant stator flux, leading to the control law: const. The stator voltage is to be controlled proportional to the stator frequency. This is valid down to some 5 % of the nominal speed, as in the low region the influence of the stator resistance has to be included. The realization of this voltage frequency control is the same as shown for the IM-SC. It should be noted, that this simple control law is presented here for general introduction. More accurate control methods are presented in a following chapter. The natural torque speed curves of the variable speed converter fed synchronous machine are parallel vertical lines as shown in the figure, according to fixed frequency for each line.

Fig. 4.48: Torque speed characteristic of the synchronous machine with voltage frequency control

90 4.5.9

4 Electrical Machines and Transformers Realization and Application

4 Electrical Machines and Transformers 4.5.10

91

Exercise 1

A pole synchronous machine is installed in a little pump storage hydro power plant. The star connected machine has the following nominal data: f1N = 50 Hz X12

nN = 375 min-1

X1h= 7.5 

The machine is operating as an overexcited generator with nominal values: a) Draw the two different types of equivalent circuit diagrams of the synchronous machine. What do it‟s elements stand for? b) What is the number of pole pairs and the value of the synchronous reactance X1 of this machine? In the following considerations the ohmic losses of the stator windings can be neglected. c) Evaluate the nominal stator current I1N (magnitude and phase), the nominal torque MN as well as the reactive power QN and the apparent power SN. d) Draw the complete current- and voltage phasor diagram of the synchronous machine (working as a generator) at the nominal operation point to scale. Evaluate the armature voltage UP and the rotor angle N as well as the magnetizing current I and the exciting current

referred to the stator of the synchronous machine. (Scale: 1cm



100 A; 1 cm

 500 V) In the case of not operating the machine at the nominal point, the boundary values are as follows: -20° <  < 35° I1max = 1.25  I1N Beside of these operating limitations the maximum field current is limited

.

e) Which value has the armature voltage, if the machine is operated at the operation limit  = 35° with maximum exciting current ? f) Plot all operation limitations for the armature voltage UP into the phasor diagram. The machine is used as a phase shifter. The limitations are still the same. g) How much reactive power can be absorbed by the synchronous machine in phase shifter operation and how much can be delivered? Mark the operation points of the armature voltage for these cases. The machine is fed via a converter with voltage frequency control. The machine is running at 50 % of the nominal speed with 50 % torque in generator mode. h) Calculate all electrical and mechanical values (U1,UP,I1, M, n) and draw the complete current- and voltage phasor diagram of the synchronous machine. (Scale: 1cm



100 A; 1 cm

 500 V)

92


4.6

Doubly-Fed-Induction Machine (IM-DF)

4.6.1

General Function

Induction machines with slip rings have been utilized in the past in fixed speed applications for example for high start up torque requirements. The adaption of the start up torque to the demands can be done via external connected rotor resistors. Actually the slip ring induction machine is utilized in highest power variable speed application, for example in variable speed pumping power stations, and in variable speed wind turbines. The sator is equipped with a standard three pahse winding as known from induction or synchronous machines. The rotor is equipped with a three phase winding of the same pole number as the stator, too. The three connections of the rotor winding can be accessed via splip rings and brushes, see in the figure. For both, either a star or a delta connection is possible. Slip rings in former times had short operation times, but actually there are systems with a warranted liftetime of two years.

Rotor

Stator

Fig. 4.49: Induction machine with slip rings and brushes


93

In this case the stator of the induction machine is directly connected to the grid. The rotor is connected to the grid via its slip rings and a frequency converter. This gives the possibility to feed the rotor with variable voltage and frequency and make variable speed operation possible. Because of the feeding of stator and rotor the machine is called doubly fed induction generator (DFIG) or doubly fed induction machine (DFIM). For wind power stations the rotor usually is connected directly to the medium voltage mains, while the rotor is fed via a low voltage converter and this is connected via a transformer to the medium voltage grid (10 k … 33 k ).

Fig. 4.50: Equivalent circuit of the doubly fed induction machine

b) No-load operation: fR  a) Area of operation

c) Big small fR

Small n, n, d) big fR

94 4.6.2

4 Electrical Machines and Transformers Equivalent Circuit

The performance of the DFIG can easily derived based on the simplified equivalent circuit diagram developed from the basic diagram of the induction machine. Therefore, the stator resistance RS and the stator stray inductance LS are neglected. The machine is fed at the stator side from the mains and at the rotor side via a converter.

Fig. 4.51:

Equivalent circuit diagram of the doubly fed induction machine for derivation of the performance

The rotor resistance can be subdivided in the real stator referred rotor resistance and the remaning part, as shown right in the figure. In general, the equivalent circuit diagram shown above, without , can be derived from that one with and only the stator resistance neglected by some special transforming, for example to be found in machine literature. The equivalent circuit diagram has been derived. Here the rotor feeding is added compared with the previous analysis.

4.6.3

Mathematical Equations

The rotor current can be determined via the voltage difference over the rotor impedance: –

With positioning the stator voltage in the real axis parallel and vertical component

and separating the rotor voltage in a this is modified to:

The equation is expanded with the conjugate complex value of the denominator to get a real denominator:

This results in the rotor current equation:

4 Electrical Machines and Transformers a)

b)

c)

d)

95 e)

f)

The equation contains six components. Two components are depending on the stator voltage (a, d), four others are depending on the rotor voltage (b, c, e, f). There is a real part of the current, in the direction of the stator voltage, this are the components a, b and c. There are imaginary components, vertical to the stator voltage direction,these are the components d, e and f.

4.6.4

Voltage and Current Phasor Diagram

The rotor current generates a voltage at the rotor resistance and rotor stray inductance. This voltage is the difference between the stator and rotor voltage, as can be seen in the circuit diagram. From this relationship the voltage and current phasor diagrams can be wrawn, as shown in the figure.

Fig. 4.52:

Voltage and current phasor diagrams of the doubly fed induction machine for different operation points; from left to right: 1) no load, no stator current; 2) stator parallel to lower stator referred rotor voltage; 3) stator parallel to higher rotor voltage; 4) phase shift between stator and higher rotor voltage; 5) phase shift between stator and higher rotor voltage

96 4.6.5

4 Electrical Machines and Transformers Power and torque

Power The different power parts of the doubly fed induction machine are compiled here: Stator power Stator losses 1

1

„+‟ for generator operation

Rotor power losses Power via converter mech

2

3

2

„+‟ for generator operation

3

„+‟ for supersynchronous operation

The active airgap power for neglected stator resistance and stator stray inductance is equal to the active stator power. It can be derived by means of conjugate complex of the imaginary part of the rotor current above:

The equivalent is valid for the reactive power in the airgap

The mains active power is the sum of the sum of stator and rotor active power and, as the losses in the machine are neglected, equal to the mechanical power with included efficiency: mains

mech

This mains active power is controlled by means of the rotor side converter via the active and the reactive part of the rotor voltage. Equivalent summation and control is possible for the reactive power, It is difficult to get a good insight to the behavior. The general behavior can be visualized by the power flow diagram of the machine.


97

There is power flow into or out of the stator and the rotor, according to the operating point. The power flow can be illustrated as follows:

Motor operation, subsynchronous

Generator Operation subsynchronous

Generator Operation supersynchronous

Fig. 4.53:

Power flow in stator and rotor of the doubly fed induction machine for different operating modes

Torque The torque can be derived from the active air gap power: mech

98


Another way to determine the machine behavior is a voltage control ratio ü and the angle between stator and rotor voltage α:

This gives for

For

the no load rotor frequency depending on the rotor voltage .

this gives see diagram voltage control

4.6.6

Speed-Control

In case of a reference of zero active and reactive stator power and the equation for the airgap power must be 0. With another assumption for the imaginary part of the rotor voltage this gives.

The rotor voltage amplitude has to be controlled proportional to the varying rotor frequency, assumed that stator voltage and frequency stay constant. This is shown in the following diagram.


Fig. 4.54:

99

Rotor voltage of the doubly fed induction machine at speed control for simplifying conditions

The typical operating range for wind turbines is 30 % around the medium speed and around rotor frequency 0 Hz, so the rotor frequency lies in the range:

Example: A 2 MW wind turbine with doubly fed induction generator has a stator nominal voltage of 6 kV and is equipped with a rotor side converter with a nominal voltage of 690 V, connected to the 6 kV mains with a transformer. With the simplifying conditions the maximum rotor no load voltage is . In this operating point the maximum current is the machine nominal current. Under the condition of the same displacement factor the rotor-side converter power is calculated to conv .

4.6.7

Realization and Application

4.6.8

Exercise (Wind turbines with doubly-fed induction machine.)

Wind turbines convert a part of the kinetic energy of the wind into electrical energy. To get best use of the windturbines, a maximum conversion to electrical energy is to be achieved. This is possible when the rotor speed is adjusted. The conversion quality is defined by means of the power coefficient cp. The relation between the power coefficient and the tip speed ratio λ (relation between the rotor circumferential speed and the wind speed) is shown in the diagram Fig. 4.55. It shows the power coefficient of a typical wind power station with three blades and typical rotor blade design. The power coefficient cP has its maximum in the range of the tip speed ratio 7 to 9. In general the power coefficient is also depending on the aerodynamic of the blade

100


Fig. 4.55:

Power coefficient

For an optimal energy coefficient, generators with an adjustable speed are installed. A possible concept is the combination of a gearbox and a doubly fed induction machine, where a power converter is connected on its rotor side. The benefit of this construction is, that the power converter has only to be constructed for the slip power of the DFIM. Therefore it is smaller and cheaper than a full inverter.

The kinetic energy of the wind

can be assumed by the area of the rotor AR passing

through the volume and the air mass The derivation for the time leads to the power of the wind.

.

According to the law of Betz, the maximum energy that can be used of the wind is only 59%. The usable energy (optimized for the tip speed ratio ) based on the total power of the wind, leads to the power coefficient cP. Modern facilities are able to have a power coefficient of cP = 0.5. Table 1 shows a part of the technical data of the turbine of the company DeWind. The given exercise relates to this turbine.


Turbine:

101

Generator:

Rotor diameter

80 m

Slip region

Hub height

80 m

Rated Frequency

50 Hz

Leaf number

3

Rated Voltage, delta

690 V

Rotor area

5027 m2

Rated Current, line

1675 A

Cut in wind speed

3 m/s

Power Factor

cos

Nominal wind speed

13.5 m/s

Cut-out wind speed

25 m/s

Functional wind speed

57.4 m/s

Rated speed

18.0 U/min

Speed range

11.1 – 20.7 U/min

Speed control

Pitch

Gear

One stage planetary gear

=1

Two stage spur gear Gear ratio

1:94.4 Table 1: Technical data of the windturbine DeWind D8 (2.000 kW)

For the considered doubly fed induction machine additional data for the calculations have to be known. These are not given by the manufacturer. Stator inductance

L1

6.0 mH

X1ζ 0m

Rotor inductance

L2

28.3 mH

X2ζ‟= 28 m

Coupling inductance

M

12.7 mH

1,15

Stator resistance

R1

Rotor resistance

R2

0.15

2,12 m

=

At 50 Hz

102


Table 2: Technical data of an induction machine with slip ring rotor, suitable for an windturbine with a nominal power of 2.000 kW

The following exercises have to be solved: a.) What peripheral speed vBlade, Tip do the tips of the tree blades have at rated speed? How large is the tip speed ratio λ at rated speed? b.) Determine for the rated operation the power of the wind PW (density of the air is ρAir = 1.2 kgm-3) and with the power coefficient cP = 0.5 the total efficiency η of the plant. c.) By means of a speed control of the turbine, the speed ratio in the optimum of λN = 5.585 is kept constant. Determine with an assumption for the efficiency η = ηN the power delivered to the grid at the turning on and off speed. d.) Determine the number of pole pairs p of the generator. e.) Draw the equivalent circuit diagram of the induction machine with eliminated stator stray inductance with the assumption of I0 = I1 +I‟2. The resistance of the stator R1 does not have to be taken into account. Determine the elements of the equivalent circuit diagram. f.) The generator shall be connected via a power switch to the grid at 11.1 min-1 rotations per minute. Name the conditions, so that there is no current flow through the stator. What are the conditions for the rotor and stator current? g.) How great is the rotor current at the rated point and what rotor voltage has to be supplied? Draw a phasor diagram and a power flow diagram.


103

For later use Drive in the rotor circuit is most often an IGBT power converter. With the converter, connected to the rotor, the voltage link on the rotor side can be switched. Pulse-like sinusoidal voltages and currents of variable frequency and amplitude can be switched into the rotor circuit. Thus it can be fed back into the grid from the rotor circuit energy. By means of a suitable rotor pre-resistance the starting current can be reduced from values of 5-6 ISN down to about 2-4 ISN for operating with the breakdown torque. Because of the power losses in these resistances the method is not appropriate for continuous operation.

Fig. 4.56: Influence of the series resistance on the speed-torque-diagram

Another operation method of the slip ring induction machine is the sub-synchronous cascade (Static Kramer Cascade). Here a grid commutated back-to-back converter is connected between rotor and grid and can take out or deliver in power to the rotor. By means of this a speed control in the sub-synchronous operation is possible with low losses, applicable for continuous operation.

104


Fig. 4.57 Speed-torque-diagram of the sub-synchronous cascade (Static Kramer Cascade)

5 Three Phase AC to Three Phase AC Frequency Converters for Generators

105

5 Three Phase AC to Three Phase AC Frequency Converters for Generators 5.1

Introduction

To feed three phase AC machines for variable speed operation, there is a need for control of the voltage in amplitude and frequency. This control today is achieved by means of frequency converters feeding the machines from the mains. The conversion process is done by switching the voltages by means of electronic switches on and off very fast. The relation of on and off times determines the short time mean of the output voltage. In the same way the frequency can be controlled. The advantage of this conversion or voltage control by means of switching instead of other methods is its excellent efficiency. The switches are power semiconductors as transistors, for example IGBT (Insulated Gate Bipolar Transistor) or IGCT (Integrated Gate Commutated Thyristor). They can switch on and switch off the current or voltage by their own. Because of this ability of self switching on and off this type of the converter equipped with these switches is called self commutated converter. The switch usually enables a current flow in one direction only. A diode is connected antiparallel to the power semiconductor that can be switched on and off, for carrying the inductive part of the current (free wheeling diode). Further energy storage and filter elements are needed as inductors and capacitors. To understand the functionality, it is indispensable to remember the mode of operating of inductors and capacitors. Both elements can work as a load or as a source. At capacitors the voltage cannot vary abruptly, for inductors the current cannot vary abruptly. The basic analysis of the power electronic converter circuits is aimed on the general operational performance which is given by the voltage and current shape at input, output and maybe inside the circuit. For this analysis of the power electronic circuits, assumptions are taken, if not specified in another way, which do in general not influence the shape of voltages and currents in a relevant way:   

5.2

Constant source- and load voltage with no internal resistance Ideal power semiconductors (zero switching time, zero on state voltage when conducting, no blocking losses) Ideal connection circuits (zero resistance and zero stray inductivity)

Voltage Control by means of switching – DC/DC-Converter

For introduction into the field of power conversion, voltage frequency conversion by means of switching, a DC/DC-converter is presented. DC/DC conversion is an essential part of every device needing dc power which is powered by battery or by dc solar energy source. DC/DC converters are for example also used in special types of Windturbines. This is the case, when the voltage of a permanent magnet synchronous machine feeding into a dc circuit via a diode

106


rectifier is to be stepped up or boosted to a higher value to be able to feed the power to the mains via an inverter. The task of a DC/DC converter is to convert a given voltage either to a higher or a lower level.

5.2.1

Buck-Converter (Step-Down-Converter)

The following diagram shows the equivalent circuit diagram of a DC/DC converter of the buck type or step-down type. This means, that the output voltage Ud2 is lower than the input voltage Ud1. Here, V1 is the on and off switchable power semiconductor. It is represented by means of the diode symbol with two gate connections, as a general symbol for power semiconductors which can be switched on and off (switchable power semiconductor). The element V2 shows the power electronic diode. The power electronic circuit additionally consists of output side or load inductance and a voltage or current load.

Fig. 5.1:

DC/DC chopper, buck type (step down), equivalent circuit diagramm; with dc voltage load (left) and capacitor and dc current load (right) (V1 power semiconductor, able to switch on and off; V2 power diode)

By means of the chopper circuit the power is transferred from voltage source Ud1 to voltage source Ud2, where the voltage is lower than of the first one. Another point of view is to regard the load as seen on the right of the figure as a capacitor, which is loaded with a current sink. The control of the mean value of the voltage ud is done by switching the power semiconductor V1 on and off. The relationship of on and off time determines the mean output voltage. This switching is usually done periodically. This can be illustrated with the circuit diagrams in the two parts of the next figure. During parts of the time period only selected sections of the entire circuit are in operation. Thus, for these parts of the time period special circuit diagrams are taken only comprising the active components. During on-time TE, the left figure is valid, power semiconductor V1 is switched on and conducting and power semiconductor V2 is blocking. The current is flowing from the source to the load. In the remaining time, period time minus on-time T – TE, here the right figure is valid, the power semiconductor V1 is switched off and blocking, as its voltage is negative. The current id2 flows through V2. No current id1 flows in the primary voltage source at this time period.


id1 Ud1

Ld V1

Ld

id2 Ud2

id 0 < t

= const.

≠0 finite, current pulsed on-time

Ud

cycle duration duty factor of pulses

TE (theoretical)

T

t

id2

conducts blocks

rises

t id1

blocks

t1 t 2

t

conducts

decreases

Fig. 5.3:

DC/DC chopper, buck type (step down), variation in time of electrical values for TE/T = 2/3; from top to down: output voltage Ud [Ud1, 0];

Voltages, average values The control law for the mean value of the output voltage of the buck chopper reads as follows:


109

The mean value of the output or load side voltage can be controlled via the proportion of the switch on time TE of the switch to the period of time T. This relation is called duty cycle a. The theoretical range and the real range are given below. The real range is more limited compared to the theoretical one due to minimum switch on and switch off times. Theoretical:

This voltage control law given above can be derived in a simple way. The voltage at the inductor Ld determines the derivation of the current:

From integrating this formula the current rise can be calculated:

In periodic operation mode, as for example shown in the figure above, the current increase or decrease in a period must be zero.

Solving for Ud2 gives the control law written above.:

Currents, average values The current control law shows the inverse transfer factor (1/a) as the voltage control law:

The mean value of the output current is depending on the input current and the inverse duty cycle a. The derivation of this formula is done starting with the condition of equal mean power at the input and output side, because of the idealized assumptions (no power losses in the DCDC converter).

Inserting the equation for the voltage control law

110


leads to the current control law written above.

Control characteristic diagram The control for the mean output voltage Ud2 and the mean output current Id2 can be shown and illustrated in diagrams. The conversion from input to the output voltage is proportional to the duty cycle, the conversion from the input to the output current is proportional to the inverse of the duty cycle a.

Ud2/Ud1

1

id2/id1

5

1 a1 Fig. 5.4:

0,2

a

1

DC/DC chopper, buck type (step down); voltage control via duty cycle (left); current conversion via duty cycle (right)

Behavior By means of the DC/DC buck converter the voltage at the output can continuously be controlled theoretically between zero and the input voltage. The conversion range in reality is limited so the duty cycle zero and one cannot be fully reached. The voltage shows a rectangular, the current a triangular wave shape with dc offset. The current shape can be smoothed by use of inductors with higher inductance.

For specialists: Range of Current Fluctuation Due to the switching of the output voltage, current fluctuations at the output side are generated. They are influenced especially by the smoothing inductor Ld. Usually these fluctuations have to be limited because of proper function of the system, electromagnetic interference (EMI) and taking into account for the inductor design. The range of fluctuations is derived here.

- Calculation of the variation of time The current increase (k=1) and decrease (k= 0) in the buck chopper circuit can be calculated as follows: | for

, otherwise

for TE, V1 switched on

(1)

linear positive slope


111

for T≠TE, V1 switched off

(2)

linear negative slope

For stationary operation it is required: (3)

- Current fluctuation range | from (1) |

This relationship can be shown in a diagram. a

a²

a – a²

0

0

0

0.2

0.04

0.16

0.5

0.25

0.25

0.8

0.64

0.16

1

1

0

equals

for

= const.

Fig. 5.5: Output current fluctuation of the DC/DC chopper, buck type (step down)

112


The output current fluctuation Δid2 is shown related to the circuit and control parameters (T, Ud, Ld). It varies strongly with the duty cycle. The maximum is found for a duty cycle of 0.5.

5.3

Basic DC/AC Converter Functions – Phase Leg

For feeding electrical machines from a DC source it is indispensable to use an inverter which converts the DC input to the common used three-phase AC at the output. Its basic function can be explained by means of one separate phase leg, sometimes called power electronic building block (pebb). The circuit diagram of the phase leg is shown in the figure.

Fig. 5.6: Circuit diagramm of a phase leg od power elctronic building block

A voltage source of the voltage value Ud is feeding the circuit. For easier understanding, the DC source is divided into two equal parts, connected in series, each with the voltage Ud/2. It is connected with the series connected on-off-switchable switches V1 and V2 and their and parallel diodes V1‟ and 2‟. The diodes are necessary to conduct inductive current components. The midpoint between these switches is led out for connecting the load with the midpoint of the dc voltage source. Different loads are possible, here an ohmic-inductive one is assumed. The preconditions are still valid, ideal switches, ideal switching, no stray inductances and ohmic resistors in the connection paths. When switching the switchable switches on and off, the voltages and currents, as for example in the next figure, are generated. Switching Switch V1 on leads to positive voltage U10, switching switch V2 on leads to a negative voltage. This output voltage is independent of the current flow direction. If V1 is switched on, for positive current it flows via V1, for negative current via V1‟. In both cases the output voltage is positive. The equivalent applies to the lower leg of the phase building block. The output ac voltage is fixed, equal to half the dc voltage. The mean value of the voltage amplitude can be controlled, when chopping during positive voltage time or negative voltage time as done with the dc buck chopper. So, some short time sections of negative voltage would be inserted to the positive voltage block, reducing its mean value. The currents for example behave according to the ohmic (dashed line, LL=0) or ohmic-inductive (full line) load circuit with a function: )


and are actively switched, and forced to conduct by the load current.

Fig. 5.7:

113

are

Output voltage and current of a phase leg with ohmic and ohmic inductive load; firing ranges (ZB), conduction ranges (LB); condition: TE(V1) = TE(V2);

In line ZB above the lowest line in the figure the firing ranges of the switchable power semiconductors are shown. Alternating V1 and V2 are fired. The output voltage has a rectangular shape. In case of ohmic loads, the current has the same shape as the voltage. It is to be analyzed, which switches are switched on and which conduct the current. In the lowest line the conduction ranges are shown. If V1 is switched on and the current is positive, the current flows via V1. Then V1 is switched off and V2 is switched on. An R-L-load is assumed. The current in the inductor cannot jump, it varies moderately and is still positive. The only way for the current to flow is now in the circuit: lower Ud/2 – 2‟ – connecting point 1 – load – connecting point 0. It is similar to the phase with current in the diode at the dc chopper. The output voltage here is negative (-Ud/2, compared to voltage zero for the chopper), though switch V1 is switched on. Equivalent performance is given when switch V2 is switched on. The aim of generating an ac voltage is to get the shape sinusoidal or nearly sinusoidal, for example equal or nearly equal to the reference voltage . Amplitude and angular frequency can be controlled. This can be achievd by partitioning the period T of the sinusoidal reference voltage in many subsectors with the time and emulate the mean value of the reference voltage in this sector by means of the phase leg circuit output voltage. The switch on tdn and switch off time of the phase leg for this sector must be controlled adequately so that the mean value of the reference and the phase leg output voltage for this sector are equal.

114

uL*(t), uL(t)


Ud ûL

uL(t)

*

tdn

*

uL (t)

uL*(tn)

Tpn 0 Fig. 5.8:

Einseitenbandmodulation T/2 t

Generating the reference voltage value by means of switching on the dc voltage Ud for fixed time tdn in the n-th interval (mean voltage equivalence method)

: degree of modulation

In the figure this process is shown for a converter with an output voltage of + and 0. For the control of the output voltage, the input DC voltage must be known. All values usually are referred to the input DC voltage:

This control is called sinusoidal pulse width modulation, as the pulse width is modulated sinusoidal. It is in general executed by means of microcontrollers.

5.4

Three Phase Voltage Fed DC/AC-Converters

The Three Phase Voltage fed DC/AC is the common type of converter able to produce three phase output voltages with variable frequency and amplitude.

5.4.1

Basic Circuit

The basic circuit consists of three phase legs or three basic single phase inverters as introduced in the previous section:


Fig. 5.9:

115

Three phase voltage fed DC/AC Converter, circuit diagram

Each phase leg can be controlled in the way described in the previous section, so three phase voltages u10, u20 and u30 can be controlled.

5.4.2

Pulse Width Modulation of the output voltage

The task of a pwm-inverter is to generate three phase output voltages uLii(t) which are well approximated to the three phase reference voltages uLi*(t). Usually the output voltages are symmetrical sinusoidal three phase voltages. This is achieved by means of pulse width modulation (pwm) of the output voltage for every phase leg in the appropriate way, based on the analysis for the phase leg above. To achieve the required output voltage and frequency, the semiconductors have to be switched on and off at the right times. Here uL denominates the line voltage (between the connection points 1, 2, 3). This pulse width modulation can be demonstrated and implemented by means of different methods. There are three different modulation methods presented here: -

Mean value equivalence modulation Sine – triangle modulation Space vector modulation

In general, with these methods the same output voltages can be generated, but for each in another way. 5.4.2.1

Mean Voltage Equivalence Method

The output voltage for each phase can be controlled according to the three phase reference voltages as explained for the phase leg. Each phase has its own pulse width modulator, the reference signals in case of sinusoidal symmetrical reference voltages are one of these voltages for each phase. The figure for creating the mean voltage equivalent values applies in an adapted way for each phase. 5.4.2.2

Sine-Triangle Modulation

The method of pulse width modulation can be described by means of a sine triangle comparison method and is often implemented in such a way. Applying sine-triangle-modulation, a sinusoidal symmetrical three phase voltage is taken as reference. A triangle-shaped signal d(t) or u∆ with the carrier frequency fT > f1 gets compared in the

116


pulse width modulator with the sinus-shaped reference value sin(t) with fundamental output frequency f1. The intersection of both signals represents the switching instant. In case that the reference value is higher than the triangle, the upper switch in the phase has to be switched on, in the other case the lower switch has to be switched on. The figure shows the performance for one phase. It is to be seen, that the phase leg voltage u10, though still very rough in shape, shows some approximation to a sinusoidal shape.

Fig. 5.10:

Sinus-Triangle-PWM of a three phase voltage fed DC/AC Converter

Here a modulation index M is defined:

Modulation sinusoidal:

M = 0 … 1.00

Modulation sinusoidal fundamental + third harmonic: M = 0 … 1.15 (only in case of three phase systemes) which determines the output voltage, here given for the line voltage:

For sine triangle pulse width modulation with a higher pulse frequency of 2.5 kHz the next figure shows the reference and the output voltage of the line voltage of three phase voltage fed inverter. The resulting current at a resistive-inductive load is shown too. The current is well smoothed by the inductive load. Similar currents can be found in converters of wind power stations. The diagram on the bottom shows the result of a fourier analysis of the output voltage. Here, the fundamental is the highest component with about 415 V. There are harmonic components around the pulse frequency and multiples, for example around 2,5 kHz with an amplitude of about 120 V.


Fig. 5.11:

117

Sinus-Triangle-PWM of a three phase voltage fed DC/AC converter with high pulse frequency; top: ac line voltage; midle: line current; bottom: fourier analysis of the line voltage

< Exercise (sine-triangle control) > 5.4.2.3

Space Vector Modulation

Space vector modulation is an alternative to the sine-triangle modulation. For a three phase system with linear dependent values of the three phases (l1+l2+l3=0) the system can be described by means of two values as the third one can be calculated by means of the linear dependence. SVM is based on the transformation of a linear dependent three-phase system into an orthogonal coordinate system. The values in the rectangular coordinate system are called space phasors. They can be interpreted as representing the distribution of flux at the perimeter (rotating field) of an electrical machine through a vector/ space vector in the maximum of the sinus-function of the field. For a three phase sinusoidal symmetrical system the space vector is a complex value of constant

118


magnitude rotating with constant velocity. Assuming a three-phase system, space vectors are universal, not only to be applied on the machine flux. Space phasors can also be used if there is no linear dependency, then an extra equation for the zero system is necessary. 5.4.2.3.1

Space Vectors for Converter Operation

The rule for the transformation of linear dependent three phase quantities in space vector representation in an orthogonal coordinate system with the coordinates α and β is given by:

The space vector for one switching state of the converter is being determined as an example. The switching state of the converter as shown in the figure is assumed, called state 1.

Fig. 5.12: Switching state 1 of the pwm converter for exemplary determination of a space vector

The line voltages have to be calculated from this switching state 1.

These voltages can be set into the transformation equations to give the α- and β-component of the space vector for switching state 1:

The real component has the value vector

, the imaginary component is equal to zero. So, the space

, sometimes also written with an underlined arrow, can be written as:


119

In a voltage fed pwm converter, always one switch in every leg has to be switched on. There are active states and two zero states. In the zero states, either all upper or all lower switches are switched on or the space vectors as well as the output voltage are zero. Never two switches in the same phase are allowed to be switched on, as there a short circuit of the dc source will occur. The switching states, the corresponding switched switches and the space phasors are written down in the following table. They can easily be derived from the circuit diagram. Voltage State

Switching State

Space Vector Components Arc (U)

Nr.

+

-

-

10

01

01

1

0

0°

+

+

-

10

10

01

2

60°

+

-

+

10

01

10

6

-60°

+

+

+

10

10

10

7

0

0

0

0

-

-

-

01

01

01

0

0

0

0

0

-

-

+

01

01

10

5

-

+

+

01

10

10

4

-

+

-

01

10

01

3

-120°

0

-180°

120°

Tab. 5.1: Switching states and space phasors of a three phase voltage fed DC/AC converter

These space vectors can be illustrated in the complex plane. The six active vectors span up a hexagon, the two zero states are in the middle.

120


6 (1…6) active basic-Space vectors 2 (0;7) Zero-states Block-pulsing

Fig. 5.13: Space vectors of the three phase voltage fed AC/DC PWM converter This results in 23=8 switching states and an equal number of discrete space vectors (for the so called two level converter considered here). The space vector describes a hexagon with the radius 2/3 Ud. For one switching state there are 8 conducting states, for example for switching state 1: Conducting valves

+

-

-

Tab. 5.2: Conducting states for switching state 1 of a three phase voltage fed DC/AC converter

In general there are 8 switching states, everyone with 8 conducting states, summed up to 64 states in total. 5.4.2.3.2

Time Discrete Space Vector Modulation

A symmetrical three phase voltage or current system is equal to a space vector of constant magnitude rotating with constant speed. To generate this reference three phase system by means of space vector modulation, this space vector has to be generated or approximated. This has to be done time discrete. So, for every switching period TS or pulse period Tp the space vector has to be approximated as exact as possible. The time discrete approximation can be seen from the next figure. In the complex plane the vectors for state 1 and 2 are shown. The circle describes the way of the vector reference value. We assume


121

an output frequency of 25 Hz. With a time period of the output signal of 40 ms there is a time of 6.7 ms to pass through a subsector. The subsector shall be represented by 6 time discrete space phasors. So, a space phasor stands for the time interval of 1,1 ms or new space phasors have to be calculated with 900 Hz frequency.

Fig. 5.14: Time discrete representation of the reference signal

Example: N: number of subsector in 60o TSektor: time of the space vector in one sector Tp: Time pass through one subsector or pulse period time fp: a new sector with 900 Hz, pulse frequency

5.4.2.3.3

Determination of the space vector in the n-th interval

The continuously rotating constant magnitude reference space vector has to be annulated by the 8 discrete space vectors. Therefore, it is differentiated, in which triangle sector of the hexagon the reference vector is situated. Then it is composed by the left, the right and zero space vector of the sector, where the reference space vector is located in for a specific time. Each space vector of this subsection is switched on for the necessary time. The sum (vectorial adding) results in the reference space vector. This is shown in the following figure.

122


: Time perion for a subsector : time to switch on space vector i Length:

Fig. 5.15:

Determination of the space vector in sector 1 of the three phase voltage fed AC/DC PWM converter

The longer the vector is switched on, for example , the longer the effective vector. For the Vector is as long as . The equation for the total space vector is:

The turn on times

,

and

can be derives from the diagram above for the n-th vector

U(n) = U(t) in the figure in the subsector 1, using the sinus theorem:

This holds analogously for all other basic sectors. PWM Rule It is recommended to turn on upper (⑦) and lower (⓪) switching states homogenously to avoid excessive loading of particular switches. In general, the switching causes losses, so the switching frequency should not be too high. On the other way, high switching frequencies lead to smoothed currents.


5.4.2.4 -

Variants of Pulse Width Modulation Synchronous or asynchronous pulse width modulation: 

Pulsing synchronous with reference period T, especially for low puls frequency :



pulsing is asynchronous with reference period T, for

-

Point of time to take voltage reference value o o

-

-

5.4.2.5

123

value of reference voltage taken in the center of pulse period (Natural Sampling) value of reference voltage taken at the beginning of pulse period (Regular Sampling)

o Modulation in case of ac one phase ac bridge circuit o

symmetric

o

asymmetric

(favored)

Rectangle – triangle modulation Mean value equivalence method Sine – triangle modulation Space vector modulation

General Remarks

Mean value comparison, sine triangle method as well as space vector method are operated in industrial converters at pulse frequencies of in general 500 Hz to 10 kHz. For wind turbines in the 1 to 3 MW range a pulse frequency of 2 to 3 kHz is used, for higher power systems a pulse frequency of down to 500 Hz. The voltage space vector U(n) for the n-th interval is generated by the current control of the converter. Assuming a pulse frequency of 2.5 kHz, every 400 μs a new time discrete phase vector has to be controlled. The three switching on times Ton1, Ton2 and Ton o,7 are generated using the equations above by counters in microcontrollers or similar components (DSP, FPGA). There is some logic behind the microcontroller to feed the right signal to the right power semiconductor. In

124


each interval, the right, left and zero basic space vector in the relevant sector are switched on for the right time to generate the voltage space vector U(n). 5.4.2.6

Realization and Implementation

The converter consists of the power section, the pulse width modulation and the control as shown in the figure. The signals exchanged between these components are shown in the figure.

Fig. 5.16:

Structure and main components in a pwm inverter

The pulse width modulation is done via counters in a microcontroller as well as the control is done in the microcontroller. There are special microcontrollers that are equipped with suitable counters. Additionally there are analog to digital converters necessary to convert the analogue measured signals from current or voltage to binary values for processing in the microcontroller. The next figure gives an idea of the hardware realization of a pulse width modulation.


Fig. 5.17:

5.4.2.7

125

Realization of a pwm control in a microcontroller

Output voltage value of the pwm converter with diode bridge feeding

The standard inverter for industrial use is fed via a diode bridge from the mains. The maximum output voltage is of interest. Line root mean square voltage Line amplitude voltage DC voltage of a voltage fed DC/AC converter fed via diode bridge

126


Line root mean square voltage of the pwm converter output The maximum output voltage when fed from diode bridges is 381 V, lower than the mains voltage of 400 V. For wind power station the feed in from the mains is done via mains connected pwm converters, the same as presented above. There, the dc link voltage can be stepped up to values higher than for feeding with a diode bridge. So, machine voltages of nominal value (here for example 400 V9 can be reached. < Exercise space vector control >

5.4.3

Composed Circuits

An inverter has been analyzed, converting a dc voltage to a three phase ac voltage. Flow of active power is possible with these converters in both directions, from dc to ac as well as from dc to ac. This is the reason, that a composition of two of these pwm converters can be connected together to convert an ac voltage with given amplitude and frequency, for example of a generator in a wind turbine, into a three phase ac voltage of different amplitude and frequenca, for example of the maisns.The next figure shows this composed circuit.

Fig. 5.18: Three phase ac to ac converter circuit, composed of two dc to three phase ac converters

In the dc link the voltage is smoothed via a capacitor to get a nearly constant dc voltage. Also for the composed circuit power flow in both directions is possible.

5.5

Power Semiconductors

Types:  

Diodes Thyristor (turn on only)

5 Three Phase AC to Three Phase AC Frequency Converters for Generators   

127

Bipolar transistor (unimportant today because of high losses) BJT bipolar junction transistor MOSFET (state of the art, up to 800V) IGBT = Insulated gate bipolar transistor (state of the art, 600V-8kV, wind power)

IGBT characterization:      

Combination of MOSFET and BJT Controlled by gate voltage Bipolar device: positive and negative charge carriers in conducting state Typical switching time 80 ns – 1µs Compared to MOSFETs lower conducting losses but higher switching losses Typical lower switching frequency (2 kHz to 16 kHz, depending on power)

Switching performance

Fig. 5.19:

  

Switching performance of a power semiconductor in typical hard switched application, Switch off, left; switch on, right

Negative gate signal for turn-off, positive for turn on (UGE, Gate Emitter) IGBT starts conducting Current peak because of the freewheeling diode

128

5 Three Phase AC to Three Phase AC Frequency Converters for Generators  

5.6

Voltage peak due to parasitic inductances (up to a few 100V) Switching losses due to the fact that first the current rises and after that the voltage drops (turn-on) and for turn-off first the voltage rises and than the current drops.

For later use

Time discrete representation of rebuilding the sinusoidal shape The control of the output voltage uL(t) is accomplished respectively for discrete time intervals TP (pulse-period). The average value of the reference voltage uL*(tn) for the n-th pulse period TPn is set by appropriate triggering of the input DC voltage for one part of the pulse period.

For controlling the modulation, the value of the input voltage Ud has to be known. The diagram is referred to Ud in general. Values used: ;

;

;

rocedure → calculate pulse width and execute. 5.6.1.1

Rectangle-Triangle Modulation

The three phase legs in the circuit of the three phase voltage fed DC/AC converter are switched to feed positive voltage in the first half of the period and to negative voltage in the second half of the period. The switching is phase shifted at 1200 between the phases. The voltage shape as given by the solid line in the figure is generated. The voltages have constant amplitude; the frequency can be varied by means of reducing the switching period. In the lower part of the figure there are the diagrams of the line voltage u12 = u10 – u20 and the resulting current in an interphase inductive load.


129

For voltage zero the current is constant, for positive voltage increasing, for negative voltage decreasing. Applying a rectangle modulation, the amplitude can be controlled. If the reference signal is higher than the triangle signal, the positive voltage is switched on, otherwise the negative voltage. The resulting reference value of the phase to midpoint voltage is shown by the dashed line; it is half the maximum voltage. Voltage control is done by comparing the reference value with a triangle function, with constant amplitude of the maximum of the voltage reference value (here referred quantities and the value equal to 1) and a frequency higher than the output frequency. Here, a very low triangle frequency is taken for example. The phase leg output voltage now is shown by the dashed line. The positive constant voltage section is not disrupted by a short negative voltage part; the mean value is 50 % of the maximum value, according to the reference value. This is valid for the negative voltage section and the other phases in a similar manner. The line voltage and the output current can be constructed in the same way as before.

130


Fig. 5.20:

Rectangle-Triangle-PWM of a three phase voltage fed DC/AC Converter

So, by the rectangle-triangle-control a controlled output voltage is possible. However, the voltages have still a rectangular shape, not a sinusoidal shape.


5.7

Exercises

5.7.1

Sine Triangle Modulation of a single phase DC/AC converter

131

A single-phase DC/AC converter is given in fig. 1. The dc link is charged to Ud = 290 V. id Ud 2

V1

V'1

V2

V'2

LL

iL uL

0

Ud 2

V4

V'4

V3

V'3 u L10

u L20

Fig.: DC/AC Inverter with inductive load

Ud = 290 V, LL = 2,9 mH, fP = 1 kHz The load has to be supplied by a sinusoidal AC voltage. An asymmetric sine-triangle control (natural sampling) should be used here. The frequency ratio between the triangular function d(t)  1 and the sinusoidal modulation function m(t) is p = fp/f = 5. The degree of modulation is m firstly. a) Draw the phase voltages uL10, uL20 and the load voltage uL into a time diagram. b) Draw the characteristic of the load current iL. When does the output current cross zero? c) Describe the conducting intervals of the semiconductors. Draw the current characteristic id of the DC source.

 = 0,5 to 0. Now the degree of modulation is reduced from m d) Determine the characteristcs of the output voltage uL and the output current iL. What ist the size of the first harmonic in the output voltage?

 > 1 (overmodulation) Now the degree of modulation is increased to m  = 1,2. Which output voltage characteristic occurs for e) Draw the output voltage uL for m   ? m f) Determine the width of an output voltage pulse during a pulse period between 0,05 T and 0,15 T analytically for Tp = 0,1 T with T = 5 ms. Calculate the mean value of the ˆ  sin( t ) for a degree of modulation m ˆ  1 . The modulation function m(t )  m modulation is asymmetrical. Draw the resulting voltages u L10, uL20 and uL during the pulse period.

132


5.7.2

Space Vector Modulation of a three phase AC/DC converter

The three phase grid voltage should be emulated by means of an inverter. Therefore the space vector modulation is applied. id V1 V1´

V3 V3´

V5 V5´

Ud/2

u1M i1 u12 i2

M

u23 i3 Ud/2

V2 V2´

V4 V4´

V6 V6´

RL

LL u3M

u10 u20 u30 0

Fig.: Three phase inverter

3  230V  2 2

DC-link Voltage

Ud 

Load voltage

 u LM  150V

Grid frequency

fN = 50 Hz

Switching frequency

fP = 5 kHz

a) Describe by means of a drawing the basic space vectors of the output voltage u i which can be produced by the inverter and name them. b) What is the maximum instantaneous value of the output voltage of the inverter? Calculate therefore the space vector components u and u for the switching state [10 01 10] and the norm of this space vector. c) In how many discrete sectors is one cycle of the space vector divided? What is the size of such a sector? d) Calculate the space vector components u and u for the 38th sector. e) Calculate the required space vectors for emulate the 38th sector and their turn-on times. f) How long are the valves V1 to V6 of the inverter for this switching period turned on?

6 Variable Speed Generator Systems (Stationary Performance)

133

6 Variable Speed Generator Systems (Stationary Performance) To extract maximum power out of the wind, wind turbines have to be controlled with variable speed. This is achieved by means of a power electronic frequency converter. This converter is feeding all or part of the generator power to the grid, depending on the generator type, induction with short circuit rotor, synchronous or doubly fed induction generator, and thus enables speed control of the generator. The basic topology of such a power electronic generator system, here for a full power frequency converter and a squirrel–cage induction generator, is shown in the next figure. The full mechanical to electrical power conversion line has to be designed for the active power required. The mechanical part of the machine has to be designed in power to fit to the rotor power, electrical part of the generator and the converter and the mains connection have also to be designed to fit together. This design is analyzed and shown in this chapter.

Fig. 6.1:

6.1

Basic topology of a power electronic generator system for wind turbines

Speed control of induction machines (repetition)

The performance of the ac machines for variable speed as analyzed in chapter 4 is summarized here in an overview manner. Stator flux and voltage control Basic control concept for speed control of ac machines in the basic speed region (constant torque) as applied for wind turbines is to keep the flux linkage in the machine constant. In a simplified consideration this leads for constant stator flux linkage to the rule, that the stator voltage U s and the stator angular frequency ωs have to be controlled proportional to each other. The control law for constant stator flux thus is written:

134


A control of other fluxes as the rotor flux or the main flux is possible but not regarded here. The basic performance is very similar to the one for controlled stator flux linkage. Torque speed diagram Controlling the machine in this way, for either of the three machine types as enumerated the torque speed diagram (M-N-diagram) and the power speed diagram (P-N-diagram) as shown in the next figure are valid.

Fig. 6.2: Torque speed and power speed diagram of power electronic generator systems for wind turbines (here for squirrel cage induction machine) The constant stator flux linkage control is applied to wind turbines. Then, in this range, the torque for nominal stator current is constant (M/MN = 1) and the power for nominal stator current increases or decreases with the speed (P/PN ~ +/-N/NN). This diagram is the base for the design of the connection of generator and rotor. This is not deepened here. The torque speed diagram is shown here for a squirrel cage induction machine. Due to the slip increasing with increasing torque, the limit curves to the field weakening range show a soft deviation from the vertical line. In case of a synchronous machine or doubly fed induction machine these figures would show really vertical lines at the transition to the field weakening range. Beyond the nominal speed the voltage at the machine terminals has to be kept constant. Thus the power is constant and the torque decreases with the inverse of the speed. This is the constant voltage or field weakening region. The wind turbines are usually not operated in this region.

Stator voltage versus stator current diagram of the ac machine Concerning the connection between generator and converter, the maximum voltages and currents are of importance. Maximum torque is generated with maximum, that is nominal stator current of the machine. (In case of the doubly fed induction machine it is nominal stator and rotor current.)


135

The voltage, applying the stator flux linkage control law, is given as:

Maximum voltage is required for nominal speed. Additionally, for squirrel cage induction machines, for maximum stator current at nominal speed, the stator frequency must be increased. This is due to the law, that the stator frequency is the sum of speed multiplied by the pole pair number and the rotor frequency.

Additionally, for all types of machines, for correct analysis the voltage drop at the stator resistance has to be included.

The figure below shows the approximated stator voltage versus stator current diagram. It has a rectangular shape. Taking into account the voltage drop at the stator resistance and in case of a squirrel cage induction machine additionally the voltage increase due to the additional rotor frequency, the exact diagram as shown below results.

Fig. 6.3:

Stator voltage versus stator frequency diagram of ac machines (at nominal speed)

136

6.2


Operation range of the frequency converter

The performance of frequency converters can also be described in voltage versus current diagrams, as well as for the input as for the output side. Here the fundamentals are considered, though not marked specially. The amplitude of the ac voltage can be written as:

k: constant term; m: modulation index; Ud: DC link voltage In reality there is an additional voltage drop due to the power semiconductors. So, more exact, the ac voltage of the converter can be written as:

(Voltage drop over power semiconductors,

threshold voltage,

ohmic

resistance; for example: 3…6 )) This results in the ac voltage versus ac current diagram of the converter shown in the figure. The current limit is constant according to the nominal current, the voltage limit varies slightly with the current.

Fig. 6.4:

Stator voltage versus stator current diagram of a frequency converter


6.3

137

Drive with frequency converter and ac machine

When combining an ac machine with a frequency converter, the nominal power should fit to each other. Each component can only be operated in its allowed range. So, for continuous operation only the common nominal operating range can be used. Both components can be the limiting factor of the drive. The figure gives an example.

Fig. 6.5: Nominal operating ranges of converter and ac machine and common range for continuo operation

6.4

Example

As an example for a drive in operation, a reversing motion drive, crane drive, is regarded. It is assumed, that there is only constant resistive force. The mass is driving in one direction, than decelerated, speed goes to zero, and afterwards accelerated into the opposite direction. This is shown in the following figure and described below.

138

Fig. 6.6:


Reversal of a drive of a crane versus time


Fig. 6.7:

139

Reversal of a drive of a crane versus time, torque speed diagram

Description of the regions: 1. Stationary operation with NN resisting torque MW = 0.5 MN ; IS 0.6 ISN ; . 2. Changing of reference value N* / NN = -1 causes transition states: The drive changes to generator operation. Duration depends on used control scheme, usually between 10 ms and 500 ms. 3. Generator operation, crane decelerates. IS = ISN, maximum value; because of generator operation; respectively;Voltage US tracks the driving speed dN/dt = (1/teta)MA – MW = (1/teta)(-1.5 MN). 4. Speed becomes zero and afterwards negative. Drive changes to motor operation with other direction of rotation; M > 0 because of slow control method. 5. MA = -MN, same direction and height. 6. N < 0, other direction of rotating, now again P = 2 NM motor operation, respectively. 7. Reference speed –NN reached, Transition to stationary operation. 8. Stationary operation with –NN, resisting torque MW = 0.5 MN; IS 0.6 ISN; (Heylandcircle).

140


6.5

Exercise

Rating of an electrical drive for a mountain railway with induction machine The drive of the mountain railway, which is shown in the figure, has to be modernized. A given induction machine in delta-connection should be used in conjunction with a frequency converter, which has to be proper designed. The speed is controlled with the method of constant stator flux linkage (constant relation between stator-voltage and -frequency, Us/s = const).

The following details of the machine are known: Machine data:

Railroad data:

gear:

Rated voltage Rated speed Rated frequency Rated torque Power factor at MN Efficiency at MN Maximum speed Breakover torque Rotor breakover frequency Mass of the wagon Max. number of occupants Weight of occupants Friction torque of the drive Max. uphill grade Rated velocity at 25 % Min. uphill grade Rated velocity at 5 % Radius of one wheel Width of the wagon Height of the wagon Air drag coefficient Density of dry air

UN NN fN MN cos N N Nmax Mk RK mW n mP MR smax vI smin vII R B H cW 

= = = = = = = = = = = = = = = = = = = = = =

400 V 1479.2 min-1 50 Hz 587.8 Nm 0.87 0.94 2.5 NON 1448.2 Nm 20.56 s-1 2000 kg 50 75 kg 50 Nm 25 % 20 km/h 5% 51 km/h 0.3 m 2m 2.5 m 0.5 1.29 kg/m3

gear transmission ratio

üG

=

8

1. Calculate the rated mechanical quantities related to the machine side of the gear. Draw them into an M(N)-diagram. 2. Verify that the given machine fulfills all load conditions. Derive the limiting curve of the induction machine for this purpose and plot all operation points. 3. Evaluate all electrical quantities in the operation points as well as the electrical limiting curve of the machine.