modelin synchronous machine.pdf

8/14/2019 modelin synchronous machine.pdf

1/85

Detailed Description of

Synchronous Machine Models

Used in Simpow

Emil Johansson

Master Thesis B-EES-0201

Electric Power SystemsDepartment of Electrical Engineering

Royal Institute of Technology

Sweden


2/85

Master Thesis

Submitted to the Royal Institute of Technology, KTH,

in partial fulfilment of the requirements for the degree ofMaster of Science Electrical Engineering.

Vsters 2002


3/85

- iii -

AbstractIn this work, it is described how different synchronous machine models arederived; especially those used in the power system simulation software Simpow.

There are various ways how to describe the model of a synchronous machine, in

the literature the biggest difference is the per unit system used in the models. Otherdifferences are: the definitions of the d- and q-axis, the performance of the dq0-transform, and, of course, the number of damper windings modelled in the rotorcircuit.

In this report, the per unit system used is thoroughly described, so the risk ofmisunderstanding is minimized.

Theoretical foundations have been found for the equations used in the FORTRAN-coded synchronous machine models implemented in Simpow. In this thesis, theseequations are derived on a general basis, and are thoroughly derived for the modelsused in Simpow. There is also an explanation of relations between differentmodels.

New DSL-coded models have been programmed for the four fundamentalfrequency models in Simpow. These new models are validated, and have beenfound to have a close resemblance with the old FORTRAN-models.


4/85

- iv -


5/85

- v -

Table of Contents

1 Introduction........................................................................1

1.1 Preface ................................................................................1

1.2 Project description................................................................1

1.3 Outline .................................................................................1

1.4 Acknowledgement................................................................2

2 Theory.................................................................................3

2.1 Physical description..............................................................32.1.1 Multiple pole machines.........................................................32.1.2 Direct and quadrature axis ...................................................4

2.2 Time dependency.................................................................42.2.1 The dq0-transformation ........................................................42.2.1.1 Three-phase stator to two-phase global transform ...............61.1.1.2 Two-phase global to local transform.....................................61.1.1.3 Final dq0-transform..............................................................6

1.3 Reference systems in Simpow.............................................71.3.1 Fundamental frequency models ...........................................71.3.2 Instantaneous value models.................................................81.3.3 Conclusions .........................................................................9

3 Mathematical description ................................................11

3.1 Machine equations.............................................................11

3.2 Time dependency...............................................................13

3.3 The dq0-transformation......................................................14

3.4 Reciprocity .........................................................................15

4 Per unit representation ....................................................19

4.1 Machine per unit bases ......................................................19

4.2 Per unit flux linkage equations............................................204.2.1 General description............................................................204.2.2 dq0-axes flux linkages........................................................214.2.2.1 d-axis .................................................................................224.2.2.2 q-axis .................................................................................244.2.2.3 0-axis .................................................................................244.2.3 Conclusions .......................................................................244.2.4 Definition of the transient and sub-transient reactances .....254.2.4.1 d-axis .................................................................................254.2.4.2 q-axis .................................................................................27

4.3 Per unit voltage equations..................................................274.3.1 Stator voltages...................................................................274.3.2 Rotor voltages....................................................................274.3.2.1 Field winding ......................................................................284.3.2.2 Damper windings ...............................................................284.3.2.3 Summary of the rotor voltage equations.............................294.3.3 Conclusions .......................................................................294.3.4 Definition of the time constants ..........................................294.3.4.1 d-axis .................................................................................29

4.3.4.2 q-axis .................................................................................304.4 Per unit torque equations ...................................................304.4.1 Fundamental frequency models .........................................31


6/85

- vi -

4.4.2 Models without damper windings........................................31

4.5 Global - local per unit transformation..................................32

5 Modelling ..........................................................................33

5.1 Transient-/Machine stability ................................................33

5.2 Initial conditions at steady state..........................................345.3 Type 4 ................................................................................375.3.1 Assumptions for the classical model...................................375.3.2 Assumptions in Type 4 .......................................................375.3.3 Description of Type 4..........................................................385.3.3.1 Instantaneous value model.................................................385.3.3.1.1 Equivalent circuit 395.3.3.2 Fundamental frequency model ...........................................395.3.3.2.1 Equivalent circuit 39

5.4 Type 3A..............................................................................405.4.1 Instantaneous value model.................................................40

5.4.1.1 Equivalent circuit ................................................................415.4.2 Fundamental frequency model ...........................................42

5.5 Type 2A..............................................................................435.5.1 Instantaneous value model.................................................435.5.1.1 Equivalent circuit ................................................................435.5.2 Fundamental frequency model ...........................................44

5.6 Type 1A..............................................................................455.6.1 Instantaneous value model.................................................455.6.1.1 Equivalent circuit ................................................................465.6.2 Fundamental frequency model ...........................................47

5.7 Saturation...........................................................................48

6 Programming....................................................................51

7 Validation ..........................................................................53

8 Conclusions and future work ..........................................57

8.1 Conclusions........................................................................57

8.2 Future work ........................................................................57

Appendix A..........................................................................................59

Appendix B..........................................................................................65

Appendix C..........................................................................................67

Appendix D..........................................................................................71

References ..........................................................................................73

List of symbols ...................................................................................75


7/85

- vii -

Table of figuresFigure 2.1 Definition of generator quantities and the direct- and quadrature axes...3Figure 2.2 Schematic view of a two-pole three-phase synchronous machine .......... 5Figure 4.1 Three magnetically coupled windings...................................................20Figure 5.1 Steady-state equivalent circuit...............................................................35

Figure 5.2 Voltage diagram of steady-state equivalent circuit ............................... 35Figure 5.3 Equivalent dq0-circuits for Type 4, instantaneous value model in perunit ..................................................................................................................39

Figure 5.4 Equivalent circuit for Type 4, fundamental frequency model,symmetrical components ................................................................................40

Figure 5.5 Equivalent negative- & 0-sequence circuits, fundamental frequencymodel in SI units.............................................................................................40

Figure 5.6 Equivalent d-circuit for Type 3A, instantaneous value model in SI units........................................................................................................................42

Figure 5.7 Equivalent d-circuit for Type3A, leakage inductance instantaneousvalue model in SI units ...................................................................................42

Figure 5.8 Equivalent q-circuit for Type3A, in SI units .........................................42

Figure 5.9 Equivalent dq-circuits for Type2A, leakage inductance instantaneousvalue model in SI units ...................................................................................44

Figure 5.10 Equivalent dq-circuits for Type1A, leakage inductance instantaneousvalue model in SI units ...................................................................................47

Figure 5.11 Flux linkage - mmf diagram, showing effects of saturation................48Figure 7.1 Power systems used during validation ..................................................53Figure 7.2 Ud for both the DSL- and FORTRAN-coded Type 1A ........................54Figure 7.3 Difference between Ud for the DSL- and for the FORTRAN-coded

Type 1A ..........................................................................................................54


8/85

- viii -


9/85

1 IntroductionPreface

- 1 -

1 IntroductionThis report is one part of the result of the master thesis work in the Master ofScience Electrical Engineering Programme, at the Royal Institute of Technology.The thesis was carried out at the department of Power Systems Analysis, ABB

Utilities, in Vsters during the autumn 2001.The other parts of the work are the code for the programmed machine models and atechnical report for ABB.

1.1 Preface

Within Simpow, the power system simulation software developed by ABB, thereare four synchronous machine models in the standard library of models. Thedifferences between these models are the representation of the rotor circuits.

The models are coded using FORTRAN in the 1980s, based on equationssummarised in the technical reports [1] and [2]. The contents of the reports are

mainly these equations, without any reference to the theory behind them. Duringthe years, the models have been changed and these changes are only brieflydocumented. Additional documentation of the models is in the Simpow usersmanual [3], where the parameters that must be given to the model and the variablescalculated by the models are listed.

The models have regularly been used in power system simulations and have beenfound to give enough accuracy for most cases. For other cases, special models have

been made using the Simpow Dynamic Simulation Language, DSL. These aredocumented in special reports, [4] and [5].

1.2 Project description

Using available theory and equations, realise the four synchronous machinemodels from Simpow with DSL.

The realisation of the models is to be based on classical theory of thesynchronous machine local reference system (two-axis theory, dq0-transformation, and equivalent schemes). Further, give the physical-mathematical description of the machine included in such global referencesystems used when simulating power systems.

The realised models shall by simulation of a sufficiently number of cases bevalidated against existing models. This will give opportunity to notice eventualimperfections.

Theoretical foundation exists, but other literature should be examined. To startwith, a fundamental frequency model shall be realised and validated, and, ifthere is time, so shall also an instantaneous value model.

Documentation of the realised models, with references to appropriate literature,is a central part of the work. A clear description shall be made of the theoryand of test cases used for validation.

The work shall be documented in a technical report.

1.3 Outline

This report describes how different synchronous machine models are derived andgives a background to how different models are related.


10/85

Detailed Description of Synchronous Machine Models Used in Simpow

- 2 -

Chapter 2 describes the basic definitions used in the report. In chapter 3 thederivation of the time independent and reciprocal machine equations, in SI units, isdescribed for a general synchronous machine model. Whereas chapter 4 containsthe per unit description of these equations. Chapter 5 thoroughly describes the fourdifferent machine models used in Simpow. In chapter 6, a short introduction to theDSL-programming and the DSL-coded models is given. While in chapter 7, these

DSL-coded models are validated against the already existing FORTRAN-codedmodels. Chapter 8 consists of the conclusions drawn during the work and asummary over the work that is still to be done.

1.4 Acknowledgement

During this thesis work, I have had a lot of help from different people, but first andforemost from Mr. Jonas Persson, my 1st supervisor from the Department ofElectrical Engineering at KTH. I wish to thank him for all the time he has spentwith me, helping me to understand even the most simple of problems, even thoughhe is busy with the fulfilment of his Technology Licentiate degree. I would alsolike to thank him for his positive thinking: It is never too late to give up, Emil,

and his e-mail-mania that always is a source of joy.

I wish to thank Professor Lennart Sder, Tech. Lic. Mats Leksell and Tech. Lic.Fredrik Carlsson for their engagement in the start-up phase of my work, when Iwas somewhat confused. I would also like to thank Mr. Magnus hrstrm, my 2nd

supervisor from KTH, for his comments on my report.

At the Power Systems Analysis department at ABB Utilities I would like to thankeverybody for making me feel at home at the department, and for giving me theopportunity of attending the Simpow Basic Course, twice. Especially I would liketo thank:

Mr. Lars Lindkvist, who has been most helpful with all my programmingdifficulties, even though I havent got a Simpow release named after me yet. Andfor his daily cheering comment: Are you finished soon?

Mr. Bo Poulsen, my supervisor from ABB, who has helped and guided me throughmy work, especially for the hint towards German literature.Gott wrfelt nicht.

Mr. Tore Petersson, who is a better source of knowledge than any book I have everread.

I would also like to thank my friend, Mr. Mathias Carlsson, for his comments on

my work and the various conversations during Sunday dinner, after the Ltzen-foghas left the kitchen.

At last, I would like to thank my family, for their support through all my years ofstudy, and in particularly my mother who probably will be as relieved as I whenthis work is finished.


11/85

2 TheoryPhysical description

- 3 -

2 TheoryThroughout the report, upper-case letters are used when relating to quantities in SIunits and lower-case when relating to per unit values. The exceptions are time, t[s],speed, [rad/s], and angle, [rad, degree]. Bold letters are used when

referring to a matrix or a vector. All symbols used in the report are defined in theList of symbols that is attached at the end of this report.

2.1 Physical description

To be able to understand different models of the synchronous machine, theproperties of the machine must be understood. Here, the reader is assumed to havea basic knowledge of electrical machines, for further reading see e.g. [6] or [7]. Toavoid misunderstandings, a short description including basic definition is included.

Fundamental frequency models are normally used when simulating the transientstability of larger power systems. In Simpow these models are used in the module

called Transta. The instantaneous value models are used when simulating thedetailed behaviour of particular machines, which in Simpow is made in the Mastamodule.

An electric machine is used for energy conversion. In a generator, mechanicalenergy is converted into electrical via electromagnetism, in a motor it is the otherway around. There are no differences in the theoretical treatment of motor orgenerator, in this report generator definitions are used, i.e. the stator currents, ia, iband ic, are defined positive out of the machine, see Figure 2.1.

Figure 2.1 Definition of generator quantities and the direct- and quadrature axes

2.1.1 Multiple pole machines

Synchronous machines are mainly operated at synchronous speed; i.e. the electricalfrequency of the rotor is the same as the frequency of the stator (the net frequency).

ua

ia

ub

ib

uc ic

ifd

ufd

i1d

i1q

a

b c

1q

1d

fd

q

d

a

global reference axis


12/85


- 4 -

The relationship between electrical frequency, , and mechanical, mech, is thenumber of pole pairs.

pairspolemech

= (2.1)

This means that the higher number of poles a machine has, the slower it rotates.Due to the symmetry of the machine, the same relation exists between electricaland mechanical degrees or radians. The symmetry makes it possible to model amultiple pole machine as a single pole pair machine.

2.1.2 Direct and quadrature axis

Two axes are defined, the direct axis, d-axis, and the quadrature axis, q-axis, seeFigure 2.1. These axes are following the rotor; the d-axis is aligned with themagnetic flux vector generated by the field current, i.e. the magnetic north pole,and is lagging the q-axis by 90 electrical degrees. This is the most commondefinition, and is used by the IEEE Standard Dictionary of Electrical andElectronic Terms, [8].

2.2 Time dependency

Due to the non-uniform air gap, the reluctance of the magnetic path, , will berotor position dependent, i.e. time dependent. This is due to the fact that:

lt )( (2.2)

Where lrepresents the length of the magnetic path.

This effect is especially noticeable in a salient pole machine, but also in a roundrotor machine. This also implies that the inductances of the stator circuits, as wellas the rotor to stator mutual inductances, are time dependent, since:

1L (2.3)

WhereLrepresents the inductances of the stator circuits.

For the present modelling work, the structure of the stator will be assumed to beperfectly smooth, with evenly distributed windings, this will cause no positiondependency of the rotor inductances [9]. That is, from the rotor point of view, theair gap is not time dependent.

2.2.1 The dq0-transformation

To be able to get a time independent equation system, the stator quantities(voltages, currents and flux linkages) are transformed using the dq0-transformation. This transformation, sometimes called the Park- or the Blondeltransformation, is a transformation to rotor coordinates.

Bhler, [10], has thoroughly described how the dq0-transformation can be dividedinto two steps, see Figure 2.2.


13/85

2 TheoryTime dependency

- 5 -

ua

ia

ub

ib

uc ic

ifd

ufd

i1d

i1q

a

b

c1q

1d

fd

q

d

a

u

i

ui

ifd

ufd

i1d

i1q

1q

1d

fd

q

d

ud

id

uq

iq

ifd

ufd

i1d

i1q

d

1q

1d

fd

q

d

a) synchronous machine with the three-phase stator windings

b) synchronous machine with the three-phase stator windings transformed tothe two-phase global referencesystem

c) synchronous machine with the two-phase global reference windingstransformed to the two-phase localreference system

Figure 2.2 Schematic view of a two-pole three-phase synchronous machine


14/85


- 6 -

2.2.1.1 Three-phase stator to two-phase global transformAssuming that the global reference axis coincides with the a-axis of the machine,the first step is a transformation of the three-phase stator quantities, indices a, bandc, to a two-phase global reference system, indices and, see Figure 2.2b.

cba!" UaaUUkjUU2

++=+=!"U (2.4)

Where kis an arbitrary transformation constant and

32j

ea = (2.5)

If kis chosen as 32=k , a peak value invariant transformation is obtained.

cba UaaUU2

32 ++=!"U (2.6)

Now Uand Uin equation 2.4 becomes:

[ ]cba UUUU 21

21

32 = (2.7)

[ ]cb

UUU =3

1

(2.8)

Other choices of k can be made; a common choice is 32=k , which makes the

transform power invariant. Different advantages and disadvantages with thesetransformations are described in [9].

2.2.1.2 Two-phase global to local transformIn the second step, the time dependency is extracted from the equation system. Thetwo-phase global reference quantities are transformed to the local rotor coordinatesystem, indices dand q, see Figure 2.2c.

jqd ejUU =+= !"dq UU (2.9)

Here is the electrical displacement angle between the real-axis of the globalreference frame and the real axis of the local system, i.e. the d-axis of the rotor, seeFigure 2.1. Separating the real and imaginary parts, the d- and q-axis quantities aredescribed as:

sincos UUUd += (2.10)

cossin UUUq += (2.11)

2.2.1.3 Final dq0-transformTogether, these two steps gives, after some trigonometric calculations:

[ ])cos()cos(cos 323232 +++= cbad UUUU (2.12)

[ ])sin()sin(sin3

23

232 +++= cbaq UUUU (2.13)

To get a complete degree of freedom, a third-axis is added to the two-phase system,the 0-axis, index 0. During symmetrical conditions, the 0-sequence quantities areequal to zero.

[ ]cba UUUU ++= 31

0 (2.14)

Now, the final dq0-transformation looks like:

+

+

=

c

b

a

q

d

U

U

U

U

U

U

21

21

21

32

32

32

32

0

)sin()sin()sin(

)cos()cos()cos(

32

(2.15)


15/85

2 TheoryReference systems in Simpow

- 7 -

or shorter:

Sdq0 BUU = (2.16)

Where,

Udq0, are the stator voltages in the local rotor reference system.

USare the three-phase stator voltages.

Bis the transformation matrix.

Now, the dq0-transformation has been defined for the stator voltages, but the sameis valid for the stator currents and flux linkages, see Equations (2.17) and (2.18).

Sdq0 BII = (2.17)

Sdq0 B## = (2.18)

The inverse transform is given by:

dq0$

dq0S UBUU =

++

=

1)sin()cos(

1)sin()cos(

1)sin()cos(

32

32

32

32

(2.19)

2.3 Reference systems in Simpow

Every synchronous machine has its own local reference system, in which itsarmature currents and voltages are expressed. In the network, these quantities

belong to a common global reference system. Therefore, transformations have tobe made between these reference systems in every node where a synchronousmachine is connected.

The models described in this report are transforming the voltages from the globalto the local system, where they are used when calculating the machinecharacteristic variables: or , , d, q, fd, kd, kq, Id, Iq, Ifd,Ikd, Ikq, Tm, Te, Pand Q; which all will be discussed in this report. After the currents out of themachine, Id, Iq, are calculated, they are transformed back to the global referencesystem.

As described in the previous sub-chapter, the transformation of the voltages fromthe global- to a local reference system looks like:

jqd ejUU =+= !"dq UU (2.20)

This is valid when is the electrical displacement angle between the real-axis ofthe global reference frame and the real axis of the local system. In Simpow theglobal reference system is defined in different ways depending on whether afundamental frequency model or an instantaneous value model is made.

2.3.1 Fundamental frequency models

For a fundamental frequency model, the reference frame is a complex planerotating with the system frequency n [rad/sec]; here the electrical displacementangle is measured between the real-axis of the reference frame and the q-axis ofthe rotor. This changes the transformation equation (2.9), to:

)2

(

=j

e!"dq UU (2.21)


16/85


- 8 -

and the d- and q-axis quantities are described as:

cossin UUUd = (2.22)

sincos UUUq += (2.23)

For the inverse transform, the -and-quantities looks like:

cossin qd UUU += (2.24)

sincos qd UUU += (2.25)

In Simpow, the electrical displacement angle is defined as Equation (2.26), [4].

devitit ++= 00 (2.26)

Where

it0is the machine-node voltage angle, calculated in a power-flow

it0is the internal load angle of the machine at time t=0

devis the integrated deviation angle, and is calculated as:

=t

ndev dt

0

)( (2.27)

is the angular frequency of the machine

2.3.2 Instantaneous value models

For an instantaneous value model, the reference frame is the dq-axes of a referencemachine rotating with the angular frequency ref. Here the electrical displacementangle is measured between the d-axis of the reference machine and the d-axis ofthe present machine, i.e. between the real-axis of the global reference frame and thereal axis of the local system. This means that no changes are made in thetransformation equation (2.9), and the d- and q-axis quantities are described as:

sincos UUUd += (2.28)

cossin UUUq += (2.29)

For the inverse transform, the -and-quantities looks like:

sincos qd UUU = (2.30)

cossin qd UUU += (2.31)

In Simpow, the electrical displacement angle is defined as Equation (2.32), [4].

)( refdevrefref ++= (2.32)

Where

is defined in Equation (2.26)

ref is the reference machine-node voltage angle, calculated in a power-flow.

refis the internal load angle of the reference machine at time t = 0.

refdev is the integrated deviation angle of the reference machine, and iscalculated as:


17/85

2 TheoryReference systems in Simpow

- 9 -

=t

nrefrefdev dt

0

)( (2.33)

refis the angular frequency of the reference machine.

2.3.3 ConclusionsIn Simpow, the global to local transformation looks like:

sincoscossin UUUUUd +== (2.34)

cossinsincos UUUUUq +=+= (2.35)

and the local to global transformation looks like:

sincoscossin qdqd UUUUU =+= (2.36)

cossinsincos qdqd UUUUU +=+= (2.37)

Where

is the electrical displacement angle for an instantaneous value model, andis measured between the d-axis of the reference machine and the d-axis ofthe present machine.

is the electrical displacement angle for a fundamental frequency model,and is measured between the real-axis of the reference frame and the q-axisof the rotor.

Even though does not have to be equal to -/2, the equations in the dq0-transformation can be treated in this way. Therefore, in the dq0-transformation, theonly difference between the fundamental frequency models and the instant valuemodels is how the reference angle is calculated. This is why, in the followingequations, only the angle will be used.

It should be noted that there are often different per unit systems for the machineand for the system, this must be taken under consideration, when transforming

between the global and local reference systems. This matter will be furtherdiscussed in chapter 4.4.


18/85


- 10 -


19/85

3 Mathematical descriptionMachine equations

- 11 -

3 Mathematical descriptionIn this chapter, a general description of the mathematical theory of the synchronousmachine models is made. That is, the equations derived in this chapter are valid forthe instantaneous value models, while for the fundamental frequency models some

changes have to be made. These changes are discussed in chapter 1.

3.1 Machine equations

The performance of a machine can be described by the machine equations (boldletters are used when referring to matrices or vectors).

IL# = (3.1)

#IRUdtd+= (3.2)

emmechdtd MMJ = (3.3)

Whereis the flux linkage matrix of the machine.

L is the inductance matrix of the machine.

Iis the current matrix of the machine.

Uis the voltage matrix of the machine.

Ris the resistance matrix of the machine.

Jis the combined moment of inertia of machine and turbine/load.

mechis the rotor speed, in mechanical radians per second.

Mmis the mechanical torque applied on the machine axis.

Meis the electrical torque produced/consumed by the machine.

For a general machine, with one field winding, k damper windings in the d-axis andm damper windings in the q-axis, the flux linkage, inductance, current, voltage andresistance matrices look like:

[ ] [ ]TRSTmqkd ##### == fcba #### (3.4)

=

=RR

%SR

SRSS

mqcmqbmqamq

kdfkdckdbkdakd

fk

cmqckd

bmqbkd

amqakd

LL

LL

L00LLL

0LLLLL

0L

LL

LL

LL

Lfcfbfaf

cfccbcac

bfbcbbab

afacabaa

LLLL

LLLL

LLLL

LLLL

(3.5)

[ ] [ ]TRSTmqkd IIIII == fcba IIII (3.6)

[ ] [ ]TRST UU00U == fcba UUUU (3.7)


20/85


- 12 -

=

=

R

S

mq

kd

R

R

R00000

0R0000

00

00

00

00

Rf

a

a

a

R

R

R

R

000

000

000

000

(3.8)

In addition, Mecan be described as:

}*Im{2

3SS I#

mecheM = (3.9)

Where the indices are:

a, b, candSare referring to the three phase stator related quantities

f, kd, mqandRare referring to the two-phase rotor related quantities

fis referring to field winding related quantities

kdis referring to quantities related to the k damper windings of the d-axis

mqis referring to quantities related to the m damper windings of the q-axis

Thus,

a, b, cand Sare the flux linkages of the stator windings a, band c

S* is the complex conjugate of the stator flux linkages

Ia, Ib,IcandISare the currents in the stator windings a, band c

Ua, Ub, Ucand USare the voltages over the stator windings a, band c

f, kd, mqare the flux linkages of the rotor windingsf, kdand mq

If,Ikd,Imqare the currents in the rotor windingsf, kdand mq

Uf, Ukd, Umqare the voltages over the rotor windingsf, kdand mq

Rais the armature resistance

Rfis the resistance of the field winding

Rkdis the resistance of the d-axis damper windings

Rmqis the resistance of the q-axis damper windings

is the rotor speed, in electrical radians per second

LSSis containing the stator self and mutual inductances

LSRis containing the stator to rotor mutual inductances

LSRTis containing the rotor to stator mutual inductances

LRRis containing the rotor self and mutual inductances

As described in chapter 2.2, the matrixesLSSandLSRare time dependent.

Index k andmrefers to the k and the m damper windings of the d- resp. the q-axis.E.g. if there are 1 damper winding in the d-axis and 2 damper windings in the q-axis, the vectorIRand the matrixLRRwould look like:

[ ] [ ]TTmqkdR III qqdff IIIII 21==


21/85

3 Mathematical descriptionTime dependency

- 13 -

=

=

qq

qq

d

LL

LL

L

212

121

1

0

0

00

mq

kd

RR L0

0LL

3.2 Time dependencyThe self-inductances of the stator windings, Laa, Lbband Lcc, are assumed to havethe same maximum value,Laa. These inductances can be divided into one constantterm,Laa0, and one term dependent of the second and higher order harmonics, seeEquation (3.10). The terms with higher order harmonics can often be neglected [9].

2cos)cos( 201

0 aaaa

n

aanaaaa LLnLLL ++= >

(3.10)

Where

Laa0is the constant term of the self-inductances of the stator.

Laa2 is the maximum value of the term that is dependent of the secondorder harmonic.

Laan is the maximum value of the term that is dependent of the nth order

harmonic.

is the electrical displacement angle for the machine.

In the same way, the mutual inductances between the stator windings, Lab,LacandLbc, are assumed to have the same maximum value, Lab. These can be divided intoone constant term,Lab0, and one term dependent of the second order harmonics [9].Due to the distribution of the windings, a 60-degree phase displacement will occur.

)(2cos)2cos( 320320 =++== ababababbaab LLLLLL (3.11)

Where

Lab0 is the constant term of the mutual inductances between the statorwindings.

Lab2 is the maximum value of the term that is dependent of the secondorder harmonic.

The maximum value of the mutual inductances between the stator and the rotor,Laf, Lbf,Lcf, Lakd, Lbkd,Lckd, Lamq, Lbmq andLcmq, are assumed to be independent ofwhich stator winding that is in question, and are called:Laf,LakdandLamq. These aredependent of the fundamental frequency, with a 90-degree lead of the mutualinductances between the stator and the q-axis windings of the rotor [9].

cosdfaf LL = (3.12)

cosdkdakd LL = (3.13)

sinqmqamq LL = (3.14)

Where Ldf, Ldkdand Lqmq, are the mutual inductances between the rotor windingsand the fictive, dq0transformed, local stator windings dand q.

The time dependent matrixesLSSandLSRcan now be described as:


22/85


- 14 -

+

+

+

+

+

=

)2cos()2cos()2cos(

)2cos()2cos()2cos(

)2cos()2cos()2cos(

32223

12

232

231

2

31

231

22

000

000

000

aaabab

abaaab

ababaa

aaabab

abaaab

ababaa

LLL

LLL

LLL

LLL

LLL

LLL

SSL

(3.15)

+++

=

qmq

dkdSR

L

LL

dfL

)sin()cos()cos(

)sin()cos()cos(

)sin()cos()cos(

32

32

32

32

32

32

(3.16)

3.3 The dq0-transformation

In section 2.2.1, the dq0-transformation is described. The transformation and its

inversion are repeated here for convenience.

SSdq0 BUUU =

+

+

=

21

21

21

32

32

32

32

)sin()sin()sin(

)cos()cos()cos(

3

2

(3.17)

dq0$

dq0S UBUU =

++

=

1)sin()cos(

1)sin()cos(

1)sin()cos(

32

32

32

32

(3.18)

As defined earlier, the index dq0reefers to the stator quantities in the local rotorreference system, i.e. the d, q and 0 indices reefers to the fictive d-, q- and 0-

windings. Inserting the transformed components into the expressions for the statorvariables, the following expressions are reached after some reduction oftrigonometric terms.

=

=R

dq0qmq

dkd

dq0I

IL

L

#

00000

0000

000

00 L

L

LL

#

#

#

q

dfd

q

d

(3.19)

++=

0d

q

dtd #

#

dq0dq0Sdq0 #IRU (3.20)

)(23

qddqmech

e I#I#M =

(3.21)

Where

223

00 aaabaad LLLL ++= (3.22)

223

00 aaabaaq LLLL += (3.23)

000 2 abaa LLL = (3.24)

Ld,LqandL0, are the self-inductances of the fictive d-, q- and 0-windingsrespectively.

In the voltage equations, the last term is called the speed voltage, and is aconsequence of the transformation from a stationary to a rotating reference frame.These speed voltages are the dominant components in the voltage equations.


23/85

3 Mathematical descriptionReciprocity

- 15 -

The terms ddt

d and q

dt

d , are called the transformer voltages, and can often be

neglected during transient conditions [9].

The speed of the machine, ,is equal to the time derivative of the angle .

In the rotor flux linkage equations, the transformed currents substitute the statorcurrents and the following relations are calculated for the rotor windings.

=

=R

F

mqqmq2

3

kdfkddkd2

3

fkd

RI

I

LL

LLL

L

#

0000

000

00023

fdf

mq

kd

f LL

#

#

#

(3.25)

RRRR #IRU dtd+= (3.26)

It is obvious that the dq0-transformation results in an equation system where all theinductances are independent of the rotor position. This makes the equations easier

to solve; i.e. it makes the simulation faster.

3.4 Reciprocity

Although the dq0-transformation results in constant inductances, the mutualinductance coefficients between rotor and stator are non-reciprocal.

=

mq

kd

mqqmq2

3

kdfkddkd2

3

fkd

qkq

dkd

mq

kd

I

I

L000L0

0LL00L

0L

00

L0

0L

#

#

f

q

d

fdf

q

dfd

f

q

d

I

I

I

I

LL

L

L

LL

#

#

#

#

0

23

00

00

000

000

00

(3.27)

For example dff

d LI

#=

, but df

d

fL

I

#

23=

. An essential condition for the

existence of a static equivalent circuit is the reciprocity of the mutual inductances[11].

This difficulty arises because a peak value invariant dq0-transformation waschosen. It could easily have been avoided, for example by choosing a powerinvariant transformation. The problem can be solved in different ways, either byusing a reciprocal per unit system, see e.g. [9], or by simply changing the rotor

currents by a factor 32

, see e.g. [11], here the latter is chosen. Defining thereciprocal stator-rotor mutual inductances as:

dfdf LL 23= (3.28)

dkddkd LL 23= (3.29)

qmqqmq LL 23= (3.30)

the reciprocal rotor self inductances and resistances as:


24/85


- 16 -

ff LL 23= (3.31)

kdkd LL ff 23= (3.32)

mqmq LL ff 23= (3.33)

ff RR 23

= (3.34)kdkd RR 2

3= (3.35)

mqmq RR 23= (3.36)

and the reciprocal rotor currents as:

ff II 32= (3.37)

kdkd II 32= (3.38)

mqmq II 32= (3.39)

Inserting these reciprocal quantities, in the flux linkage Equation (3.27), gives:

=

00

000

000

00

000

mq

kd

mqdmq

kdfkddkd

fkd

dkq

dkd

mq

kd

I

I

L000L0

0LL00L

0L

00

L0

0L

#

#

f

q

d

fdf

q

dfd

f

q

d

I

I

I

I

LL

L

L

LL

#

#

#

#

(3.40)

It is now obvious that the mutual inductance coefficients between rotor and statorare reciprocal. Since there is no reason to use the non-reciprocal values of thestator-rotor mutual inductances, the rotor self inductances and resistances or therotor currents, the primes in Equation (3.40) will be omitted in the followingcalculations of this report.

Dividing the matrix into the d-axis and the q-axis flux linkages, and including theeffects of the leakage inductances, the flux linkage equations can be rewritten as:

[ ][ ]

[ ]

+

+

+

=

kdd&kd&kkdfkddkd

fkd

dkd

kd ILLLLL

L

L

#

f

d

ff$fdf

dfd$d$d

f

d

I

I

LLLL

LLLL

#

#

)(

)(

)(

(3.41)

[ ]

+

+=

mq&mq&mqmqqmq

qmq

mq ILLLL

L

#

q$q$qqq ILLL#

)(

)((3.42)

Where

Ldis the leakage inductance of the d-winding

Lfis the leakage inductance of the field winding

Lkdis the leakage inductance of the damper windings in the d-axis

Lqis the leakage inductance of the q-winding

Lmqis the leakage inductance of the damper windings in the q-axis

The leakage inductances of the d- and q-windings,Ld, andLq, are equal, and are

often calledLl, but in Simpow, they are called La, why henceforth this will also bethe case in this report, i.e.

$q$da LLL == (3.43)


25/85

3 Mathematical descriptionReciprocity

- 17 -

In the case when the d-axis rotor currents are zero,If= Ikd= 0, the flux linkage thatwill be mutually coupled between the d-axis circuits is:

daddad ILLIL# )( =+ (3.44)

In this case, the flux linkages in the field and d-damper windings are:

d

ddff

IIL#

dkdkd L# == (3.45)

When SI units are concerned, there is equal mutual flux, i.e. in this case:

kd#==+ fdad #IL# . This is also the case with some per unit systems, but not all

[12]. As a consequence of the equal mutual flux, the mutual inductances are alsoequal. Usually this quantity is calledLad,

dkdL=== dfadad LLLL )( (3.46)

In the same way, it is proved that:

fkd&kdkd LLL === )()( ffad LLL (3.47)

and in the q-axis,

qmq&mqmq LLL === )()( aqaq LLL (3.48)

Now the final reciprocal flux linkage and voltage equations, expressed in SI units,are described as:

=

mq

kd

mqaq

kdadad

ad

aq

ad

mq

kd

II

L000L00LL00L

0L

00

L0

0L

##

f

q

d

fad

q

add

f

q

d

I

I

I

I

LL

L

L

LL

#

#

#

#

000

00

000

000

00

(3.49)

+

+

=

0

0

0

0

000

000

000

000

000

d

q

f

q

d

f

q

d

f

a

a

a

f

q

d

#

#

#

#

#

#

dt

d

I

I

I

I

R

R

R

R

U

U

U

U

mq

kd

mq

kd

mq

kd

mq

kd

#

#

I

I

R00000

0R0000

00

00

00

00

U

U

(3.50)

Where

Ldis the self-inductance of the fictive d-winding.

Lqis the self-inductance of the fictive q-winding.

L0is the self-inductance of the fictive 0-winding.

Lfis the self-inductance of the field winding.

Lkdis the self- and mutual inductances of the kdamper windings of the d-axis.

Lmqis the self- and mutual inductances of the mdamper windings of the d-axis.

Lad

is the mutual inductance between the rotor and the stator circuits in thed-axis.


26/85


- 18 -

Laqis the mutual inductance between the rotor and the stator circuits in theq-axis.

If redefining the matrixes used earlier, with the new reciprocal quantities, theequations can be described in a shorter manor, as:

=

R

dq0

RR%dq0R

dq0Rdq0dq0

R

dq0

I

I

LL

LL

#

#

+

+

=

0#

#

I

I

R0

0R

U

U

R

dq0

R

dq0

R

a

R

dq0d

q

#

#

dt

d

The changes from the original equations, do not affect the torque equation morethan the change from stator quantities to dq0quantities [10], which is expressed as:


with

)(2

3qqqd

meche I#I#M =

(3.52)


27/85

4 Per unit representationMachine per unit bases

- 19 -

4 Per unit representationWhen dealing with power systems, quantities are often presented in per unit values.For synchronous machines, different per unit systems are used, therefore it isimportant to clearly describe how the per unit system used is defined.

4.1 Machine per unit bases

The following per unit bases are used in the machine per unit system:

nSbase UU 32= (4.1)

n

nSbase

U

SI

32= (4.2)

nnbase f 2== (4.3)

n

SbaseSbase

%U# = (4.4)

SbaseSbasenSbase IUSS 23== (4.5)

Sbase

SbaseSbase

I

UZ = (4.6)

Sbase

SbaseSbase

I

#L = (4.7)

nmech

Smechbase

= (4.8)

mech

SbaseSbase

nmech

SbaseSbase

mechbase

nSbase

I#IUSM 2

323

=== (4.9)

n

Smechbase

S

JH

2

2

1 = (4.10)

Where

Unis the nominal phase-phase RMS voltage of the machine.

Snis the nominal power of the machine.

nis the system frequency in [rad/s]fnis the system frequency in [Hz]

mechis the mechanical frequency in [mech. rad/s]

is the electrical frequency in [rad/s]

baseis the system base frequency

Smechbaseis the mechanical base frequency of the machine

H is the per unit inertia constant.


28/85


- 20 -

4.2 Per unit flux linkage equations

4.2.1 General description

When three windings are magnetically coupled as in Figure 4.1, the flux linkage ineach of them can be described as:

313212111 ILILIL# ++= (4.11)

323112222 ILILIL# ++= (4.12)

223113333 ILILIL# ++= (4.13)

Where

iis the flux linkage in winding i

Liis the self-inductance in winding i

Lijis the mutual inductance between winding iandj

Iiis the current flowing through winding i

Figure 4.1 Three magnetically coupled windings

To simplify calculations, the quantities are often presented in per unit; thefollowing per unit system is used by Bhler, [10], and Laible, [13].

=

=

3

2

1

3113

2112

1

3

2

1

3

33

3

223

3

113

2

323

2

22

2

112

1

313

1

212

1

11

3

2

1

1)1(

1)1(

11

i

i

i

x

x

x

i

i

i

#IL

#IL

#IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

(4.14)

Where

I1

L11

U1

dt

d

I2

L22

U2dt

d

I3

L33

U3

dt

d

L12

L23

L31


29/85

4 Per unit representationPer unit flux linkage equations

- 21 -

inis the base flux linkage of winding i

Iinis the base current of winding i

x&is the per unit value of the reactance of winding 1

ijis the decrement factor of windingj relative winding i

iis the translation (or screening) factor of winding i

Here the upper-case letters are used when relating to quantities in SI units andlower-case when relating to per unit values.

The per unit value of the inductance of winding &, l&, has the same value as the perunit value of the reactance of winding &,x&, and is defined as:

11

11

1

111 x

U

IL

#

ILl

n

nn

n

n === (4.15)

Where

nis the base frequency of the system

U&nis the base voltage of winding 1

The decrement factor is defined as:

ji

ijij

LL

L2

1= (4.16)

The translation factor is defined as:

jki

ikiji

LL

LL= (4.17)

To get this description of the flux linkages, the different base factors has to be

defined as:

12

12

L

#I nn = (4.18)

13

13

L

#I nn = (4.19)

nn IL# 222 = (4.20)

nn IL# 333 = (4.21)

4.2.2 dq0-axes flux linkagesThe flux linkages of the dq0-axes was reciprocally described using SI units inchapter 3.4 as:

=

mq

kd

mqaq

kdadad

ad

aq

ad

mq

kd

I

I

L000L0

0LL00L

0L

00

L0

0L

#

#

f

q

d

fad

q

add

f

q

d

I

I

I

I

LL

L

L

LL

#

#

#

#

000

00

000

000

00

(4.22)

The magnetic coupling is dividing the flux linkage matrix into d-, q- and 0-axisparts. A machine with one d-axis winding and two q-axis windings, the d-, q- and0- axis matrixes can be written as:


30/85


- 22 -

=

d

f

d

&dadad

adfad

adadd

d

f

d

I

I

I

LLL

LLL

LLL

#

#

#

11

(4.23)

=

q

q

q

qaqaq

aqqaq

aqaqq

q

q

q

II

I

LLLLLL

LLL

##

#

2

1

2

1

2

1 (4.24)

[ ] [ ][ ]000 IL# = (4.25)

Comparing this with the general per unit description,

=

=

3

2

1

3113

2112

1

3

2

1

3

33

3

223

3

113

2

323

2

22

2

112

1

313

1

212

1

11

3

2

1

1)1(

1)1(

11

i

i

i

x

x

x

i

i

i

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

(4.26)

the following analysis is made for the d-, q- and 0-axis flux linkages.

4.2.2.1 d-axisIf d, f and &d replace the indices &, 2 and 3 respectively, and considering thedefinitions used, the flux linkages of the d-axis can be described as:

=

=

d

f

d

dddd

fddf

d

d

f

d

dbase

dbased

dbase

fbase

dfdbase

dbasedd

fbase

dbasedf

fbase

fbase

ffbase

dbasedf

dbase

dbasedd

dbase

fbase

dfdbase

dbased

d

f

d

i

i

i

x

x

x

i

i

i

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

1111

1

11

11

11

11

11

1 1)1(

1)1(

11

(4.27)

Where

ibaseis the base flux linkage of winding i

Iibaseis the base current of winding i

xdis the per unit value of the reactance of the d-winding

ijis the decrement factor of windingj relative winding i

iis the translation (or screening) factor of winding i

The per unit base current and flux linkage of the d-winding,Idbaseand dbase, are thesame as the per unit bases for the machine,ISbaseand Sbase, i.e.

Sbasedbase II = (4.28)

Sbasedbase = (4.29)

The per unit value of the d-winding inductance, ld, has the same value as the perunit value of the reactance,xd, and is defined as:

d

Sbase

d

Sbase

Sbasedn

Sbase

Sbasedd x

LL

U

IL

#

ILl ====

1 (4.30)

Where

nis the base frequency of the system.


31/85


- 23 -

USbaseis the base voltage of the machine.

LSbaseis the base inductance of the machine.

The decrement factors and translation factors are defined as:

fd

dfdf

LL

L2

1= (4.31)

dd

dddd

LL

L

1

21

1 1= (4.32)

ddf

dfdff

LL

LL

1

1= (4.33)

dfd

dddfd

LL

LL

1

111 = (4.34)

Using the definition of equal mutual inductances, Ldf = Ld&d = Lf&d = Lad, anddividing all inductances with the base inductance of the machine, LSbase, all

inductances can be described as the corresponding per unit reactance value.

dSbase

d

fSbase

f

adSbase

ad

dSbase

d

xL

L

xL

L

xL

L

xL

L

111

1

1

1

=

=

=

=

(4.35)

This makes the decrement factors and the translation factors somewhatunnecessary, and the flux linkages can be described as:

=

d

f

d

d

ad

d

ad

f

ad

f

ad

d

d

f

d

i

i

i

x

x

x

x

x

x

x

x

x

'

'

'

1

11

2

2

1

1

1

11

(4.36)

To get this description of the flux linkages, the different base factors has to bedefined as:

ad

Sbase

adSbase

Sbase

ad

Sbase

df

Sbasefbase

x

I

xL

#

L

#

L

#I ==== (4.37)

ad

Sbase

ad

Sbase

dd

Sbasedbase

x

I

L

#

L

#I ===

11 (4.38)

Sbasead

ffbaseffbase #

x

xIL# == (4.39)

Sbasead

ddbaseddbase #

x

xIL# 1111 == (4.40)


32/85


- 24 -

4.2.2.2 q-axisIn the same way as for the d-axis, the flux linkages of the q-axis can be describedas:

=

=

q

q

q

q

aq

q

aq

q

aq

q

aq

q

q

q

q

qbase

qbaseq

qbase

qbaseqq

qbase

qbaseqq

qbase

qbaseqq

qbase

qbaseq

qbase

qbaseqq

qbase

qbaseqq

qbase

qbaseqq

qbase

qbaseq

q

q

q

i

ii

x

x

x

x

x

x

x

x

x

i

ii

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

#

IL

2

1

22

211

2

2

1

2

22

2

121

22

1

221

1

11

11

22

11

2

1

1

1

11

(4.41)

where

qSbase

qSbase

Sbaseqn

Sbase

Sbaseqq x

LL

U

IL

#

ILl ====

1 (4.42)

qSbase

q

qSbase

q

aqSbase

aq

xL

L

xL

L

xLL

22

11

1

1

1

=

=

=

(4.43)

aq

Sbase

aq

Sbase

qq

Sbaseqbase

x

I

L

#

L

#I ===

11 (4.44)

aq

Sbase

aq

Sbase

qq

Sbaseqbase

x

I

L

#

L

#I ===

22 (4.45)

Sbaseaq

qqbaseqqbase #

x

xIL#

1111 == (4.46)

Sbaseaq

qqbaseqqbase #

x

xIL#

2222 == (4.47)

4.2.2.3 0-axisIn the zero axis, there is only one winding, therefore the per unit value flux linkageof the zero axis is equal to

000 ix= (4.48)where

00001

xL

L#

ILl

SbaseSbase

Sbase === (4.49)

4.2.3 Conclusions

Rearranging the equations into the stator dq0 quantities and the rotor f, kd, mqquantities, the flux linkages matrix can be formulated, see Equation (4.38).

For a model with fewer windings in the d- and/or q-axes, the flux linkages can be

described using Equation (4.38), if the rows and columns of the matrix, that arecorresponding to the windings that are not included, are deleted.


33/85


- 25 -

=

q

q

d

f

q

d

q

aq

q

aq

q

aq

q

aq

d

ad

d

ad

f

ad

f

ad

q

d

q

q

d

f

q

d

i

i

i

i

i

i

i

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

2

1

1

0

22

211

211

2

20

2

1

1

0

10000

10000

00100

00100

000000

110000

001100

(4.50)

Here

xdis the direct axis synchronous reactance.

xais the armature leakage reactance, often calledxl.

xad= xd- xais the mutual reactance between the d-axis windings.

xfis the field winding reactance.

x&dis the damper winding reactance of damper &in the d-axis.

xqis the quadrature axis synchronous reactance.

xaq= xq- xais the mutual reactance between the q-axis windings.

x&qis the damper winding reactance of damper &in the q-axis.

x2qis the damper winding reactance of damper 2in the q-axis.

4.2.4 Definition of the transient and sub-transient reactances

To be able to define the field winding reactance and the different damper windingreactances, the transient- and subtransient reactances of the d-and q-axis have to bedefined.

4.2.4.1 d-axisThe subtransient reactance is defined as the initial stator flux linkage per unit ofstator current, with all the stator circuits shorted and previously unenergised [12],i.e. during this subtransient time, the stator flux linkages are described as:

qqq

ddd

ix

ix

==

(4.51)

where

xd is the subtransient reactance of the d-axis

xqis the subtransient reactance of the q-axis

Directly after the voltage is applied on the terminals, the rotor flux linkages, f,&d, &qand 2q, are still zero, since they can not change instantly [12]. The fluxlinkages of the d-axis during this subtransient time, looks like:


34/85


- 26 -

=

=

d

f

d

d

ad

d

ad

f

ad

f

ad

dd

d

f

d

i

i

i

x

x

x

x

x

x

x

x

x

1

11

2

2

1

1

1

11

0

0

(4.52)

This gives the rotor currents, ifand i&d, during the subtransient time as:

d

addf

adfadd

d

addf

addadf

ixxx

xxxi

ixxx

xxxi

21

2

1

21

21

=

=

(4.53)

Inserting these equations in the expression for d, gives the subtransient stator fluxlinkage of the d-axis:

dad

addf

adfddd ix

xxx

xxxx

+= 2

21

1 2 (4.54)

From the definition of the subtransient reactance, it is obvious that:

22

1

1 2 ad

addf

adfddd x

xxx

xxxxx

+= (4.55)

When the damper current has decayed to zero, or if there is no damper winding, themachine is in the transient stage, and the stator flux linkages are in [12] described

as:

qqq

ddd

ix

ix

=

=

(4.56)

where

xd is the transient reactance of the d-axis

xq is the transient reactance of the q-axis

In the same way as in the subtransient case, but with i&d= 0, the transient statorflux linkage in the d-axis is calculated as:

df

addd ixxx

=

2

(4.57)

From the definition of the transient reactance, it is obvious that:

f

addd

x

xxx

2

= (4.58)

The field winding reactance is now defined as in Equation (4.59):

2

dd

adf

xx

xx

= (4.59)

Inserting this into the equation for the subtransient reactance, gives the definitionof the damper winding reactance,x&d, as:


35/85

4 Per unit representationPer unit voltage equations

- 27 -

))((1

dd

adadadd

xx

xxxxxx

= (4.60)

4.2.4.2 q-axisIn the same way, the reactances of the q-axis damper windings can be defined. For

the subtransient period, both damper windings are active, and during the transientperiod, the current through the second winding has decayed to zero. Therefore, thesubtransient and the transient reactances of the q-axis damper windings aredescribed as:

22

21

12 2 aq

aqqq

aqqqqq x

xxx

xxxxx

+= (4.61)

q

aqqq

x

xxx

1

2

= (4.62)

The reactance of the first damper winding,x&q,is now defined as:

2

1qq

aqq

xx

xx

= (4.63)

Inserting this into the subtransient reactance equation, gives the definition of thesecond damper winding reactance,x2qas:

))((2

qq

aqaqaqq

xx

xxxxxx

= (4.64)

For a model with only one damper winding in the q-axis, the sub-transientreactance is used in Equation (4.63), instead of the transient reactance [3], [13].

4.3 Per unit voltage equations

4.3.1 Stator voltages

In chapter 3.4, the stator voltages, in SI units, are described as:

++=

0d

q

dtd #

#

dq0dq0Sdq0 #IRU (4.65)

Using the per unit bases for the machine, the stator voltages can be described as:

Sbased

q

Sbasedtd

SbaseSbaseSbase ##IZU

++=

0dq0dq0Sdq0 'iru (4.66)

Dividing both sides with USbase, gives the stator voltages in per unit, as:

based

q

dtd

base

++=

0

1dq0dq0Sdq0 'iru (4.67)

4.3.2 Rotor voltages

In chapter 3.4, the rotor voltages, in SI units, are described as:


36/85


- 28 -

RRRR #IRU dtd+= (4.68)

Separating the system into equations:

mqmqmq

kdkdkd

#IR

#IR

dt

d

dtd

#dt

dIRU ffff

+=

+=

+=

0

0 (4.69)

4.3.2.1 Field windingThe field winding voltage can be written as:

fbaseffbasefffbasef #dt

dIiRUu += (4.70)

Defining Ufbaseas:

fbaseffbase IRU = (4.71)

Using this and the Equation (4.27), fbase= LfIfbase, gives:

ff

f

fbasef

fbase

fbase

fbase

R

L

IR

#

U

#=== (4.72)

Dividing both sides of Equation (4.65) with Ufbase, the per unit field voltage isdescribed as:

ffffdt

diu += (4.73)

wherefis the time constant of the field winding

4.3.2.2 Damper windingsThe damper winding voltages can be described as:

mqbase#'IiR

#'IiR

mqmqbasemqmq

kdbasekdkdbasekdkd

dt

d

dt

d

+=

+=

0

0(4.74)

Dividing the equations with RkdIkdbase andRmqImqbase respectively, and using the

Equations (4.28), (4.34) and (4.35) to define:

kdkd

kd

kdbasekd

kdbase (R

L

IR

#== (4.75)

and

mqmq

mq

mqbasemq

mqbase(

R

L

IR

#== (4.76)

gives


37/85

4 Per unit representationPer unit voltage equations

- 29 -

mqmqmq

kdkdkd

'(i

'(i

dt

d

dt

d

+=

+=

0

0(4.77)

where

kdare the time constants for the damper windings in the d-axis

mqare the time constants for the damper windings in the q-axis

4.3.2.3 Summary of the rotor voltage equationsFrom the sections 4.3.2.1 and 4.3.2.2, the per unit rotor voltages are summarisedas:

+

=

mq

kd

mq

kd

mq

kd

'

'

(

(

i

i

ffff

dt

diu

0

0 (4.78)

or shorter

RRRR '(iu dtd+= (4.79)

4.3.3 Conclusions

For a machine with one field winding, one damper winding in the d-axis and twodamper windings in the q-axis, the per unit voltage equations derived in sections4.3.1 and 4.3.2, are assembled as:

base

d

q

q

q

d

f

q

d

q

q

d

f

q

q

d

f

q

d

a

a

a

f

q

d

dt

d

i

i

i

i

i

i

i

rr

r

u

uu

u

base

base

base

+

+

=

0

0

0

0

0

1000000

0100000

0010000

0001000

000000

000000

000000

0

0

0

2

1

1

0

2

1

1

1

1

1

2

1

1

00

(4.80)

Where

rais the armature resistance.

fis the time constant of the field winding.

&dis the time constant of the first d-damper winding.

&qis the time constant of the 1stq-damper winding.

2qis the time constant of the 2nd

q-damper winding.

4.3.4 Definition of the time constants

To be able to define time constants of the windings in the d- and q-axis, thetransient- and open circuit time constants of the d-and q-axis are introduced.

In Simpow, the following definitions are used for the time constants of thewindings in the d- and q-axis.

4.3.4.1 d-axisIn Laible, [13], the time constants of the d-axis are defined as:


38/85


- 30 -

ddf 10 = (4.81)

df

ad

d

df

dd

xx

x

1

0

1

01

1

==

(4.82)

whered0 is the transient open circuit time constant of the d-axis.

d0 is the subtransient open circuit time constant of the d-axis.

f&dis the decrement factor of winding &d relative windingf.

This definition is approximate, and is only valid when f >> &d, i.e. when

d

d

f

f

R

L

R

L

1

1>> , which mostly is the case [13].

In the case with no damper winding is modelled in the d-axis, 0df = .

4.3.4.2 q-axisAnalogously, in the q-axis, the time constants are defined as:

qqq 201 = (4.83)

qq

aq

q

qq

qq

xx

x

21

0

21

02

1

==

(4.84)

where

q0 is the transient open circuit time constant of the q-axis.

q0 is the subtransient open circuit time constant of the q-axis.

f&q2qis the decrement factor of winding 2q relative winding &q

This definition is approximate, and is only valid when &q >> 2q, i.e. when

q

q

q

q

R

L

R

L

2

2

1

1>> .

In the case with only one damper winding modelled in the q-axis, 01 qq = .

4.4 Per unit torque equations

In chapter 3.4, the torque is described, in SI units, as:


with

)(2

3qqqd

meche I#I#M =

(4.86)

Using the same per unit bases for the machine as the previous sub-chapter, with theadditional bases:


39/85

4 Per unit representationPer unit torque equations

- 31 -

n

Smechbase

mech

SbaseSbase

nmech

SbaseSbase

Smechbase

nbase

nmech

Smechbase

S

JH

I#IUSM

2

23

23

2

1

=

===

=

(4.87)

WhereH is the per unit inertia constant. The torque can be written as:

baseemn

Smechbasedtd

Smechbase

n MmmHS

)(22

=

(4.88)

Dividing both sides withMbase, gives the torque in per unit, as:

emn

dtd mmH =

2 (4.89)

In the above equation,

dtd

ndtd

nndtd

n

ndtd

ndtd 11 ==

=

= (4.90)

where

is the rotor speed deviation

is the deviation angle of the rotor

The per unit electrical torque, me, is now:

dqqde iim = (4.91)

4.4.1 Fundamental frequency modelsFor the fundamental frequency models, an extra term must be added to theelectrical torque, the negative-sequence breaking torque, me2, [9]. That is, for thetransient stability simulations, the per unit electrical torque is changed to:

2222 )()()( irriimiim adqqdedqqde +=+= (4.92)

where

r2is the negative sequence resistance.

rais the armature resistance.

i2is the negative sequence current.

4.4.2 Models without damper windings

The damping torque is linearly dependent of the speed deviations of the machine[12], and is the result of an electrical effect. I.e. if an excess torque changes thespeed of the machine, this relative movement generates currents in the rotorcircuits. Losses caused by these currents will strive to counteract the change inmotion [15].

When no damper windings are modelled, no currents will occur in the rotorcircuits, i.e. no electrical damping torque will be included in the model.

To obtain damping in the simpler models without rotor damper circuits, a substitutefor the electrical damping torque must be implemented in the torque equation [9].This damping torque is called md, and is defined as:


40/85


- 32 -

nd Dm

= [p.u.] (4.93)

whereDis the damping factor.

In these cases, the per unit torque equation is described as:

demn

dtd

mmmH =

2 (4.94)

4.5 Global - local per unit transformation

At the node where the machine is connected to the net, the quantities are describedusing system per unit bases. A per unit transformation constant is therefore neededwhen performing a global-local transformation in per unit.

In Simpow, the global voltages are transformed to the local system, and the localcurrents are transformed to the global system, why the transformation constants CUand CIare defined as:

n

basenodeU

UUC =

basenetwn

basenodenI

SU

USC =

where

Unodebaseis the nominal phase-phase voltage for the machinenode

Snetwbaseis the 3-phase network power base

The transformation constants are used in the transformation equations in thefollowing manner:

for the global to local transformation:

UUd CUUCUUU sincoscossin +== (4.95)

UUq CUUCUUU cossinsincos +=+= (4.96)

and the local to global transformation looks like:

IqdIqd CIICIII sincoscossin =+= (4.97)

IqdIqd CIICIII cossinsincos +=+= (4.98)

Where

is the electrical displacement angle for an instantaneous value model.

is the electrical displacement angle for a fundamental frequency model.

For the fundamental frequency models, transforms are made between the globaldq-system and the local dq0-system. For the instantaneous value models,transforms are made between the global positive-, negative- and 0-sequencesystems and the local dq0-system.


41/85

5 ModellingTransient-/Machine stability

- 33 -

5 ModellingWithin Simpow, there are four different synchronous machine models in thestandard library; a list of them is shown below.

Type 1. Model with four rotor windings including saturation; one fieldwinding, two d-axis and one q-axis damper windings

Type 1A. As Type 1, but without saturation

Type 2. Model with three rotor windings including saturation; one fieldwinding, one d-axis and one q-axis damper windings


Type 3. Model with one rotor windings including saturation; 1 fieldwinding


Type 4. Model with constant voltage behind transient reactanceIn this chapter, first the differences between the fundamental frequency, andinstantaneous value models are discussed, then the initial conditions for themachine models are described. At last, the four different synchronous machinemodels are discussed, starting the simplest model, Type 4, and continuing with themore and more advanced models, Type 3A, 2A and 1A.

5.1 Transient-/Machine stability

When the machine stability of a power system is being simulated, instantaneousvalue models of the power system components are used. In Simpow, thissimulation mode is called Masta. For transient stability simulations, fundamental

frequency models are used; this mode is in Simpow called Transta.

For the fundamental frequency models, the following changes are made:

The transformer voltages, ddt

d and q

dt

d , are neglected.

The models are only used for the positive sequence quantities, i.e. the zero axis

voltage and flux linkage equations, 0001

dt

diru

na += and 000 ix= , are

neglected.

The negative and zero sequences are modelled as two impedance circuits, with

the resistance and reactance: r2, x2and r0, x0respectively. The electrical torque is modelled with an extra term, the negative-sequence

breaking torque: 2222 )( irrm ae = .

With the transformer voltages neglected, the stator voltage equations only containsfundamental frequency components, and appear as algebraic equations. When alsothe zero axis voltage equation is neglected, the fundamental frequency models willhave three state variables less than the instantaneous value models. The machinemodels discussed in the following sub-chapters are of order 5, 6, 8 and 9, referredto the instantaneous value order; i.e. the fundamental frequency models are of order2, 3, 5 and 6. In Simpow, these models are named the synchronous machine modelsType 4, Type 3, Type 2and Type &respectively.


42/85


- 34 -

5.2 Initial conditions at steady state

At time zero, the initial conditions are calculated. From the load-flow calculation,the initial node voltage and angle, ut0 andi, and the active- reactive power outfrom the generator node,pt0, qt0, are known.

At steady state, all time derivatives are zero. The machine is assumed to run under

symmetrical conditions,0= u0=i0=0. When deriving the stator circuit quantities,ud0, uq0, d0, q0, id0and iq0,saliency is neglected, i.e.xd= xq, but when deriving therotor circuit quantities, f0, &d0, &q0, 2q0, and if0, saliency is included. With theseconsiderations in mind, the voltage equations derived in the previous chapters, seeEquation (4.80), will be changed to:

n

d

q

q

q

d

f

q

d

a

a

a

f

q

d

i

i

i

i

i

i

i

r

r

r

u

u

u

+

=

00

0

0

0

10000000100000

0010000

0001000

000000

000000

000000

00

0

00

0

02

01

01

0

00

0

0

0

0

0

(5.1)

Where the extra index, 0, is referring to the initial moment of time.

Obviously, the currents in the damper windings are zero, i.e. i1d0 = i1d0 = i1d0.This was expected, since the damper windings only affect the machine duringunstable conditions. This changes the flux linkage equation (4.46) to:

=

0

0

0

2

21

211

2

2

02

01

01

0

0

0

00

00

0

10

00

10

f

q

d

q

aq

q

aq

d

ad

d

ad

f

ad

d

d

q

q

d

f

q

d

i

i

i

x

x

x

x

x

x

x

xx

x

x

x

(5.2)

The initial node voltage is defined as

00 qd juu +=t0u (5.3)

Inserting the voltages from Equation (5.1), gives:

)( 0000 dn

qaqn

da irjir

++=t0u (5.4)

Now, the machine is assumed to run at synchronous speed, i.e. = n, and withthe flux linkages from Equation (5.2), ut0can be written as:

))(()( 00000 fddqaqdda iixirjixir ++=t0u (5.5)

Rearranging, and introducing the initial steady-state current out from the generatorto the node, it0= id0+ jiq0, and from Equation (5.1), uf0= if0. This gives the finalequation for the node voltage as:

0)( fda jujxr += t0t0 iu (5.6)

where


43/85

5 ModellingInitial conditions at steady state

- 35 -

uf0is the initial steady-state field voltage

The initial steady-state model can thus be viewed as the initial field voltage, uf0,behind an impedance, ra + jxd, see Figure 5.1.

Figure 5.1 Steady-state equivalent circuit

From the steady-state equivalent circuit, the internal load angle, i, and the dq-voltages, udand uq, can be calculated [9]. Using the steady-state equivalent circuit,a voltage diagram is drawn, see Figure 5.2, and the Equations (5.7-5.11) are stated.

Figure 5.2 Voltage diagram of steady-state equivalent circuit

itd uu sin00 = (5.7)

itq uu cos00 = (5.8)

Wheresiniand cosican be described as:

0

00 sincossinf

itaitdi

u

irix

= (5.9)

0

000 sincoscosf

itditati

u

ixiru

++= (5.10)

The initial field winding voltage is derived using the Pythagorean theorem.

2

00

2

0000 )sincos()sincos( itaitditditatf irixixiruu +++= (5.11)With the initial power, out from the generator into the node, defined as:

ut0i

uf0i+i

xdra

it0

t0, qt0

~

rait0sin()

xdit0sin()

q

rait0cos() xdit0cos()

ut0rait0

xdit0

it0

uf0

i

d

uq0

ud0


44/85


- 36 -

))(( 00000000 qdqdtttt jiijuuiujqp +==+ (5.12)

this gives the real and imaginary power as:

ittt iup cos000 = (5.13)

ittt iuq sin000 = (5.14)

The initial field voltage, uf0, the dq-voltages, ud0, uq0, and the internal load angle, i,can be written as:

( )

+++

+

+=

20

020

022

2

20

0

2

20

000 21

t

tq

t

tada

t

t

t

ttf

u

qx

u

prxr

u

q

u

puu (5.15)

0

0

0

0

0

00f

t

ta

t

td

tdu

u

qr

u

px

uu

= (5.16)

0

0

0

0

00

00f

t

td

t

tat

tqu

uqx

upru

uu

++

= (5.17)

)arctan(0

0

q

di

u

u= (5.18)

Equations (5.14)-(5.17) are the initial settings of the machine, and all othervariables can be calculated from these.

The initial values of the currents id0and iq0can be derived from the power equation,(5.12), as:

20

002

0

000

20

002

0

000

t

td

t

tqq

t

tq

t

tdd

u

qu

u

pui

uqu

upui

=

+=

(5.19)

From Equation (5.1), the flux linkages, d0and q0, are described as:

+

=

0

0

0

0

0

0

0

0

q

d

a

an

d

qn

q

d

i

i

r

r

u

u

Equation (5.2) can be rearranged, with the magneto motive forces of the d- and q-axis, Wd0and Wq0, separated from the first two rows:

+

=

02

01

01

0

0

0

0

0

0

02

01

01

0

0

0

000

000

000

100

00

00

qq

aq

qq

ad

dd

ad

df

ad

q

d

f

q

d

f

ad

a

a

q

q

d

f

q

d

Wx

x

Wx

x

Wx

x

Wx

x

W

W

i

i

i

x

x

x

x

(5.20)


45/85

5 ModellingType 4

- 37 -

From row 1 and 2 in (5.20), Wd0 and Wq0 are extracted, so the unknown fluxlinkages, f0, &d0, &q0and 2q0, can be calculated.

The magneto motive forces expressed as a function of currents, are:

aqqq

addfd

xiW

xiiW

00

000

=

=(5.21)

Without saturation, equations (5.16) and (5.17) gives the final value of ud0and uq0,and the other variables can be calculated as in equations (5.18) (5.21).

In the case when saturation is included, equations (5.16) and (5.17) gives startvalues for ud0 and uq0. The initial state can only be obtained after an iteration

procedure by means of equations (5.18) (5.22), with the change in Equation(5.20) from magneto motive forces to saturation functions, i.e. change Wdand Wqtofd(Wd)andfq(Wq).

2220 qdt uuu += (5.22)

The concept of saturation is discussed in chapter 5.7.

5.3 Type 4

In classical theory, the machine is modelled as a constant voltage behind a transientreactance; this is generally called the classical model. The Type 4 model inSimpow, is a modified classical model, and can be used for transient stabilitystudies. However, in most cases, the more accurate models, Type &and 2, are to

prefer, since in these models there is a more adequate representation of theelectrical damping [15].

5.3.1 Assumptions for the classical model

For a typical classical model, as can be found in the literature, see section 5.3.1Classical Model in Kundur [9], the following assumptions are made:

1. The flux linkages of the field winding and of the first q-damper winding areconstant.

2. The effect from other damper circuits is neglected.

3. Transient saliency is ignored, i.e.xd = xq.

4. The constant voltage and reactance are not affected by speed variations.

5. Armature resistance,Ra, can be neglected.

In section 4.9.3.3.1 Classical Model IEEE Brown Book [9], the fo

modelin synchronous machine.pdf

Documents

Transcript of modelin synchronous machine.pdf