detailed description of synchronous machine models used in

Detailed Description ofSynchronous Machine Models

Used in SimpowEmil Johansson

Master Thesis B-EES-0201

Electric Power SystemsDepartment of Electrical Engineering

Royal Institute of TechnologySweden

Master Thesis

Submitted to the Royal Institute of Technology, KTH,in partial fulfilment of the requirements for the degree of

Master of Science Electrical Engineering.

Västerås 2002

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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, inthe 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.

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- v -

Table of Contents1 Introduction ........................................................................11.1 Preface ................................................................................11.2 Project description................................................................11.3 Outline .................................................................................11.4 Acknowledgement................................................................2

2 Theory.................................................................................32.1 Physical description..............................................................32.1.1 Multiple pole machines.........................................................32.1.2 Direct and quadrature axis ...................................................42.2 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 ..............................................................61.3 Reference systems in Simpow .............................................71.3.1 Fundamental frequency models ...........................................71.3.2 Instantaneous value models.................................................81.3.3 Conclusions .........................................................................9

3 Mathematical description ................................................113.1 Machine equations .............................................................113.2 Time dependency...............................................................133.3 The dq0-transformation ......................................................143.4 Reciprocity .........................................................................15

4 Per unit representation ....................................................194.1 Machine per unit bases ......................................................194.2 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 .................................................................................274.3 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 .................................................................................294.3.4.2 q-axis .................................................................................304.4 Per unit torque equations ...................................................304.4.1 Fundamental frequency models .........................................31

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4.4.2 Models without damper windings........................................314.5 Global - local per unit transformation ..................................32

5 Modelling ..........................................................................335.1 Transient-/Machine stability ................................................335.2 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 395.4 Type 3A..............................................................................405.4.1 Instantaneous value model.................................................405.4.1.1 Equivalent circuit ................................................................415.4.2 Fundamental frequency model ...........................................425.5 Type 2A..............................................................................435.5.1 Instantaneous value model.................................................435.5.1.1 Equivalent circuit ................................................................435.5.2 Fundamental frequency model ...........................................445.6 Type 1A..............................................................................455.6.1 Instantaneous value model.................................................455.6.1.1 Equivalent circuit ................................................................465.6.2 Fundamental frequency model ...........................................475.7 Saturation...........................................................................48

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

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

8 Conclusions and future work ..........................................578.1 Conclusions........................................................................578.2 Future work ........................................................................57

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

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

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

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

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

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

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

unit .................................................................................................................. 39Figure 5.4 Equivalent circuit for Type 4, fundamental frequency model,

symmetrical components ................................................................................ 40Figure 5.5 Equivalent negative- & 0-sequence circuits, fundamental frequency

model in SI units ............................................................................................. 40Figure 5.6 Equivalent d-circuit for Type 3A, instantaneous value model in SI units

........................................................................................................................ 42Figure 5.7 Equivalent d-circuit for Type3A, leakage inductance instantaneous

value model in SI units ................................................................................... 42Figure 5.8 Equivalent q-circuit for Type3A, in SI units ......................................... 42Figure 5.9 Equivalent dq-circuits for Type2A, leakage inductance instantaneous

value model in SI units ................................................................................... 44Figure 5.10 Equivalent dq-circuits for Type1A, leakage inductance instantaneous

value model in SI units ................................................................................... 47Figure 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

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1 IntroductionPreface

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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, ABBUtilities, in Västerås 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 PrefaceWithin 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 1980�s, based on equationssummarised in the technical reports [1] and [2]. The contents of the reports aremainly 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 user�smanual [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 havebeen 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 machine

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

Detailed Description of Synchronous Machine Models Used in Simpow

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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, theseDSL-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 AcknowledgementDuring 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 Söder, 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 Öhrström, 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 haven�t 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 würfelt 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 onmy work and the various conversations during Sunday dinner, after the Lützen-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.

2 TheoryPhysical description

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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, ω[rad/s], and angle, δ φ ϕ θ [rad, degree]. Bold letters are used whenreferring 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 descriptionTo 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 modulecalled 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 machinesSynchronous 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

ibuc ic

ifd

ufd

i1d

i1q

θ

a

b c

1q

1d

fd

q

da

ω

δ

global reference axis


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

pairspolemechωω = (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 axisTwo 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 dependencyDue to the non-uniform air gap, the reluctance of the magnetic path, ℜ, will berotor position dependent, i.e. time dependent. This is due to the fact that:

lt ∝ℜ )( (2.2)

Where l represents 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:

ℜ∝ 1L (2.3)

Where L represents 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-transformationTo 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.

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

2 TheoryTime dependency

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ua

ia

ub

ibuc ic

ifd

ufd

i1d

i1q

θ

a

b

c1q

1d

fd

q

da

ω

uα

iα

uβiβ

ifd

ufd

i1d

i1q

θ

α

1q

1d

fd

q

dα

ω

udid

uq

iq

ifd

ufd

i1d

i1q

θ

d

1q

1d

fd

q

dα

ω

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


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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, b andc, to a two-phase global reference system, indices α and β, see Figure 2.2b.

[ ]cbaβα UaaUUkjUU 2++=+=αβU (2.4)

Where k is an arbitrary transformation constant and

32πjea −= (2.5)

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

[ ]cba UaaUU 232 ++=αβU (2.6)

Now Uα and Uβ in equation 2.4 becomes:

[ ]cba UUUU 21

21

32 −−=α (2.7)

[ ]cb UUU −=3

1β (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 d and q, see Figure 2.2c.

θjqd ejUU −=+= αβdq UU (2.9)

Here θ 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:

[ ]θθ βα sincos UUUd += (2.10)

[ ]θθ βα cossin UUUq +−= (2.11)

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

[ ])cos()cos(cos 32

32

32 ππ θθθ ++−+= cbad UUUU (2.12)

[ ])sin()sin(sin 32

32

32 ππ θθθ ++−+−= 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

UUU

UUU

21

21

21

32

32

32

32

0

)sin()sin()sin()cos()cos()cos(

32 πθπθθ

πθπθθ(2.15)

2 TheoryReference systems in Simpow

- 7 -

or shorter:

Sdq0 BUU = (2.16)

Where,

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

US are the three-phase stator voltages.

B is 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 BΨΨ = (2.18)

The inverse transform is given by:

dq01

dq0S UBUU −=��

�

�

��

�

�

+−+−−−

−=

1)sin()cos(1)sin()cos(1)sin()cos(

32

32

32

32

πθπθπθπθ

θθ(2.19)

2.3 Reference systems in SimpowEvery synchronous machine has its own local reference system, in which itsarmature currents and voltages are expressed. In the network, these quantitiesbelong 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: δ or θ, ω, Ψd, Ψq, Ψfd, Ψkd, Ψkq, 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:

θjqd ejUU −=+= αβdq UU (2.20)

This is valid when θ 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 modelsFor a fundamental frequency model, the reference frame is a complex planerotating with the system frequency ωn [rad/sec]; here the electrical displacementangle δ 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

( πδ −−=

jeαβdq UU (2.21)


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

[ ]δδ βα cossin UUU d −= (2.22)

[ ]δδ βα sincos UUU q += (2.23)

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

[ ]δδα cossin qd UUU += (2.24)

[ ]δδβ sincos qd UUU +−= (2.25)

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

devitit δδφδ ++= 00 (2.26)Where

φit0 is the machine-node voltage angle, calculated in a power-flow

δit0 is the internal load angle of the machine at time t=0

δdev is the integrated deviation angle, and is calculated as:

� −=t

ndev dt0

)( ωωδ (2.27)

ω is the angular frequency of the machine

2.3.2 Instantaneous value modelsFor an instantaneous value model, the reference frame is the dq-axes of a referencemachine rotating with the angular frequency ωref. Here the electrical displacementangle θ 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:

[ ]θθ βα sincos UUUd += (2.28)

[ ]θθ βα cossin UUUq +−= (2.29)

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

[ ]θθα sincos qd UUU −= (2.30)

[ ]θθβ cossin qd UUU += (2.31)

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

)( refdevrefref δδφδθ ++−= (2.32)Where

δ is defined in Equation (2.26)

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

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

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

2 TheoryReference systems in Simpow

- 9 -

� −=t

nrefrefdev dt0

)( ωωδ (2.33)

ωref is the angular frequency of the reference machine.

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

[ ] [ ]θθδδ βαβα sincoscossin UUUUUd +=−= (2.34)

[ ] [ ]θθδδ βαβα cossinsincos UUUUUq +−=+= (2.35)

and the local to global transformation looks like:

[ ] [ ]θθδδα sincoscossin qdqd UUUUU −=+= (2.36)

[ ] [ ]θθδδβ cossinsincos qdqd UUUUU +=+−= (2.37)

Where

θ 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.

δ 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 θ does not have to be equal to δ -π/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 θ 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 transformingbetween the global and local reference systems. This matter will be furtherdiscussed in chapter 4.4.


- 10 -

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

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

ILΨ ⋅= (3.1)

ΨIRU dtd+⋅−= (3.2)

emmechdtd MMJ −=ω (3.3)

Where

ΨΨΨΨ is the flux linkage matrix of the machine.

L is the inductance matrix of the machine.

I is the current matrix of the machine.

U is the voltage matrix of the machine.

R is the resistance matrix of the machine.

J is the combined moment of inertia of machine and turbine/load.

ωmech is the rotor speed, in mechanical radians per second.

Mm is the mechanical torque applied on the machine axis.

Me is 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:

[ ] [ ]TRST

mqkd ΨΨΨΨΨ == fcba ΨΨΨΨ (3.4)

��

��

�−=

��

�

�

��

�

�

−−−−−−−−−

=RR

ΤSR

SRSS

mqcmqbmqamq

kdfkdckdbkdakd

fk

cmqckd

bmqbkd

amqakd

LLLL

L00LLL0LLLLL0L

LLLLLL

Lfcfbfaf

cfccbcac

bfbcbbab

afacabaa

LLLLLLLLLLLLLLLL

(3.5)

[ ] [ ]TRST

mqkd IIIII == fcba IIII (3.6)

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


- 12 -

��

��

�

−=

��

�

�

��

�

�

−−

−=

R

S

mq

kd

RR

R000000R000000000000

Rf

a

a

a

RR

RR

000000000000

(3.8)

In addition, Me can be described as:

}*Im{23

SS IΨω

ωmecheM = (3.9)

Where the indices are:

a, b, c and S are referring to the three phase stator related quantities

f, kd, mq and R are referring to the two-phase rotor related quantities

f is referring to field winding related quantities

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

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

Thus,

Ψa, Ψb, Ψc and ΨΨΨΨS are the flux linkages of the stator windings a, b and c

ΨΨΨΨS* is the complex conjugate of the stator flux linkages

Ia, Ib, Ic and IS are the currents in the stator windings a, b and c

Ua, Ub, Uc and US are the voltages over the stator windings a, b and c

Ψf, ΨΨΨΨkd, ΨΨΨΨmq are the flux linkages of the rotor windings f, kd and mq

If, Ikd, Imq are the currents in the rotor windings f, kd and mq

Uf, Ukd, Umq are the voltages over the rotor windings f, kd and mq

Ra is the armature resistance

Rf is the resistance of the field winding

Rkd is the resistance of the d-axis damper windings

Rmq is the resistance of the q-axis damper windings

ω is the rotor speed, in electrical radians per second

LSS is containing the stator self and mutual inductances

LSR is containing the stator to rotor mutual inductances

LSRT is containing the rotor to stator mutual inductances

LRR is containing the rotor self and mutual inductances

As described in chapter 2.2, the matrixes LSS and LSR are time dependent.

Index k and m refers 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 vector IR and the matrix LRR would look like:

[ ] [ ]TTmqkdR III qqdff IIIII 21==

3 Mathematical descriptionTime dependency

- 13 -

��

�

�

��

�

�

=��

��

�=

qq

qq

d

LLLL

L

212

121

1

00

00

mq

kdRR L0

0LL

3.2 Time dependencyThe self-inductances of the stator windings, Laa, Lbb and 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].

θθ 2cos)cos( 201

0 aaaan

aanaaaa LLnLLL +≈+= �>

(3.10)

Where

Laa0 is 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 orderharmonic.

θ is the electrical displacement angle for the machine.

In the same way, the mutual inductances between the stator windings, Lab, Lac andLbc, 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( 320320ππ θθ −−−=++−== 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 and Lcmq, are assumed to be independent ofwhich stator winding that is in question, and are called: Laf, Lakd and Lamq. 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].

θcosdfaf LL = (3.12)

θcosdkdakd LL = (3.13)

θsinqmqamq LL = (3.14)

Where Ldf, Ldkd and Lqmq, are the mutual inductances between the rotor windingsand the fictive, dq0 transformed, local stator windings d and q.

The time dependent matrixes LSS and LSR can now be described as:


- 14 -

��

�

�

��

�

�

+−−−−−−−+−−−+−

+

+��

�

�

��

�

�

−−−−−−

=

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

32

2231

2

232

231

2

31

231

22

000

000

000

πθπθπθπθπθπθπθπθθ

aaabab

abaaab

ababaa

aaabab

abaaab

ababaa

LLLLLL

LLL

LLLLLLLLL

SSL

(3.15)

��

�

�

��

�

�

−��

�

�

��

�

�

+++−−−=

qmq

dkdSR

LLL

dfL

)sin()cos()cos()sin()cos()cos(

)sin()cos()cos(

32

32

32

32

32

32

πθπθπθπθπθπθ

θθθ(3.16)

3.3 The dq0-transformationIn section 2.2.1, the dq0-transformation is described. The transformation and itsinversion are repeated here for convenience.

SSdq0 BUUU =��

�

�

��

�

�

+−−−−+−

=

21

21

21

32

32

32

32

)sin()sin()sin()cos()cos()cos(

32 πθπθθ

πθπθθ(3.17)

dq01

dq0S UBUU −=��

�

�

��

�

�

+−+−−−

−=

1)sin()cos(1)sin()cos(1)sin()cos(

32

32

32

32

πθπθπθπθ

θθ(3.18)

As defined earlier, the index dq0 reefers 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

dq0 II

LL

Ψ00000

0000000

00 LL

LL

ΨΨΨ

q

dfd

q

d

(3.19)

ω��

�

�

��

�

�−++−=

0d

q

dtd Ψ

Ψ

dq0dq0Sdq0 ΨIRU (3.20)

)(23

qddqmech

e IΨIΨM −=ω

ω (3.21)

Where

223

00 aaabaad LLLL ++= (3.22)

223

00 aaabaaq LLLL −+= (3.23)

000 2 abaa LLL −= (3.24)

Ld, Lq and L0, 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.

3 Mathematical descriptionReciprocity

- 15 -

The terms ddtd ψ and qdt

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

neglected during transient conditions [9].

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

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

��

��

�

��

�

�

��

�

�

−−−

=��

�

�

��

�

�

=R

F

mqqmq23

kdfkddkd23

fkd

R II

LLLLLL

Ψ0000

0000002

3fdf

mq

kd

f LL

ΨΨΨ

(3.25)

RRRR ΨIRU 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 easierto solve; i.e. it makes the simulation faster.

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

��

�

�

��

�

�

��

�

�

��

�

�

−−−

−−

−

=

��

�

�

��

�

�

mq

kd

mqqmq23

kdfkddkd23

fkd

qkq

dkd

mq

kd

II

L000L00LL00L0L00

L00L

ΨΨ

f

q

d

fdf

q

dfd

f

q

d

IIII

LLL

LLL

ΨΨΨΨ

0

23

00

00000000

00

(3.27)

For example dff

d LIΨ =∂∂ , but df

d

f LIΨ

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 rotorcurrents by a factor 3

2 , 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:


- 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 23´= (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

II

L000L00LL00L0L00

L00L

ΨΨ

f

q

d

fdf

q

dfd

f

q

d

IIII

LLL

LLL

ΨΨΨΨ

(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:

[ ][ ]

[ ] ��

�

�

��

�

�

��

�

�

��

�

�

+−−+−−

+−−=

��

�

�

��

�

�

kddεkdεkkdfkddkd

fkd

dkd

kd ILLLLLLL

Ψf

d

ffεfdf

dfdεdεd

f

d

II

LLLLLLLL

ΨΨ

)()(

)(

ε (3.41)

[ ][ ] �

�

��

��

��

�

+−−+−−

=��

��

�

mqεmqεmqmqqmq

qmq

mq ILLLLL

Ψqεqεqqq ILLLΨ

)()(

(3.42)

Where

Lεd is the leakage inductance of the d-winding

Lεf is the leakage inductance of the field winding

Lεεεεkd is the leakage inductance of the damper windings in the d-axis

Lεq is the leakage inductance of the q-winding

Lεεεεmq is the leakage inductance of the damper windings in the q-axis

The leakage inductances of the d- and q-windings, Lεd, and Lεq, are equal, and areoften called Ll, but in Simpow, they are called La, why henceforth this will also bethe case in this report, i.e.

εqεda LLL == (3.43)

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 ILLILΨ )( −−=+ (3.44)

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

d

ddff

I

ILΨ

dkdkd LΨ −=

−=(3.45)

When SI units are concerned, there is equal mutual flux, i.e. in this case:kdΨ==+ fdad ΨILΨ . 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 called Lad,

dkdL==−= dfadad LLLL )( (3.46)

In the same way, it is proved that:

fkdεkdkd LLL =−=−= )()( ffad LLL ε (3.47)

and in the q-axis,

qmqεmqmq 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

L000L00LL00L0L00

L00L

ΨΨ

f

q

d

fad

q

add

f

q

d

IIII

LLL

LLL

ΨΨΨΨ

000

00000000

00

(3.49)

ω

��

�

�

��

�

�−

+

��

�

�

��

�

�

+

��

�

�

��

�

�

��

�

�

��

�

�

−−

−−=

��

�

�

��

�

�

0000

000000000000

000

d

q

f

q

d

f

q

d

f

a

a

a

f

q

d

ΨΨ

ΨΨΨΨ

dtd

IIII

RR

RR

UUUU

mq

kd

mq

kd

mq

kd

mq

kd

ΨΨ

II

R000000R000000000000

UU

(3.50)

Where

Ld is the self-inductance of the fictive d-winding.

Lq is the self-inductance of the fictive q-winding.

L0 is the self-inductance of the fictive 0-winding.

Lf is the self-inductance of the field winding.

Lkd is the self- and mutual inductances of the k damper windings of the d-axis.

Lmq is the self- and mutual inductances of the m damper windings of the d-axis.

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


- 18 -

Laq is 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

RRΤdq0R

dq0Rdq0dq0

R

dq0

II

LLLL

ΨΨ

ω��

�

�

��

�

�−+�

�

��

�+�

�

��

��

��

�

−=�

�

��

�

0ΨΨ

II

R00R

UU

R

dq0

R

dq0

R

a

R

dq0d

q

ΨΨ

dtd

The changes from the original equations, do not affect the torque equation morethan the change from stator quantities to dq0 quantities [10], which is expressed as:


with

)(23

qqqdmech

e IΨIΨM −=ω

ω (3.52)

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 basesThe 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 fπωω 2== (4.3)

n

SbaseSbase ω

UΨ = (4.4)

SbaseSbasenSbase IUSS 23== (4.5)

Sbase

SbaseSbase I

UZ = (4.6)

Sbase

SbaseSbase I

ΨL = (4.7)

nmech

Smechbase ωω

ωω = (4.8)

ωω

ωωωω mech

SbaseSbase

nmech

SbaseSbase

mechbase

nSbase

IΨIUSM 23

23

=== (4.9)

n

Smechbase

SJH

2

21 ω= (4.10)

Where

Un is the nominal phase-phase RMS voltage of the machine.

Sn is the nominal power of the machine.

ωn is the system frequency in [rad/s]

fn is the system frequency in [Hz]

ωmech is the mechanical frequency in [mech. rad/s]

ω is the electrical frequency in [rad/s]

ωbase is the system base frequency

ωSmechbase is the mechanical base frequency of the machine

H is the per unit inertia constant.


- 20 -

4.2 Per unit flux linkage equations

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

313212111 ILILILΨ ++= (4.11)

323112222 ILILILΨ ++= (4.12)

223113333 ILILILΨ ++= (4.13)

Where

Ψi is the flux linkage in winding i

Li is the self-inductance in winding i

Lij is the mutual inductance between winding i and j

Ii is 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 Bühler, [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

iii

xx

x

iii

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

µσµσ

ψψψ

(4.14)

Where

I1

L1Ψ1U1

dtd

I2

L2Ψ2U2

dtd

I3

L3Ψ3U3

dtd

L12

L23

L31

4 Per unit representationPer unit flux linkage equations

- 21 -

Ψin is the base flux linkage of winding i

Iin is the base current of winding i

x1 is the per unit value of the reactance of winding 1

σij is the decrement factor of winding j relative winding i

µi is 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 1, l1, has the same value as the perunit value of the reactance of winding 1, x1, and is defined as:

11

11

1

111 x

UIL

ΨILl

n

nn

n

n === ω (4.15)

Where

ωn is the base frequency of the system

U1n is the base voltage of winding 1

The decrement factor is defined as:

ji

ijij LL

L2

1−=σ (4.16)

The translation factor is defined as:

jki

ikiji LL

LL=µ (4.17)

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

12

12 L

ΨI nn = (4.18)

13

13 L

ΨI nn = (4.19)

nn ILΨ 222 = (4.20)

nn ILΨ 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

II

L000L00LL00L0L00

L00L

ΨΨ

f

q

d

fad

q

add

f

q

d

IIII

LLL

LLL

ΨΨΨΨ

000

00000000

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:


- 22 -

��

�

�

��

�

�

��

�

�

��

�

�

−−−

=��

�

�

��

�

�

d

f

d

1dadad

adfad

adadd

d

f

d

III

LLLLLLLLL

ΨΨΨ

11

(4.23)

��

�

�

��

�

�

��

�

�

��

�

�

−−−

=��

�

�

��

�

�

q

q

q

qaqaq

aqqaq

aqaqq

q

q

q

III

LLLLLLLLL

ΨΨΨ

2

1

2

1

2

1 (4.24)

[ ] [ ][ ]000 ILΨ −= (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

iii

xx

x

iii

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

ΨIL

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

µσµσ

ψψψ

(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 1d replace the indices 1, 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

fbasedf

dbase

dbasedd

fbase

dbasedf

fbase

fbasef

fbase

dbasedf

dbase

dbasedd

dbase

fbasedf

dbase

dbased

d

f

d

iii

xx

x

iii

ΨI

LΨI

LΨI

L

ΨI

LΨI

LΨI

L

ΨIL

ΨI

LΨIL

1111

1

11

11

11

11

11

1 1)1(1)1(

11

µσµσ

ψψψ

(4.27)

Where

Ψibase is the base flux linkage of winding i

Iibase is the base current of winding i

xd is the per unit value of the reactance of the d-winding

σij is the decrement factor of winding j relative winding i

µi is the translation (or screening) factor of winding i

The per unit base current and flux linkage of the d-winding, Idbase and Ψdbase, are thesame as the per unit bases for the machine, ISbase and ΨSbase, i.e.

Sbasedbase II = (4.28)

Sbasedbase Ψ=Ψ (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:

dSbase

dSbase

Sbasedn

Sbase

Sbasedd x

LL

UIL

ΨILl ==== 1ω (4.30)

Where

ωn is the base frequency of the system.


- 23 -

USbase is the base voltage of the machine.

LSbase is the base inductance of the machine.

The decrement factors and translation factors are defined as:

fd

dfdf LL

L2

1−=σ (4.31)

dd

dddd LL

L

1

21

1 1−=σ (4.32)

ddf

dfdff LL

LL

1

1=µ (4.33)

dfd

dddfd LL

LL

1

111 =µ (4.34)

Using the definition of equal mutual inductances, Ldf = Ld1d = Lf1d = Lad, anddividing all inductances with the base inductance of the machine, LSbase, allinductances 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

iii

xx

xx

xx

xxx

ψψψ

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

IxL

ΨLΨ

LΨI ==== (4.37)

ad

Sbase

ad

Sbase

dd

Sbasedbase x

ILΨ

LΨI ===

11 (4.38)

Sbasead

ffbaseffbase Ψ

xx

ILΨ == (4.39)

Sbasead

ddbaseddbase Ψ

xxILΨ 1

111 == (4.40)


- 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

iii

xx

xx

xx

xxx

iii

ΨI

LΨI

LΨI

L

ΨI

LΨI

LΨI

L

ΨI

LΨI

LΨI

L

2

1

22

211

2

2

1

2

22

2

121

22

1

221

1

11

11

22

11

2

1

1

1

11

ψψψ

(4.41)

where

qSbase

qSbase

Sbaseqn

Sbase

Sbaseqq x

LL

UIL

ΨILl ==== 1ω (4.42)

qSbase

q

qSbase

q

aqSbase

aq

xL

L

xL

L

xL

L

22

11

1

1

1

=

=

=

(4.43)

aq

Sbase

aq

Sbase

qq

Sbaseqbase x

ILΨ

LΨ

I ===1

1 (4.44)

aq

Sbase

aq

Sbase

qq

Sbaseqbase x

ILΨ

LΨ

I ===2

2 (4.45)

Sbaseaq

qqbaseqqbase Ψ

xx

ILΨ 1111 == (4.46)

Sbaseaq

qqbaseqqbase Ψ

xx

ILΨ 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=ψ (4.48)

where

00001 x

LL

ΨILl

SbaseSbase

Sbase === (4.49)

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


- 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

iiiiiii

xx

xx

xx

xx

xx

xx

xx

xx

xx

x

2

1

1

0

22

211

211

2

20

2

1

1

0

10000

10000

00100

00100

000000110000001100

ψψψψψψψ

(4.50)

Here

xd is the direct axis synchronous reactance.

xa is the armature leakage reactance, often called xl.

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

xf is the field winding reactance.

x1d is the damper winding reactance of damper 1 in the d-axis.

xq is the quadrature axis synchronous reactance.

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

x1q is the damper winding reactance of damper 1 in the q-axis.

x2q is the damper winding reactance of damper 2 in the q-axis.

4.2.4 Definition of the transient and sub-transient reactancesTo 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

ixix

´´´´

==

ψψ

(4.51)

where

xd´´ is the subtransient reactance of the d-axis

xq´´ is the subtransient reactance of the q-axis

Directly after the voltage is applied on the terminals, the rotor flux linkages, ψf,ψ1d, ψ1q and ψ2q, are still zero, since they can not change instantly [12]. The fluxlinkages of the d-axis during this subtransient time, looks like:


- 26 -

��

�

�

��

�

�

��

�

�

��

�

�

−

−

−

=��

�

�

��

�

�

=��

�

�

��

�

�

d

f

d

d

ad

d

ad

f

ad

f

ad

dd

d

f

d

iii

xx

xx

xx

xxx

1

11

2

2

1

1

1

11

00ψ

ψψψ

(4.52)

This gives the rotor currents, if and i1d, during the subtransient time as:

daddf

adfadd

daddf

addadf

ixxx

xxxi

ixxxxxxi

21

2

1

21

21

−

−=

−−=

(4.53)

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

dadaddf

adfddd ix

xxxxxx

x��

�

�

��

�

�

−

−+−= 2

21

1 2ψ (4.54)

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

22

1

1 2´´ ad

addf

adfddd x

xxxxxx

xx−

−+−= (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] describedas:

qqq

ddd

ixix

´´

==

ψψ

(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 i1d = 0, the transient statorflux linkage in the d-axis is calculated as:

df

addd i

xxx

��

�

�

��

�

�−=

2ψ (4.57)

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

f

addd x

xxx2

´ −= (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, x1d, as:

4 Per unit representationPer unit voltage equations

- 27 -

´´´)´´)(´(

1dd

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. Forthe 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

xxxxxx

xx−

−+−= (4.61)

q

aqqq x

xxx

1

2

´ −= (4.62)

The reactance of the first damper winding, x1q, 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, x2q as:

´´´)´´)(´(

2qq

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

ω��

�

�

��

�

�−++−=

0d

q

dtd Ψ

Ψ

dq0dq0Sdq0 ΨIRU (4.65)

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

ωψψ

Sbased

q

Sbasedtd

SbaseSbaseSbase ΨΨIZU��

�

�

��

�

�−++−=

0dq0dq0Sdq0 ψiru (4.66)

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

based

q

dtd

base ωωψ

ψ

ω��

�

�

��

�

�−++−=

0

1dq0dq0Sdq0 ψiru (4.67)

4.3.2 Rotor voltagesIn chapter 3.4, the rotor voltages, in SI units, are described as:


- 28 -

RRRR ΨIRU dtd+= (4.68)

Separating the system into equations:

mqmqmq

kdkdkd

ΨIR

ΨIR

dtd

dtd

ΨdtdIRU ffff

+=

+=

+=

0

0 (4.69)

4.3.2.1 Field windingThe field winding voltage can be written as:

fbaseffbasefffbasef ΨdtdIiRUu ψ+= (4.70)

Defining Ufbase as:

fbaseffbase IRU = (4.71)

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

ff

f

fbasef

fbase

fbase

fbase

RL

IRΨ

UΨ

τ=== (4.72)

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

ffff dtdiu ψτ+= (4.73)

where

τf is the time constant of the field winding

4.3.2.2 Damper windingsThe damper winding voltages can be described as:

mqbaseΨψIiR

ΨψIiR

mqmqbasemqmq

kdbasekdkdbasekdkd

dtd

dtd

+=

+=

0

0(4.74)

Dividing the equations with RkdIkdbase and RmqImqbase respectively, and using theEquations (4.28), (4.34) and (4.35) to define:

kdkd

kd

kdbasekd

kdbase τRL

IRΨ == (4.75)

and

mqmq

mq

mqbasemq

mqbase τRL

IRΨ

== (4.76)

gives

4 Per unit representationPer unit voltage equations

- 29 -

mqmqmq

kdkdkd

ψτi

ψτi

dtd

dtd

+=

+=

0

0(4.77)

where

ττττkd are the time constants for the damper windings in the d-axis

ττττmq are 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

ψψ

ττ

ii

ffff

dtd

iu ψτ

00 (4.78)

or shorter

RRRR ψτiu dtd+= (4.79)

4.3.3 ConclusionsFor 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

dtd

iiiiiii

rr

r

uuuu

base

base

base

ωω

ψψ

ψψψψψψψ

ττττω

ω

ω

��

�

�

��

�

�−

+

��

�

�

��

�

�

��

�

�

��

�

�

+

��

�

�

��

�

�

��

�

�

��

�

�

−−

−

=

��

�

�

��

�

�

00000

1000000010000000100000001000000000000000000000

000

2

1

1

0

2

1

1

1

1

1

2

1

1

00

(4.80)

Where

ra is the armature resistance.

τf is the time constant of the field winding.

τ1d is the time constant of the first d-damper winding.

τ1q is the time constant of the 1st q-damper winding.

τ2q is the time constant of the 2nd q-damper winding.

4.3.4 Definition of the time constantsTo 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:


- 30 -

ddf 10´ τττ −= (4.81)

df

ad

d

df

dd

xxx

1

0

1

01

1

´´´´

−== τ

σττ (4.82)

where

τd0´ is the transient open circuit time constant of the d-axis.

τd0´´ is the subtransient open circuit time constant of the d-axis.

σf1d is the decrement factor of winding 1d relative winding f.

This definition is approximate, and is only valid when τf >> τ1d, i.e. when

d

d

f

f

RL

RL

1

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

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

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

qqq 201 ´ τττ −= (4.83)

qq

aq

q

qq

qq

xxx

21

0

21

02

1

´´´´

−==

τστ

τ (4.84)

where

τq0´ is the transient open circuit time constant of the q-axis.

τq0´´ is the subtransient open circuit time constant of the q-axis.

σf1q2q is the decrement factor of winding 2q relative winding 1q

This definition is approximate, and is only valid when τ1q >> τ2q, i.e. when

q

q

q

q

RL

RL

2

2

1

1 >> .

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

4.4 Per unit torque equationsIn chapter 3.4, the torque is described, in SI units, as:


with

)(23

qqqdmech

e IΨIΨM −=ω

ω (4.86)

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

4 Per unit representationPer unit torque equations

- 31 -

n

Smechbase

mech

SbaseSbase

nmech

SbaseSbase

Smechbase

nbase

nmech

Smechbase

SJH

IΨIUSM

2

23

23

21 ω

ωω

ωωωω

ωω

ωω

=

===

=

(4.87)

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

baseemn

Smechbasedtd

Smechbase

n MmmHS)(

22 −=

ωωω

ω(4.88)

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

emn

dtd mmH −=ωω2 (4.89)

In the above equation,

θω

ωωω

ωωωω

ωω

dtd

ndtd

nndtd

n

ndtd

ndtd 11 =∆=∆=−= (4.90)

where

∆ω is the rotor speed deviation

θ is the deviation angle of the rotor

The per unit electrical torque, me, is now:

dqqde iim ψψ −= (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 −+−=+−= ψψψψ (4.92)

where

r2 is the negative sequence resistance.

ra is the armature resistance.

i2 is the negative sequence current.

4.4.2 Models without damper windingsThe 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:


- 32 -

nd Dm

ωω∆⋅= [p.u.] (4.93)

where D is the damping factor.

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

demn

dtd mmmH −−=ωω2 (4.94)

4.5 Global - local per unit transformationAt 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 CI are defined as:

n

basenodeU U

UC =

basenetwn

basenodenI SU

USC =

where

Unodebase is the nominal phase-phase voltage for the machinenode

Snetwbase is 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 θθδδ βαβα sincoscossin +=−= (4.95)

[ ] [ ] UUq CUUCUUU θθδδ βαβα cossinsincos +−=+= (4.96)

and the local to global transformation looks like:

[ ] [ ] IqdIqd CIICIII θθδδα sincoscossin −=+= (4.97)

[ ] [ ] IqdIqd CIICIII θθδδβ cossinsincos +=+−= (4.98)

Where

θ is the electrical displacement angle for an instantaneous value model.

δ 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.

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 reactance

In 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 stabilityWhen 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, fundamentalfrequency models are used; this mode is in Simpow called Transta.

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

• The transformer voltages, ddtd ψ and qdt

d ψ , are neglected.

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

voltage and flux linkage equations, 0001 ψω dt

dirun

a +−= and 000 ix−=ψ , are

neglected.

• The negative and zero sequences are modelled as two impedance circuits, withthe resistance and reactance: r2, x2 and r0, x0 respectively.

• The electrical torque is modelled with an extra term, the negative-sequencebreaking torque: 2

222 )( 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 2 and Type 1 respectively.


- 34 -

5.2 Initial conditions at steady stateAt time zero, the initial conditions are calculated. From the load-flow calculation,the initial node voltage and angle, ut0 and φi, 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 undersymmetrical conditions,ψ0 = u0 =i0 =0. When deriving the stator circuit quantities,ud0, uq0, ψd0, ψq0, id0 and iq0, saliency is neglected, i.e. xd = xq, but when deriving therotor circuit quantities, ψf0, ψ1d0, ψ1q0, ψ2q0, 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

iiiiiii

rr

r

u

uu

ωω

ψψ

��

�

�

��

�

�−

+

��

�

�

��

�

�

��

�

�

��

�

�

−−

−

=

��

�

�

��

�

�

00000

1000000010000000100000001000000000000000000000

000

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

0010

f

q

d

q

aq

q

aq

d

ad

d

ad

f

ad

d

d

q

q

d

f

q

d

iii

xxxx

xx

xxxx

xx

ψψψψψψ

(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 ψωωψ

ωω +−+−−=t0u (5.4)

Now, the machine is assumed to run at synchronous speed, i.e. ω = ωn, and withthe flux linkages from Equation (5.2), ut0 can 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

5 ModellingInitial conditions at steady state

- 35 -

uf0 is 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, δi, and the dq-voltages, ud and 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 δsin00 = (5.7)

itq uu δcos00 = (5.8)

Where sinδi and cosδi can be described as:

0

00 sincossinf

itaitdi u

irix ϕϕδ −= (5.9)

0

000 sincoscosf

itditati u

ixiru ϕϕδ ++= (5.10)

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

200

20000 )sincos()sincos( itaitditditatf irixixiruu ϕϕϕϕ −+++= (5.11)

With the initial power, out from the generator into the node, defined as:

ut0∠φi

uf0∠δi +φi

xdra

it0

pt0, qt0

~

rait0sin(ϕ)

xdit0sin(ϕ)

q

rait0cos(ϕ) xdit0cos(ϕ)

ut0 rait0

xdit0

it0

uf0

ϕ

δi

d

uq0

ud0


- 36 -

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

this gives the real and imaginary power as:

ittt iup ϕcos000 = (5.13)

ittt iuq ϕsin000 = (5.14)

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

( )��

�

�

��

�

�+++

��

�

�

��

�

�

��

�

�

��

�

�+�

�

�

�

��

�

�+= 2

0

020

0222

20

02

20

000 21

t

tq

t

tada

t

t

t

ttf u

qxuprxr

uq

upuu (5.15)

0

0

0

0

0

00f

t

ta

t

td

td uuqr

upx

uu−

= (5.16)

0

0

0

0

00

00f

t

td

t

tat

tq uuqx

upru

uu++

= (5.17)

)arctan(0

0

q

di u

u=δ (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 id0 and iq0 can 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

uqu

upui

uqu

upui

−=

+=(5.19)

From Equation (5.1), the flux linkages, ψd0 and ψq0, are described as:

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0

0

0

0

0

0

00

q

d

a

an

d

qn

q

d

ii

rr

uu

ωω

ωω

ψψ

Equation (5.2) can be rearranged, with the magneto motive forces of the d- and q-axis, Wd0 and Wq0, separated from the first two rows:

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01

01

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02

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000000000

100

0000

qq

aq

qq

ad

dd

ad

df

ad

q

d

f

q

d

f

ad

a

a

q

q

d

f

q

d

Wxx

Wxx

Wxx

Wxx

WW

iii

xx

xx

ψψψψψψ

(5.20)

5 ModellingType 4

- 37 -

From row 1 and 2 in (5.20), Wd0 and Wq0 are extracted, so the unknown fluxlinkages, ψf0, ψ1d0, ψ1q0 and ψ2q0, 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 ud0 and 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 iterationprocedure 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 Wd and Wqto fd(Wd)and fq(Wq).

2220 qdt uuu += (5.22)

The concept of saturation is discussed in chapter 5.7.

5.3 Type 4In 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 1 and 2, are toprefer, since in these models there is a more adequate representation of theelectrical damping [15].

5.3.1 Assumptions for the classical modelFor 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 followingadditional assumptions are made:

6. The shaft mechanical power remains constant.

7. Damping is non-existent.

If the system is stable when these assumptions are made, stability will likely occurif any or all of the above factors are taken under consideration. However, if thesystem would be unstable with the assumptions made, the system may in fact bestable [9].

5.3.2 Assumptions in Type 4In Type 4, the same assumptions are made as for the classical model, with thefollowing remarks for the last three points in the list above

5. The armature resistance, Ra, is included, but the default value is set to zero.


- 38 -

6. A turbine can be connected to the model, but if no turbine is connected, theshaft mechanical power is constant.

7. Damping is included in the model, by the means of the damping coefficient, D,but the default value is set to zero.

Beside these remarks,

• Saliency is included, if xq is given a value that differs from xd.

• Both a synchronous and a transient reactance are modelled in the d-axis, if xd isgiven a value that differs from xd´.

Together, these changes make the Type 4 a little different from the typical classicalmodel.

To obtain a classical model in Simpow, as defined in the previous section, usingthe Type 4 model, the user should specify the quantities: Sn, H, Un, xd´, xd and xq.With xd = xq and xd´ = xd - ε. Here ε is a positive constant, due to a mathematicalconstraint in Simpow, so that xd´ < xd, e.g. ε = 10-6.

5.3.3 Description of Type 4Type 4 is a model with neither a field winding nor any damper windings.Therefore, as stated in chapter 4.4.2, the only way to obtain damping in this model,is to include the substitute electrical damping torque md, i.e. to include the dampingcoefficient, D.

5.3.3.1 Instantaneous value modelWith these considerations in mind, the machine equations derived in chapter 1, willfor the instantaneous value Type 4 model be changed to:

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q

d

q

d

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ixx

iiii

xx

x ψ

ψψψψ

(5.23)

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ff

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a

f

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d

dtd

iiii

rr

r

uuuu

n

n

n

ψψ

ψψψψ

τω

ω

ω

(5.24)

dem mmmdt

dH −−=∆ω2 (5.25)

ωδ ∆=dtd (5.26)

In the field winding voltage equation, (5.24), the term 0ff dtd ψτ is equal to zero,

but is included to show the resemblance with higher order models.

It is obvious to see that there are five state variables for this model, ψd, ψq, ψ0, ∆ωand δ; i.e. the Type 4 instantaneous value model is a 5th order model.

With the exception that there is no d-damper winding, i.e. x1d = ∞, the constantfield winding flux linkage is calculated as in the steady-state model, see Equation(5.2).

5 ModellingType 4

- 39 -

5.3.3.1.1 Equivalent circuitThe d-, q- and 0-axis equivalent circuits for the instantaneous value Type 4 model,can now be viewed as in Figure 5.3.

Figure 5.3 Equivalent dq0-circuits for Type 4, instantaneous value model in per unit

All models discussed in this chapter will have the same 0-sequence model, i.e. thesame equivalent 0-circuit; hence, it is only viewed here.

5.3.3.2 Fundamental frequency modelImplementing in Equations (5.23)-(5.26), the changes between the instantaneousvalue model and the fundamental frequency model, stated in chapter 5.1, the lattermodel is derived as:

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01000000´

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f

q

d

q

d

f

q

d

ixxi

ii

xx ψ

ψψψ

(5.27)

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d

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d

a

a

f

q

d

dtd

iii

rr

uuu

ψψ

ψτ (5.28)

dem mmmdt

dH −−=∆ω2 (5.29)

ωδ ∆=dtd (5.30)

with

2eqddqe miim +−= ψψ (5.31)

In the field winding voltage equation, (5.28), the term 0ff dtd ψτ is equal to zero,

but is included to show the resemblance with higher order models.

As stated earlier, this model has only two state variables: ∆ω and δ. Thus, the Type4 fundamental frequency model is of order 2.

5.3.3.2.1 Equivalent circuitIf xd´ = xd = xq, an equivalent circuit can be drawn for the positive sequencequantities of this fundamental frequency model, see Figure 5.4. Where u = ud + juqand i = id + jiq.

id

xd´

ra

ψq +-

ψdud

dtd

i0

x0

r0

ψ0u0

dtdxq

ra

iq ωψd-+

ψquq dtd


- 40 -

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

For the fundamental frequency models, the negative- and 0-sequence componentsare modelled as two impedance circuits with the resistance and reactance: r2, x2 andr0, x0 respectively, see Figure 5.5.

Figure 5.5 Equivalent negative- & 0-sequence circuits, fundamental frequency model in SI units

All models discussed in this chapter have the same model for the unsymmetricalcomponents, i.e. the same equivalent negative- and 0-sequence circuits; hence, theyare only viewed here.

5.4 Type 3AIn this chapter, the synchronous machine model Type 3A is described. This modelis a simplification of Type 3, where the saturation effect is included. In chapter 5.7,the effects of the saturation are described.

To improve the classical model, a field winding can be included. In this way, thetime variations of the direct axis reactance will be considered. Thus, the constantfield winding flux linkage, in Type 4, is changed to be dependent of the d-axiscurrent and of the changing field winding current, this also makes the d-axis fluxlinkage to be dependent of the field winding current. In addition to these changes,speed variations are included in the model.

Still, no damper windings are included in the model, i.e. the only way to obtaindamping in this model, is to include the substitute electrical damping torque md.

5.4.1 Instantaneous value modelImplementing this in the machine equations for the Type 4 model, theinstantaneous value Type 3A model, will look like:

ra

i

u -+

uf0

xd

i2

r2

u2

x2

i0

r0

u0

x0

5 ModellingType 3A

- 41 -

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d

iiii

xx

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x

020

0100

000000100

ψψψψ

(5.32)

n

d

q

f

q

d

ff

q

d

a

a

a

f

q

d

dtd

iiii

rr

r

uuuu

n

n

n

ωωψ

ψ

ψψψψ

τω

ω

ω

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01

1

1

00(5.33)

dem mmmdt

dH −−=∆ω2 (5.34)

ωδ ∆=dtd (5.35)

There are one more state variable for this model than for the Type 4 model, ψf, i.e.the Type 3A instantaneous value model is a 6th order model.

5.4.1.1 Equivalent circuitTo be able to draw an equivalent circuit, the flux linkage and voltage equations arealso stated in SI-units, see Equations (5.36) and (5.37).

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(5.36)

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RR

RR

UUUU

(5.37)

Where the upper-case letters are used when relating to quantities in SI units.

As described in chapter 3.4, the mutual inductance can be described as thedifference between the self-inductance and the leakage inductance. Therefore theself-inductance can be described as the sum of the mutual inductance and theleakage inductance, and the flux linkage equations can be described as:

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(5.38)

The d-axis equivalent circuit can be described either as in Figure 5.6, or by usingthe leakage inductance as in Figure 5.7. This latter circuit is often easier tounderstand, and it is easy to declare the equations of the circuit, therefore thefollowing models will be described this way, without showing the leakageinductance in the flux linkage equations.

The q-axis is modelled in the same way as Type 4, see Figure 5.3, with thedifference that the speed variations are included. An equivalent circuit for the q-axis is viewed in Figure 5.8.


- 42 -

La Lεf

Lad

Ra Rf

IdωΨq+-

dtdΨd dt

d ΨadUd Uf

If

Id-If

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

Figure 5.7 Equivalent d-circuit for Type3A, leakage inductance instantaneous value model in SI units

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

The equivalent 0-circuit for Type 3A is viewed in Figure 5.3.

5.4.2 Fundamental frequency modelFrom the set of equations for the instantaneous value model, see Equations (5.32)-(5.35), and implementing the changes stated in chapter 5.1, the fundamentalfrequency model is derived.

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0010

2ψψψ

(5.39)

nd

q

f

f

f

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d

a

a

f

q

d

dtd

iii

rr

uuu

ωωψ

ψ

ψτ

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(5.40)

dem mmmdt

dH −−=∆ω2 (5.41)

ωδ ∆=dtd (5.42)

with

Id

Ld

Ra

ωΨq+-

ΨdUd

dtd

If

Lf

Rf

Uf

Lad

Lq

Ra

Iq ωΨd-+

ΨqUq dtd

5 ModellingType 2A

- 43 -


In the same manner as for the instantaneous value model, it is verified that thismodel has one more state variable than the Type 4 model. Thus, the Type 3Afundamental frequency model is of order 3.


Further improvements are made if damper windings in the d- and q-axis areincluded in the model. In the case of one damper winding in each axis, the fastchanging conditions, i.e. the subtransient state, and the saliency of the rotor will beconsidered. This model is normally used for water turbine generators, i.e. salientpole machines.

5.5.1 Instantaneous value modelImplementing this in the machine equations for the Type 3A model, theinstantaneous value Type 2A model, will look like Equations (5.44)-(5.47). There isno additional damping torque, because damper windings are included in the model.

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0100

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000001000001100

ψψψψψψ

(5.44)

n

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d

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a

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00 (5.45)

em mmdt

dH −=∆ω2 (5.46)

ωδ ∆=dtd (5.47)

There are two more state variables for this model than for the Type 3A model, ψ1dand ψ1q, i.e. the Type 2A instantaneous value model is an 8th order model.

5.5.1.1 Equivalent circuitTo be able to draw an equivalent circuit, the flux linkage and voltage equations arealso stated in SI-units, see Equations (5.48 and 5.49).


- 44 -

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q

d

f

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qaq

dadad

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adadd

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d

IIIIII

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(5.48)

ω

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d

dtd

IIIIII

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(5.49)


The equivalent d-and q-circuits for the instantaneous value model is viewed, usingthe leakage inductances, in Figure 5.9.

Figure 5.9 Equivalent dq-circuits for Type2A, leakage inductance instantaneous value model in SI units


5.5.2 Fundamental frequency modelFrom the set of equations for the instantaneous value model, see Equations (5.40�5.43), and implementing the changes stated in chapter 5.1, the fundamentalfrequency model is derived, see Equations (5.50�5.54).

La Lfd

Lad

L1d

Ra Rfd

R1d

IdωΨq+-

dtdΨd

dtd ΨadUd Ufd

Ifd

Id-If-I1d

I1d

La

Laq

Ra

Iq ωΨd-+

Lε1q

R1q

dtdΨq dt

dΨaqUq

Iq-I1q

I1q

5 ModellingType 1A

- 45 -

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d

iiiii

xx

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xx

xx

1

1

1

211

2

2

1

1

1000

010

010

10000110

ψψψψψ

(5.50)

n

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q

q

d

f

q

d

f

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d

f

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d

a

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(5.51)

em mmdt

dH −=∆ω2 (5.52)

ωδ ∆=dtd (5.53)

with


In the same manner as for the instantaneous value model, it is verified that thismodel has two more state variables than the Type 3A model. Thus, the Type 2Afundamental frequency model is of order 5.


If one extra damper winding is added in the q-axis, the accuracy in modelling solidiron rotor generators will be increased. Solid iron rotor is often used in large steamturbine generators, and provides multiple paths for circulating eddy currents. [9],[1] and [12]. This model is called Type1 in Simpow.

Additional damper windings may be included both in the d- and q-axis, see [4] and[5], this further increases the accuracy in the models.

5.6.1 Instantaneous value modelThe extra q-damper winding changes the equations from the instantaneous valuemodel Type 2 to:


- 46 -

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0

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1

1

2

1

1

00

(5.56)

em mmdt

dH −=∆ω2 (5.57)

ωδ ∆=dtd (5.58)

There are one more state variable for this model than for the Type 2A model, ψ2q,i.e. the Type 1A instantaneous value model is a 9th order model.

5.6.1.1 Equivalent circuitAs for the other models, the equivalent circuit is made from the SI-units equations,see Equations (5.55) and (5.56).

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(5.59)

ω

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IIIIIII

RR

RR

RR

R

UUUUUUU

(5.60)


5 ModellingType 1A

- 47 -

The equivalent circuits for the instantaneous value model can be viewed using theleakage inductances, see Figure 5.10. This model is in Simpow referred to assynchronous machine model Type 1A.

Figure 5.10 Equivalent dq-circuits for Type1A, leakage inductance instantaneous value model in SI units


5.6.2 Fundamental frequency modelImplementing the changes stated in chapter 5.1, the fundamental frequency modelis derived, see Equations 5.61 - 5.65.

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1000

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1100000110

ψψψψψψ

(5.61)

n

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2

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2

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1(5.62)

em mmdt

dH −=∆ω2 (5.63)

ωδ ∆=dtd (5.64)

with

La Lfd

Lad

L1d

Ra Rfd

R1d

IdωΨq+-

dtdΨd dt

d ΨadUd Ufd

Ifd

Id-If-I1d

I1d

La

Laq

Ra

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L2qL1q

R1q

dtdΨq

dtdΨaq

Uq

Iq-I1q

I1q I2q


- 48 -


In the same manner as for the instantaneous value model, it is verified that thismodel has one more state variable than the Type 2A model. Thus, the Type 1Afundamental frequency model is of order 6.

5.7 SaturationModels of Type 1-3 are modelled both with and without consideration of thesaturation in the d- and q-axis. In the previous chapters, models of Type1A-3A aredescribed, i.e. models were the saturation effect is excluded.

As the field current exceeds a certain level, a non-linear relationship occursbetween the flux linkage and the current [9], i.e. between the flux linkage and themagneto motive force. An example of this effect can be seen in Figure 5.11. Whenthis level is exceeded, the flux linkages will be smaller for a model wheresaturation is included than for a model where saturation is not taken underconsideration.

Figure 5.11 Flux linkage - mmf diagram, showing effects of saturation

There are several different ways to model this saturation effect. In Simpow, thesaturation function can be modelled as a linear relation between the saturationfactor and the flux linkage, S(ψ), or by using a self-designed saturation table withcorresponding values of the magneto motive force, W, and the saturation function,f(W), as plotted in the figure above. While in e.g. the power system simulationsoftware, PSS/E, either quadratic or exponential saturation can be modelled, S(ψ).There are different opinions which of the models that is the most realistic,however, the differences in the outcome are relatively small [18].

Due to the non-linearity, the flux linkage equations will include the saturationfunction, as described in Equation (5.66).

0

1

2

0 1 2W [p.u.]

ψ [p.u.]

f(W)

f(W)=W

5 ModellingSaturation

- 49 -

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ψψψψψψψ

(5.66)

Where

aqqqqq

adddfd

xiiiW

xiiiW

−+=

−+=

21

1 (5.67)

Wd, Wq are the magneto motive forces in the d-axis and q-axis.

fd(Wd), fq(Wq) are the saturation functions for the d-axis and q-axis.

If fd(Wd) = Wd and fq(Wq)= Wq, then no saturation is included, and the Equation(5.66) will be the same as the flux linkage equations described in sections 5.4-5.6.

For the models with fewer windings, the rows and columns of the matrix,corresponding to the windings that are not included, are to be neglected.

In the manner that Equation (5.66) is written, it is easy to realise that the differencebetween the flux linkage and the mmf, is a leakage flux depending on the leakagereactance and the current. It can also be seen that there are no forces acting in the0-axis direction.


- 50 -

6 ProgrammingSaturation

- 51 -

6 ProgrammingDr. Kjell Anerud, at ABB Utilities department Power Systems Analysis, hasdeveloped the Dynamic Simulation Language used in Simpow. In this chapter, ashort description of the DSL-programming is made, since it is a far too complicatedmatter to be detailed described in this report.

When a power system is simulated in Simpow, a DSL-file can be used whenmodelling different components of the system, an example of such a component isa regulator. When calling the DSL-file, model specific data of the component hasto be given by the user, e.g. rated power, rated voltage, etc.

When a DSL-file is used when modelling a synchronous machine, system specificdata is sent to the DSL-file, which in return gives the machine response. Inside theDSL-file, the machine equations derive the characteristics of the machine. Anexample of how these equations are defined in the DSL-code, is given below.

WR: (TMX+TE)=2.*H*.D/DT.WR

DSL-code: Expression from the Type 3, synchronous machine model.

The first part of this expression, defines WR as the variable to be derived from thetorque equation:

(TMX+TE)=2.*H*.D/DT.WR

DSL-code: Torque equation from the Type 3, synchronous machine model.

Here, the following definitions has been made:

TMX is the per unit value of the mechanical torque applied on the machineaxis, which in this report has been called mm.

TE is the per unit value of the electrical torque, which in this report hasbeen called me and has the opposite sign.

H is the per unit inertia constant, and is one of the model specific data ofthe synchronous machine, that has to be given by the user.

.D/DT. is the Laplace-operator, i.e. d/dt in this report.

WR is the per unit value of the machine rotor speed, which in this reporthas been called ω/ωn.

In Simpow, it is possible to choose if the output, TM, from the turbine ismechanical power or mechanical torque, why the mechanical torque, TMX, has tobe calculated as:

IF(ISP.EQ.1.AND.WR.NE.0.)THENTMX=TM/WR

ELSETMX=TM

ENDIF

DSL-code: If statement from the Type 3, synchronous machine model.

Here the if-statement, (ISP.EQ.1.AND.WR.NE.0.), has the same meaning as(ISP=1 & WR≠0). Where ISP has the value 0, if the output from the turbine ismechanical torque, and the value 1, if the output from the turbine is mechanicalpower. I.e. the mechanical torque of the synchronous machine, TMX, is equal to


- 52 -

TM, if the output from the turbine is mechanical torque, or if the machine is atstandstill. Otherwise, TMX is equal to the fraction TM/WR.

The mechanical torque of the synchronous machine is also an output variable fromthe machine, which can be plotted by the user. In this case it is called MT and is onthe per unit base Sn/ωn, i.e. to get the real per unit value of the machine in question,MT must be divided by the pole pair number of the machine.

Four different DSL-files have been created, modelling the different synchronousmachines described in the previous chapters. These are named: Type1A.dsl,Type2A.dsl, Type3A.dsl and Type4.dsl, and are containing the fundamentalfrequency models of the same Type as the name of the files, where the �A� standsfor that no saturation is included in these models.

This was a short description of the structure of DSL, and of the content of the DSL-files developed during this thesis. Further comments can be seen in the code in thedifferent programmes.

For a more detailed description of the Dynamic Simulation Language used inSimpow, see the Simpow manual, [3].

7 ValidationSaturation

- 53 -

7 ValidationWhen trying to validate the DSL coded machine models, the old FORTRANmodels have stood as the reference. A small power system has been used for thispurpose, consisting of: a synchronous machine, a line and a swing-bus, see Figure7.1.

Figure 7.1 Power systems used during validation

In Simpow it is possible to include numerous independent power systems in thesame run, therefore two mirrored power systems have been simulated. Allcharacteristics of the systems are the same; the only difference is the machinemodel. In one of the systems, a DSL-coded machine model is used, and in theother, a FORTRAN-coded model is used. During simulations, two differentdisturbances has been added to the system, a three-phase to ground fault and a one-phase to ground fault. In this way, both symmetrical and unsymmetrical conditionsof the system are monitored. Various outputs can be viewed from the models, a listof variables that can be plotted for the existing synchronous machine models isattached in and in Appendix A.

During the simulations, specific data for the system and for the models has beenused; these in data files can be viewed in Appendix B.

To view the differences between the DSL- and FORTRAN-coded models, variousoutputs have been compared. In Figure 7.2, the direct axis voltage is plotted forboth the DSL- and the FORTRAN-coded Type 1A model.


- 54 -

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Ud for both DSL- and FORTRAN-models

Time (s)

Vol

tage

(p.u

.)

Figure 7.2 Ud for both the DSL- and FORTRAN-coded Type 1Aduring and after a three-phase fault at the generator node

Obviously, no apparent difference can be seen; therefore, the differences betweenthe voltages are plotted in Figure 7.3.

0 5 10 15 20 25 30 35 40-5

-4

-3

-2

-1

0

1

2

3

4x 10-4 Difference in Ud between DSL and FORTRAN-models

Time (s)

Vol

tage

(p.u

.)

Figure 7.3 Difference between Ud for the DSL- and for the FORTRAN-coded Type 1Aduring and after a three-phase fault at the generator node

The maximum difference between the models are about 0.00045 p.u., this creates amaximum deviation of about 0.065 %, hence, a rather small difference. In Table7.1, the differences of the currents, voltages and speed are assembled.

7 ValidationSaturation

- 55 -

A:

maximum difference

B:

minimum value whilemaximum difference

C=A/B:

maximum deviation

id 0,010293 0,777 1,32%iq 0,00085 0,4123 0,21%ud 0,00044 0,718 0,06%uq 0,0032 0,607 0,53%

nωω 0,000162 0,998 0,02%

Table 7.1 Assembly of differences between the DSL- and for the FORTRAN-coded Type 1A

for a three-phase fault at the generator node

In Table 7.2, the eigenvalues are assembled for the different machine models. Theindices A and B refer to the corresponding eigenvalues of the DSL and FORTRANmodels, and obviously, they do not have any significant difference.

Type1A Type2A Type3A Type4

λ1-A(-0.189615, 1.021520) (-0.186307, 1.016390) (-0.120430, 1.010864) (0, 1.011586)

λ1-B(-0.189615, 1.021520) (-0.186307, 1.016390) (-0.120430, 1.010864) (0, 1.011586)

λ2-A(-0.189616, -1.021520) (-0.186307, -1.016390) (-0.120430, -1.010864) (0, -1.011586)

λ2-B(-0.189616, -1.021520) (-0.186307, -1.016390) (-0.120430, -1.010864) (0, -1.011586)

λ3-A(-0.051660, 0.000000) (-36.428078, 0.000000) (-0.051662, 0.000000)

λ3-B(-0.051660, 0.000000) (-36.428085, 0.000000) (-0.051662, 0.000000)

λ4-A(-2.145103, 0.000000) (-36.428085, 0.000000)

λ4-B(-2.145103, 0.000000) (-36.428097, 0.000000)

λ5-A(-22.065731, 0.00000) (-38.227016, 0.000000)

λ5-B(-22.065739, 0.00000) (-38.227020, 0.000000)

λ6-A(-36.429890, 0.000000)

λ6-B(-36.429901, 0.00000)

Table 7.2 Eigenvalues of the power system, for the different machine models

From the tables and the plots, it is obvious that the differences between the DSL-written models and the FORTRAN models are small. Therefore, the conclusiondrawn is that there is a strong resemblance between the models, although furthersimulations should be made to fully state the validity of the DSL-models.In Appendix C the DSL-file for the machine model Type 1A is attached, and inAppendix D a block diagram of the model is viewed.


- 56 -

8 Conclusions and future workConclusions

- 57 -

8 Conclusions and future work

8.1 ConclusionsThe following conclusions are made:

• It has been found that the equations summarised in the technical reports [1] and[2], describing the underlying equations of the synchronous machine models inSimpow, have theoretical foundations. References to [1] and [2], have beenfound in Bühler, [10], and Laible, [13], after a vast literature research. Apartfrom this, a recommended source of information is Kundur [9], although he hasa different reference system.

• The synchronous machine models have been documented and explained indetail in this technical report, including thoroughly derivation of the equationswith references to different sources in the literature.

• The four synchronous machine fundamental frequency models from Simpowhave successfully been realised with DSL. Although further validation isneeded, the simulations made show a strong resemblance with the alreadyexisting FORTRAN models.

It is unnecessary to use the simple and less correct models, Type 3 and 4, in asimulation, just to save computer time. With today�s computers, the more accuratemodels, Type 1 and 2, could be used with extended reliability, and withoutnoticeable extension of the simulation time.

8.2 Future workIn future work, the following paragraphs could be considered:

• Including saturation in the DSL-coded models, regarding the parameters:DTAB, QTAB, V1D, V2D, SE1D, SE2D, V2Q, SE1Q, SE2Q, TAB, V1, V2,SE1 and SE2, see the Simpow users manual [3]

• Including the following parameters, implemented in the already existingmodels, see also the Simpow users manual [3].

CNODE - for control of the machine from another node.

PF, QF - for generation/consumption of only a part of the power at the node.

NCON, TET0, F0, U0 - for modelling of a machine not connected to thesystem.

VREG, TURB, XTURB, LOAD, 1BREAKER - identification numbers forthe regulator, turbines, load and breaker associated with the machine.

FREEZE - for freezing the state of the machine when it is disconnected.

REXP - for frequency dependent resistance.

• Further validation of the DSL models.

• Possibly, validation of the FORTRAN models.

• Realisation of the instantaneous value models in DSL.

• Updating the Simpow user manual.


- 58 -

Appendix A

- 59 -

Appendix AList of variables that can be plotted for the existing synchronous machine models,by Tore Petersson

September 25, 2001/ T.P.

Dynpost: Synchronous machines

Variables which can be plotted for the synchronous machine models of type 1, 1A,2, 2A, 3, 3A and 4

General informationFor synchronous machine models of type 1 and 1A all the listed variables can beplotted. For the other models listed above, the damping-circuit variables availableare dependent on the type; reference is made to the type descriptions in theDynpow part of the SIMPOW manual.

Furthermore, for type 4 no field-circuit variable can be plotted.

Per unit base values

In defining the per unit (p.u.) base values in the succeeding subsections, thefollowing parameters are used:

S netw = The network three-phase base power in MVA, i.e. the specifiedSN value in Optpow/General

Unetw = The machine node phase-to-phase base voltage, in kV RMS,i.e. the specified UB value in Optpow/Nodes. In usual caseswhere the machine is connected to a step-up transformer andwith respect to facilitate interpreting simulation results of somevariables it is recommended to choose UB equal to the kVrating UN of the machine.

Inetw = Snetw / ( 3 Unetw ), in kA RMS

Sn = The machine MVA rating, i.e. the specified SN value inDynpow/Synchronous machines

Un = The machine kV RMS rating, i.e. the specified UN value inDynpow/Synchronous machines

In = The machine kA RMS rating, i.e. Sn / ( 3 Un )


- 60 -

ω n = The nominal angular frequency of the machine and the powersystem, in el. rad/s, i.e. usually 2π50 or 2π60 el. rad/s

ω mech n = The nominal angular rotor speed of the machine considered, inmech. rad/s

The above listed synchronous machine models use a so-called non-reciprocal rotorp.u. system.

The p.u. base values of the field-circuit plot variables IF and UF

are defined as follows:

- One p.u. field current is the field current which would theoretically be required toproduce one p.u. stator voltage, i.e. rated voltage, on the air-gap line at open-circuit rated speed steady-state conditions.

- One p.u. field voltage is the corresponding field voltage at the field windingtemperature to consider ( usually 75 or 100 degrees centigrade).

The p.u. base values of the damper circuit currents IDD, IQ1 and IQ2 are definedin the same way as for the field current IF.

The p.u. base values of the �m.m.f.� plot variables WD and WQ and the magneticflux linkages FID, FIQ, FIF, FIDD, FIQ1, FIQ2 are determined by the p.u. basevalues of the rotor currents and of the p.u. base values of the inductances.

Sign conventions

Generator or source sign convention is used, meaning that stator currents andpowers are positive out of the machine. Positive torque and speed directions arethat of normal rotor rotation.

The torque variables MT, ME, MEP, MEN and the speed variable SPEED

The p.u. base value Sn /ω n MNm is referred to the equivalent two-pole machinemodel used. The physical torque values in p.u. are the same but then on base Sn/ω mech n MNm. Similarly SPEED is expressed in p.u. on ω n as referred to theequivalent two-pole machine model used. The physical rotor speed in p.u. is thesame but then on base ω mech n .

The plot variables are divided into three groups: Transta only, Masta only, Transtaand Masta.

Transta

Magnitude of stator voltages, in p.u. on base Unetw / 3 kV RMS (or in kV RMS) :

U = Positive-sequence voltage

UN = Negative-sequence voltage

U0 = Zero-sequence voltage

UA,UB,UC = Phase voltages

Appendix A

- 61 -

Magnitude of stator currents, in p.u. on base In kA RMS (or in kA RMS) :

I = Positive-sequence current

IN = Negative-sequence current

I0 = Zero-sequence current

IA,IB,IC = Phase currents

P = Positive-sequence active power output, in p.u. on base Sn MVA(or in MW)

Q = Positive-sequence reactive power output, in p.u. on base SnMVA (or in Mvar)

MEP = Electromagnetic torque produced by positive-sequence statorcurrents, in p.u. on base Sn /ω n MNm

MEN = Electromagnetic torque produced by negative-sequence statorcurrents, in p.u. on base Sn /ω n MNm

Additional variables available for special purposes, for instance for analysis of relayprotection matters:

Stator voltage components, in p.u. on base Unetw / 3 kV RMS (or kV RMS) andreferred to the common (global) real-imaginary reference frame:

UR, UI = Real and imaginary components of positive-sequence statorvoltage

UNR, UNI = Real and imaginary components of negative-sequence statorvoltage

U0R, U0I = Real and imaginary components of zero-sequence statorvoltage

Stator current components, in p.u. on base In kA RMS (or kA RMS) and referred tothe common (global) real-imaginary reference frame:

IR, II = Real and imaginary components of positive-sequence statorcurrent

INR, INI = Real and imaginary components of negative-sequence statorcurrent

I0R, I0I = Real and imaginary components of zero-sequence statorcurrent

Equivalent impedance values as determined by the ratio of voltage phasor tocurrent phasor (U/I) in p.u. on Unetw / ( 3 In ) Ohm ( or in Ohm):

Per phase impedance values

ZA, ZB, ZC = Impedance (magnitude)

RA, RB, RC = Resistance

XA, XB, XC = Reactance


- 62 -

Sequence impedances

Z, ZN, Z0 = Positive, negative and zero-sequence impedance (magnitude)

R, RN, R0 = Positive, negative and zero-sequence resistance

X, XN, X0 = Positive, negative and zero-sequence reactance

Masta

Stator voltages, in p.u. on base 3/2 Unetw kV (or in kV):

U = Voltage, calculated as 20

22 uuu qd ++

UZ = �Zero-sequence� voltage (u0)

UA, UB, UC = Phase voltages

Stator Currents, in p.u. on base 2 In kA (or in kA) :

I = Current , calculated as 20

22 iii qd ++

IZ = �Zero-sequence� current (i0)

IA, IB, IC = Phase currents

P = Instantaneous active power output, calculated as ud id + uq iq +2u0 u0, in p.u. on base Sn MVA (or in MW )

Q = Instantaneous �Reactive power� output, calculated as uq id - ud iq, in p.u. on base Sn MVA (or in Mvar). It is a meaningful quantityonly for steady-state balanced conditions.

Transta and MastaIF = Field current, in p.u (For p.u. base value, see General

information above.)

UF = Field voltage, in p.u. (For p.u. base value, see Generalinformation above.)

MT = Mechanical torque, in p.u. on base Sn /ω n MNm

ME = Total electromagnetic torque, in p.u. on base Sn / ω n MNm

SPEED = Rotor speed, in p.u. on base ω n

TETA = Rotor angle , in el.degrees (KVAR) or el. radians (VAR)

TETA is the difference angle between the q-axis of the machineconsidered and the q-axis of the reference machine asspecified in Dynpow/General

Appendix A

- 63 -

Additional variables available for detailed machine analysis:

Stator variables in p.u. on machine base values and referred to the local referenceframe, i.e. the machine d-q reference frame:

Stator voltages in p.u. on base Un / 3 kV RMS in Transta and 3/2 Un kV inMasta

UDL = Direct-axis stator voltage

UQL = Quadrature-axis stator voltage

Stator currents in p.u. on base In kA RMS in Transta and 2 In kA in Masta

IDL = Direct-axis stator current

IQL = Quadrature-axis stator current

Other variables in p.u. (For p.u. base values, see General information above.)

IDD = Direct�axis damper circuit current of machines of type 1, 1A, 2and 2A

IQ1 = Quadrature-axis damper circuit 1 current of machines of type 1,1A, 2 and 2A

IQ2 = Quadrature-axis damper circuit 2 current of machines of type 1and 1A

WD = Variable to which the m.m.f. related to the direct-axis stator torotor mutual inductance is proportional (WD = IF + IDD � IDL*Lad )

WQ = Variable to which the m.m.f. related to the quadrature-axisstator to rotor mutual inductance is proportional (WQ = IQ1 +IQ2 �IQL* Laq )

FID = Direct-axis stator flux linkages

FIQ = Quadrature-axis stator flux linkages

FIF = Field flux linkages

FIDD = Direct-axis damper circuit flux linkages

FIQ1 = Quadrature-axis damper circuit 1 flux linkages

FIQ2 = Quadrature-axis damper circuit 2 flux linkages


- 64 -

Appendix B

- 65 -

Appendix BSimpow test case files

Optpow (Load flow) file

evaluation of dsl models compared to fortran models** DSL-FORTRAN.OPTPOW **GENERALSN=2220ENDNODESBUS1 UB=24BUS2 UB=24ENDLINESBUS1 BUS2 TYPE=11 X=0.65ENDPOWER CONTROLBUS1 TYPE=NODE RTYP=PQ P=1998 Q=666BUS2 TYPE=NODE RTYP=SW U=23.88 FI=0END

NODESBUS11 UB=24BUS12 UB=24ENDLINESBUS11 BUS12 TYPE=11 X=0.65ENDPOWERBUS11 TYPE=NODE RTYP=PQ P=1998 Q=666BUS12 TYPE=NODE RTYP=SW U=23.88 FI=0END

END


- 66 -

Dynpow (Dynamic simulation) file

!evaluation of dsl models compared to fortran models!two separate systems, each containing:!one synchronous machine, one line and one swingbus**CONTROL DATATETL=1000TEND 90 xtrace 1DDSL=3!!!! HMIN 0.00005!!!! HMAX 0.0005ENDGENERALNREF 2FN 60 60REF BUS2 BUS12END

NODESBUS2 TYPE 1BUS12 TYPE 1END

SYNCHRONOUS MACHINEdslgen BUS1 TYPE=DSL/TYPE1A/ SN 2220 UN 24 H 3.5 RA 0.003

XD 1.81 XQ 1.76 XDP 0.3 XA 0.16 TD0P 8 XQP=0.65 XDB=0.23XQB=0.25 TQ0P=1 TD0B=0.03 TQ0B=0.07R0=0.2 X0=0.8 R2=0.03 X2=0.2

fortrangen BUS11 TYPE=1A SN 2220 UN 24 H 3.5 RA 0.003XD 1.81 XQ 1.76 XDP 0.3 XA 0.16 TD0P 8 XQP=0.65 XDB=0.23XQB=0.25 TQ0P=1 TD0B=0.03 TQ0B=0.07R0=0.2 X0=0.8 R2=0.03 X2=0.2

END

DSL-TYPESTYPE1A(BUS,SN,UN,H,D/0/,RT/0/,XT/0/,RA,XD,XQ,XDP,

XQP,XDB,XQB,XA,TD0P,TQ0P,TD0B,TQ0B,U0,FI0,P0,Q0,UF0,UF,VC,TM0,TM,W,TETR,ICON,ISP/I:0/,SPE/I:1/,R0,X0,R2,X2)

TYPE2A(BUS,SN,UN,H,D/0/,RT/0/,XT/0/,RA,XD,XQ,XDP,XDB,XQB,XA,TD0P,TD0B,TQ0B,U0,FI0,P0,Q0,UF0,UF,VC,TM0,TM,W,TETR,ICON,ISP/I:0/,SPE/I:1/,R0,X0,R2,X2)

TYPE3A(BUS,SN,UN,H,D/0/,RT/0/,XT/0/,RA,XD,XQ,XDP,XA,TD0P,U0,FI0,P0,Q0,UF0,UF,VC,TM0,TM,W,TETR,ICON,ISP/I:0/,SPE/I:1/,R0,X0,R2,X2)

TYPE4(BUS,SN,UN,H,D/0/,RT/0/,XT/0/,RA,XD,XQ,XDP,U0,FI0,P0,Q0,UF0,UF,VC,TM0,TM,W,TETR,ICON,ISP/I:0/,R0,X0,R2,X2)

_CONSTANT_FIELD(UF0,UF)_CONSTANT_TORQUE(TM0,TM)ENDFAULTF1 NODE=BUS1 TYPE=3PSGF2 NODE=BUS1 TYPE=1PSGF11 NODE=BUS11 TYPE=3PSGF12 NODE=BUS11 TYPE=1PSGEND

RUN INSTRUCTION!AT 0.1 INST CONNECT FAULT F1!AT 0.1 INST CONNECT FAULT F11!AT 0.15 INST DISCONNECT FAULT F1!AT 0.15 INST DISCONNECT FAULT F11

AT 0.1 INST CONNECT FAULT F2AT 0.1 INST CONNECT FAULT F12AT 0.15 INST DISCONNECT FAULT F2AT 0.15 INST DISCONNECT FAULT F12

ENDEND

Appendix C

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Appendix CDSL-file for synchronous machine model Type 1A.

!model with one field winding and one d- and two q-axis damperwindings

PROCESS TYPE1A(NODE1,SN,UN,H,D,RT,XT,RL,XD,XQ,XDP,& XQP,XDB,XQB,XL,TD0P,TQ0P,TD0B,TQ0B,& U0,FI0,P0,Q0,UF0,UF,VC,MECH0,MECH,SPEED,TETR,ICON,& ISP,SPE,R0,X0,R2,X2)EXTERNAL SN,UN,H,D,RT,XT,RL,XD,XQ,XDP,XQPEXTERNAL XDB,XQB,XL,TD0B,TQ0P,TD0P,TQ0BEXTERNAL U0,FI0,P0,Q0,UFEXTERNAL MECH,TETR,ISP,SPEEXTERNAL R0,X0,R2,X2INTEGER ICON,ISP,SPEREAL VC,UF0/*/,MECH0/*/,TETR,XQP,XQB,TQ0B,TQ0P,TMREAL U0,FI0,P0,Q0,TD0B,XDBREAL U2,PX,QX,FAC1,FAC2REAL SN,UN,H,D,RT,XT,RL,XL,XD,XQ,XDP,TD0P,UF,MECHREAL R0,X0,R2,X2AC NODE1AC_CURRENT I/NODE1/REAL RA/*/,XA/*/,XAD/*/,XAQ/*/,XRF/*/,TF/*/,UB/*/,W0/*/REAL SIGMAFD/*/,TD/*/,XDD/*/,RF/*/REAL UBASE/*/,IBASE/*/,SIGMAQ1Q2/*/,XQ1/*/,XQ2/*/,TQ1/*/,TQ2/*/REAL UPRE,UPIM,SINT,COST,PSID,PSIQ,IF,TE,PSIDD,PSIQ1,PSIQ2REAL UD,UQ,WD,WQ,FWD,FWQ,UDX/*/,UQX/*/REAL G2/*/,B2/*/,G0/*/,B0/*/,QQ,G1,B1,G00,B00,IN2,TE2REAL TETA,SPEED,PSIF,ID,IQ,IDD,IQ1,IQ2,TETAI,FISTATE TETA,SPEED,PSIF,ID,IQ,PSIDD,PSIQ1,PSIQ2,IDD,IQ1,IQ2STATE TETAI,FI,ICON/0/,WD,WQPLOT IDL/ID/,IQL/IQ/,SPEED,TE2,TE,UD,UQ,TETAPLOT ID,IQ,IDD,IQ1,IQ2,WD,WQ

IF (ISP.EQ.1) THEN !i.e. if mech=mechanical power (in DSL_TYPE: TM)TM=MECH/SPEED

ELSE !i.e. if mech=mechanical torque (in DSL_TYPE: TM)TM=MECH

ENDIF

IF (START) THEN!negative and zero sequencesQQ=R2*R2+X2*X2IF(QQ.EQ.0)THENG2=0.B2=0.ELSEG2=R2/QQB2=-X2/QQENDIFQQ=R0*R0+X0*X0IF(QQ.EQ.0)THENG0=0.B0=0.ELSEG0=R0/QQB0=-X0/QQENDIF

!armature resistance and leakage reactanceRA=RL+RTXA=XL+XT!reactance definitionsXAD=XD-XAXAQ=XQ-XAXRF=XAD**2/(XD-XDP)

UB=UBASE(NODE1) !machine node base voltageUBASE=UB/UN !global-local voltage transformation factorW0=.D/DT.FI0(NODE1)IBASE=UBASE*SN/PBASE !global-local current transformation factorSPEED=1

!definitions of the initial conditionsFAC1=RA**2+XQ**2U2=U0**2PX=P0/U2QX=Q0/U2FAC2=1.+(PX**2+QX**2)*FAC1+2.*(PX**2+QX**2)FAC2=SQRT(FAC2)

!first initial voltage and currentIF(START00)THEN


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UDX=U0*(PX*XQ-QX*RA)/FAC2UQX=U0*(1.+PX*RA+QX*XQ)/FAC2ID=PX*UDX+QX*UQXIQ=PX*UQX-QX*UDXELSEPX=ID*UDX+IQ*UQXQX=ID*UQX-IQ*UDXID:PX-P0=0IQ:QX-Q0=0UDX:SQRT(UDX**2+UQX**2)-U0=0UQX:WQ+XAQ*IQ=0.ENDIF

!dq-components of the flux linkageIF (SPE.EQ.1) THEN

PSID=(UQX+RA*IQ)/SPEEDPSIQ=-(UDX+RA*ID)/SPEED

ELSEPSID=(UQX+RA*IQ)PSIQ=-(UDX+RA*ID)

ENDIF

!mmf in the d resp q axisFWD=PSID+ID*XAFWQ=PSIQ+IQ*XA!no saturationWD=FWDWQ=FWQ

IF (DISC.AND.(UQX.NE.0.OR.UDX.NE.0)) THENTETAI=ATAN2(UDX,UQX)

ELSETETAI: UQX*SIN(TETAI)=UDX*COS(TETAI)

ENDIF

!external load angleTETA=TETAI+FI0SINT=SIN(TETA)COST=COS(TETA)

!field current, resistance, voltageIF=WD+ID*XADRF=(XAD+XRF)/TD0P/W0UF0=IF!field flux linkagePSIF=IF*(1-XAD/XRF)+XAD/XRF*FWD

!mechanical power or torque (in DSL_TYPE: TM0)!TR NKH p.12 or Kundur p.100IF (ISP.EQ.1) THEN

MECH0=(PSID*IQ-PSIQ*ID)/SPEEDELSE

MECH0=PSID*IQ-PSIQ*IDENDIF

! TYPE 1A-VARIABLES:!reactance of winding in d-axisXDD=XAD+(XDP-XA)*(XDB-XA)/(XDP-XDB)!field winding decrement factorSIGMAFD=1-XAD**2/XRF/XDD!time constant of d-damper windingTD=TD0B/SIGMAFD!time constant of field windingTF=TD0P-TD!reactance of 1:st q-damper windingXQ1=XAQ**2/(XQ-XQP)!reactance of 2:nd q-damper windingXQ2=XAQ+(XQP-XA)*(XQB-XA)/(XQP-XQB)!q-damper windings decrement factorSIGMAQ1Q2=1-XAQ**2/XQ1/XQ2!time constant of 2:nd q-damper windingTQ2=TQ0B/SIGMAQ1Q2!time constant of 1:st q-damper windingTQ1=TQ0P-TQ2

!flux linkages of the damper windingsPSIDD=XAD/XDD*FWDPSIQ1=XAQ/XQ1*FWQPSIQ2=XAQ/XQ2*FWQ

!real-/imag. voltage in machine per unitUPRE=UPRE(NODE1)*UBASEUPIM=UPIM(NODE1)*UBASE

ELSE !else if(start)!real-/imag. voltage in machine per unitUPRE=UPRE(NODE1)*UBASEUPIM=UPIM(NODE1)*UBASE!voltage angle

Appendix C

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SINT=SIN(TETA)COST=COS(TETA)!coordinate transformationUD=UPRE*SINT-UPIM*COSTUQ=UPRE*COST+UPIM*SINT!flux linkages (with and without speed included in the model)IF (SPE.EQ.1) THEN

PSIQ=-(UD+ID*RA)/SPEEDPSID=(UQ+IQ*RA)/SPEED

ELSEPSIQ=-(UD+ID*RA)PSID=(UQ+IQ*RA)

ENDIF

!field winding currentIF=(PSIF-(PSID+XA*ID)*XAD/XRF)/(1-XAD/XRF)

!negative & zero sequenceG1=UBASE*G2B1=UBASE*B2G00=UBASE*G0B00=UBASE*B0INRE(I)=(-UNRE(NODE1)*G1+UNIM(NODE1)*B1)INIM(I)=(-UNIM(NODE1)*G1-UNRE(NODE1)*B1)I0RE(I)=(-U0RE(NODE1)*G00+U0IM(NODE1)*B00)I0IM(I)=(-U0IM(NODE1)*G00-U0RE(NODE1)*B00)IN2=INRE(I)**2+INIM(I)**2

!negative seq. torqueTE2=-(R2-RA)*IN2

!electrical torqueTE=IQ*PSID-ID*PSIQ+TE2!deviation angleTETA: (SPEED-1)*W0-.D/DT.TETA=0!machine speedSPEED: TM-TE-.D/DT.SPEED*2*H-D*(SPEED -1)=0

!field winding flux linkagePSIF: UF-IF-.D/DT.PSIF*TF=0

!mmf in the d resp q axisFWD=PSID+ID*XAFWQ=PSIQ+IQ*XA!no saturationWD=FWDWQ=FWQ

!currents of d-and q-axisID: IF+IDD-ID*XAD-WD=0IQ: IQ1+IQ2-IQ*XAQ-WQ=0

! damper winding voltage and flux linkage equations for TYPE 1APSIDD: 0=IDD+TD*.D/DT.PSIDDPSIQ1: 0=IQ1+TQ1*.D/DT.PSIQ1PSIQ2: 0=IQ2+TQ2*.D/DT.PSIQ2IDD: PSIDD=IDD*(1-XAD/XDD)+XAD/XDD*FWDIQ1: PSIQ1=IQ1*(1-XAQ/XQ1)+XAQ/XQ1*FWQIQ2: PSIQ2=IQ2*(1-XAQ/XQ2)+XAQ/XQ2*FWQ

ENDIF !end if(start)

!posistive sequenceVC=SQRT(UPRE**2+UPIM**2)IPRE(I)=ID*SINT+IQ*COSTIPIM(I)=IQ*SINT-ID*COSTIPRE(NODE1)=IPRE(I)*IBASEIPIM(NODE1)=IPIM(I)*IBASE

!neg. & zero seq.IF(.NOT.START)THEN

INRE(NODE1)=INRE(I)*IBASEINIM(NODE1)=INIM(I)*IBASEI0RE(NODE1)=I0RE(I)*IBASEI0IM(NODE1)=I0IM(I)*IBASE

ENDIFEND


- 70 -

Appendix D

- 71 -

Appendix DBlock diagram for synchronous machine model Type 1A, by Jonas Persson.


- 72 -

References

- 73 -

References[1] Johansson, K-E, Transta- Transient Stability Program, ASEA, 1977.

[2] Johansson, K-E, Dynpow- Dynamic Power System Simulation, ASEA,1981.

[3] SIMPOW Power System Simulation & Analysis Software Users ManualRelease 10, February 2001, Binder 1, pp. 44-49.

[4] Petersson, T. SIMPOW Synchronous machine model Type ST33Mathematical description, ABB Power Systems, 1999.

[5] Petersson, T. SIMPOW Synchronous machine model Type SP1Mathematical description, ABB Power Systems, 1999.

[6] Sen, P.C. Principles of Electric Machines and Power Electronics 2nd ed.,John Wiley & Sons, 1997.

[7] Fitzgerald, A.E., et. al. Electric Machinery 5th ed., McGraw-Hill, 1990.

[8] IEEE Standard 100-1996, IEEE Standard Dictionary of Electrical andElectronic Terms 6th ed.

[9] Kundur, P. Power System Stability and Control, McGraw-Hill, 1994.

[10] Bühler, H. Einfürung in die Theorie geregelter Drehstromantriebe- Band1:Grundlagen, Birkhäuser, 1977.

[11] Concordia, C. Synchronous Machines Theory And Performance, Wiley,1951.

[12] Anderson, P.M., Fouad, A.A. Power System Control and Stability 1st ed.,Iowa State University Press, 1977.

[13] Laible, Th. Die Theorie der Synchronmaschine im nichtstationärenBetrieb, Springer, 1952.

[14] Park, R.H. Two-Reaction Theory of Synchronous Machines-Part II, AIEETrans., Vol. 52, June 1933, pp. 352-355.

[15] Kovács, P.K. Transient Phenomena in Electrical Machines, Elsevier,1984.

[16] Tore Petersson, Senior Specialist, Power Systems Analysis departmentABB Utitlities, private discussion December 2001 Västerås.

[17] IEEE Std 399-1980 IEEE Brown Book, Recommended Practice for PowerSystem Analysis.

[18] Persson, J. Conversion of PSS/E-files to SIMPOW-format focusing ondynamic models, ABB Power Systems, 1999.


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List of symbols

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List of symbols

AbbreviationsDSL Dynamic Simulation Language used in Simpow when

programming e.g. synchronous machine models. Developed byKjell Anerud

dq0-axis direct-, quadrature- and zero axis following the rotation of therotor of a machine

dq0-transforme transformation from three-phase stator quantities to rotorquantities

FORTRAN FORmula TRANslation, the first high-level computer language.Developed by Jim Backus

IEEE Institute of Electrical and Electronics Engineers, IncKTH Kungliga Tekniska Högskolan, the Royal Institute of

TechnologyMasta module of Simpow, where instantaneous models are simulatedp.u. system per unit system, unit-less system, often used in power systems

analysisSimpow Power system simulation software, developed by ABBSI-units Le Système International d�Unités, adopted by the Eleventh

General Conference on Weights and Measures, held in Paris in1960, for a universal, unified, self-consistent system ofmeasurement units based on the MKS (meter-kilogram-second)system

Transta module of Simpow, where fundamental frequency models aresimulated

Sub-indices0 the zero axis of the rotor reference system0 zero sequence0 extra index at time zero1 positive sequence2 negative sequenceα phase α, of the two-phase global reference systemβ phase β, of the two-phase global reference systemε leakage inductancea phase a, of the three-phase systema armature (leakage) resistance, inductance, reactanceb phase b, of the three-phase systemc phase c, of the three-phase systemd the direct axis of the rotor reference systemdq0 the dq0-transformed three-phase stator reference systemnetw base per unit bases for the networknode base per unit bases for the machine noden index for nominal quantities of the synchronous machineq the quadrature axis of the rotor reference systemR the three-phase rotor reference systemS the three-phase stator reference systemSbase per unit bases for the synchronous machine systemt generator node quantities

Super-indices´´ subtransient


- 76 -

´ transient* complex conjugate

Symbolsδ,θ electrical displacement angle, Transta resp. Mastaµ translation or screening factorσ decrement factorτ time constantω frequency, rad/s∆ω rotor speed deviationωmech mechanical speedΨ, ψ flux linkage, SI-units resp p.u.ℜ reluctanceCI current transformation constantCU voltage transformation constantD damping factor, p.u.f frequency, Hzf saturation functionH per unit inertia constantI, i current, SI-units resp p.u.J moment of inertia, SI-unitsL,l inductance, SI-units resp p.u.M, m torque, SI-units resp p.u.P,p active power, SI-units resp p.u.Q,q reactive power, SI-units resp p.u.R,r resistance, SI-units resp p.u.S imaginary power, SI-unitsU, u voltage, SI-units resp p.u.W magneto motive forceX,x reactance, SI-units resp p.u.Z,z impedance, SI-units resp p.u.

detailed description of synchronous machine models used in

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