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  • 3.1

    Chapter 3 Modeling Of a Synchronous Machine

    In Chapter 2 we have discussed about small-signal and transient stability of a

    synchronous machine, connected to an infinite bus, represented by a classical model.

    We have seen that there are several disadvantages of representing a synchronous

    generator by a classical model. In this Chapter we will look at detailed modeling of a

    synchronous machine.

    3.1 Representation of Synchronous Machine Dynamics

    While modeling a synchronous machine, different ways of representation,

    conventions and notations are followed in the available literature. Hence, at the outset

    the notations and conventions used for representing a synchronous machine should be

    clear. In this course, IEEE standard (1110-1991) IEEE guide to synchronous

    machine modeling has been followed for representing the synchronous machine.

    (a)

  • 3.2

    (b)

    Fig. 3.1: Synchronous machine (a) sectional view (b) Stator and rotor windings with

    mmf along the respective axis.

    The conventions and notations used along with their significance will be

    explained in this Chapter. To model and mathematically represent a synchronous

    machine first all the windings that need to be included in the model should be

    identified. Consider sectional view of the synchronous machine shown in Fig. 3.1 (a).

    The synchronous machine in Fig. 3.1 is a two pole salient machine. A general model

    with n poles will be dealt latter in this Chapter.

    The conductors anda a represent the sectional view of one turn of a-phase

    stator winding. The dot in the conductor a represents current coming out of the

    conductor and represents in to the conductor. By applying right hand thumb rule at

    conductor anda a it can be observed that the mmf due to the conductors

  • 3.3

    anda a lie along axis marked A-axis. Similarly, the mmf due to ,b b and ,c c lie

    along B and C axis, respectively. As an electrical circuit the stator can be represented

    as three windings corresponding to three-phases, as shown in Fig. 3.1 (b). The three-

    phase instantaneous ac voltages and currents in the stator windings are represented as

    , ,a b cv v v and , ,a b ci i i . According to the generator convention, currents out of the

    stator windings are considered as positive where as currents into the rotor windings

    are considered as positive.

    The rotor field is excited by a dc voltage represented as fdv with a field current

    fdi . The mmf generated by the rotor field excitation lies normal to the pole surface,

    along the direct axis or d-axis. The d-axis of the rotor is at angle m with respect to

    the stator a a mmf axis that is A-axis. Angle m is in mechanical radians and in

    case of two poles machine the electrical and mechanical angle are one and the same.

    But in case of multiple poles the electrical angle is related to the mechanical angle

    through the number of poles i.e. / 2s mP where P is the number of poles and s

    is the rotor angle in electrical radians. In case of two poles machine the electrical and

    mechanical rotor angle will be same as is the case for the synchronous machine shown

    in Fig. 3.1. The analysis holds true for multiple pole machine as well but with the

    additional condition that / 2s mP . For rest of the Chapters we will be expressing

    the angle in terms of electrical radians unless specified other wise. The axis in

    quadrature (leading or lagging by 90 ) with respect to the d-axis is called as

    quadrature axis or q-axis. The q-axis can either be represented as leading d-axis or

    lagging d-axis. Both the conventions are followed in the literature. However, here q-

    axis is taken as leading d-axis according to the IEEE 1110-1991 standard.

    Representing damper windings needs clarification. The damper windings are

    copper bars placed usually in the slots of the pole face. The ends of the copper bars

    are shorted forming a closed path for the currents to flow. The magnetic field

    generated by these damper windings, due to currents circulating through these

    windings, will be along the d-axis. However, the rotor core itself may act as closed

    path for induced currents during non-synchronous operations. Hence, to properly

    account for the action of the damper windings and damping effect of rotor core three

    damper windings are considered. One damper winding represented as 1d , with a

  • 3.4

    voltage 1dv and current 1di , is considered whose mmf is along d-axis. Two damper

    windings represented as 1 , 2q q , with a voltage 1 2,q qv v and current 1 2,q qi i are

    considered whose mmf is along q-axis. In the d-axis and q-axis rotor windings the

    current in to the winding is considered as positive.

    For very accurate representation of synchronous machine, even more number

    of damper windings may be considered along d and q axis. According to the number

    of windings considered along each axis a model number is give as following [1]

    Table 3.1: Classifications of synchronous machine model based on number of

    windings in each axis

    Number of windings in q-axis

    0 1 2 3

    Model

    1.0

    Model

    1.1

    Model

    1.2

    Model 2.1 Model

    2.2

    Number

    of

    windings

    in

    d-axis

    1

    2

    3 Model 3.3

    The first number in the model number given in Table 3.1 represents number of

    windings in d axis and second number represents number of windings in q axis.

    There should be at least one winding, field winding, in the d axis. Hence, the first

    model 1.0 means that rotor is represented by one field winding, zero d-axis damper

    winding and zero q-axis damper windings. From the view point of complexity, in the

    representation of many windings along d axis and q axis, the maximum number

    of winding that can be represented along any axis is fixed at 3. The model which is

    shown in Fig. 3.1 (b) is 2.2 that is one field winding, one damper winding along d-

    axis and two damper windings along q-axis. Model 2.2 is widely used in many

    industry grade transient stability simulation softwares.

  • 3.5

    3.1.1 Stator and rotor winding voltage equations

    Applying KVL at the stator windings the following equations can be written

    aa s a

    dv r idt

    (3.1)

    bb s b

    dv r idt

    (3.2)

    cc s c

    dv r idt

    (3.3)

    where, sr is the stator resistance and is assumed to be same in all the three

    phases. The flux linkages in a, b, and c phases are represented as , ,a b c . The rate

    of change of flux linkages in phase a, b and c lead to an induced emf (electro-motive

    force) which is equal to the terminal phase voltage plus the drop in the stator

    resistance (since we are using generator convention), as can be seen from equations

    (3.1) to (3.3). Now applying KVL at the d and q axis rotor windings will give the

    following expressions

    fdfd fd fd

    dv r i

    dt

    (3.4)

    11 1 1

    dd d d

    dv r idt

    (3.5)

    11 1 1

    qq q q

    dv r i

    dt

    (3.6)

    22 2 2

    qq q q

    dv r i

    dt

    (3.7)

    Where, 1 1 2, , ,fd d q qr r r r and 1 1 2, , ,fd d q q are the rotor field, 1d, 1q and 2q winding

    resistances and flux linkages, respectively.

  • 3.6

    3.1.2 Stator and rotor windings flux linkage equations

    The flux linkages of different windings can be expressed in terms of current

    through the windings and inductance of the windings as:

    1 1 21

    1 1 21

    1 1 22

    abc abcss srrotor

    fdafd a d a q a qa aa ab ac a

    db ba bb bc b bfd b d b q b q

    qc ca cb cc c cfd c d c q c q

    qiL Li

    iL L L LL L L i

    iL L L i L L L L

    iL L L i L L L L

    i

    (3.8)

    abc ss abc sr rotorL i L i (3.9)

    In equation (3.8) the diagonal elements of the matrix ssL represent the self

    inductance of a, b, c windings and off-diagonal elements represent the mutual

    inductance among a, b, c phases. The matrix srL represents the mutual inductance

    between the stator and rotor windings. A similar expression for flux linkage of the

    rotor windings can be written as

    1

    1 1 11 1 1 1

    1 1 1 1 1 1 1 2

    2 2 2 2 2 1 2 2

    0 0

    0 0

    0 0

    0 0abc

    rotor rs rr

    fdfd fd dfd fda fdb fdca

    dfd d dd da db dcb

    q qa qb qc q q q qc

    q qa qb qc q q q qi

    L L

    L LL L Li L LL L Li

    L L L L Li

    L L L L L

    1

    1

    2

    rotor

    fd

    d

    q

    q

    i

    iiii

    (3.10)

    rotor rs abc rr rotorL i L i (3.11)

    In the matrix rrL , the mutual inductance between the d-axis windings ( ,1fd

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