small signal stability of smib without amortisseur.pdf

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    Complete set of electrical equations in per unit

    In the following equations, two q-axis amortisseur circuits are considered, and the subscript

    1

    and 2 are used to identify them. Only one d-axis amortisseur circuit is considered, and it isindentified by the subscript 1.Per unit stator voltage equations:

    = 8.1 = + 8.2 = 8.3

    Per unit rotor voltage equations:

    = + 8.40 = + 8.50 = + 8.60 = + 8.7

    Per unit stator flux linkage equations:

    = + + + 8.8 = + + + 8.9 = 8.10

    Per unit rotor flux linkage equations:

    = + 8.11

    = + 8.12 = + 8.13 = + 8.14

    Per unit air-gap torque:

    = 8.15

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    In the above equations, we have assumed that the per unit mutual inductance = . Thisimplies that the stator and rotor circuits in the q-axis all link a single mutual flux represented by

    . This is acceptable because the rotor circuits represent the overall rotor body effects, and theactual winding with physically measurable voltages and current do not exist.For power system stability analysis, the machine equations are normally solved with all

    quantities expressed in per unit, with the exception of time. Usually time is expressed inseconds, in which case the per unit p in above equations is replaced by 1 !" .Synchronous Machine Representation in Stability Studies

    Simplifications essentials for large-scale studies

    For stability analysis of large system, it is necessary to neglect the following from Equations(8.1) and (8.2) for stator voltage:

    The transformer voltage terms, #$ . The effect of speed variations.

    The reasons for and the effects of these simplifications are discussed below.

    Neglect of Stator &' termTransformer voltages are lesser than speed voltages, and only due to transformer voltage terms

    the voltage equation (8.1) and (8.2) are differential equations. If we neglect transformer voltage

    terms, then stator voltage equation become algebraic equation.

    Stator term represents stator circuit transients. Stator is usually connected to transmissionline and transmission line has very low time constants i.e. associated transients quickly dies out,

    so in stability studies, so we generally ignore transmission line transients. On the other hand, if

    we consider stator transients and ignore transmission line transients than it is inconsistent,

    therefore we also neglect the stator transient #$ .Neglecting the Effect of Speed Variations on Stator Voltages

    Another simplifying assumption normally made is that the per unit value of is equal to 1.0 inthe stator voltage equations. This is not the same as saying that speed is constant; it assumes thatspeed changes are small and do not have a significant effect on the voltage.

    The assumption of per unit = 1.0 (i.e., = ) *# ) in the stator voltage equations doesnot contribute to computational simplicity in itself. The primary reason for making this

    assumption is that it counterbalances the effect of neglecting , terms so far as the lowfrequency rotor oscillations are concerned.

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    With per unit = 1.0, the stator voltage equations reduced to

    = 8.16

    = 8.17Relationship between per unit / and /The terminal electric power in per unit is given by

    = + Substituting for and from Equations (8.16) and (8.17) gives

    = + = + 8.18=

    The air-gap power, measured behind , is given by = + 8.19= 8.20The per unit air gap power so computed is in fact the power at synchronous speed and is equal

    to the per unit air-gap torque . Normally, = , however, the assumption of = 1.0 puin the stator voltage equation is also reflected in the torque equation, making = . This fact isoften overlooked.

    Simplified Model with Amortisseurs Neglected

    The first order of simplification to the synchronous machine model is to neglect the

    ammortisseur effects. This minimizes data requirements since the machine parameters related to

    the amortisseurs are often not readily available. In addition, it may contribute to reduction in

    computational effort by reducing the order of the model and allow large integration steps in time

    domain simulations.

    With the amortisseurs neglected, the stator voltage Equations (8.16) and (8.17) are unchanged.

    The remaining equations (8.4) to (8.15) simplify as follow.

    Flux linkages:

    = + 8.21

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    = 8.22

    = + 8.23

    Rotor voltage:

    = + = 8.24

    Equation (8.24) is now the only differential equation associated with the electrical characteristics

    of the machine. In the above equation all quantities, including time, are in per unit.

    Alternative form of machine equations

    In the literature on synchronous machines, Equations (8.21) to (8.24) are often written in terms

    of the following variables:

    56 = = Voltage proportional to 57 = :;;: = Voltage proportional to 5 = :;:= :;;:

    Then

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    7 =

    Substituting in Equation (8.26) gives

    57 = 56 7 8.27Multiplying Equation (8.24) by

    :;;: throughout, we have ? :;;: @ = :

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    =

    = J + J

    = J + 56 Therefore, 56 = + J + Multiplying by j, we have

    K56 = K + KJ + KIn terms of phasor notation,

    56 = + KJ + 8.29From equation (8.27) withJ = 7 ,

    57 = + J + J + J= + J +

    Multiplying by j gives

    K57

    = K +KJ + K

    In terms of phasor notation,

    57 = +KJ + 8.30We see that phasorsPQand PR7 both lie along the q-axis and PR7 also lies along q-axis.w.k.t. the following Equation and substituting 56 forJ, we get

    56 = 5 +KJ J 8.31Figure (8.1) shows the phasor diagram representingPR7 , 56 and 5 given by equations (8.29),(8.30) and (8.31). 5 = + K = +

    5 7 = 5 + + KJ7 SPR7 =q-axis component of5 7

    = + + KJ7

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    PQ = 5 +KJ J

    Figure (8.1) Synchronous machine phasor diagram in terms ofUV7 , UV and UWConstant flux linkage model (Classical model):

    For studies in which the period of analysis is small as compared to the machine model isoften simplified by assuming

    57(or

    ) constant throughout the study period. The assumption

    eliminates the only differential equation associated with the electrical characteristics of the

    machine.

    A further approximation to simplify the machine model is to ignore transient saliency by

    assumingJ7 = J7 , and to assume that the flux linkage also remains constant. With theseassumptions, the voltage behind the transient impedance + KJ7 has a constant magnitude.The per unit flux linkages identified in the d-axis are given by

    = + 8.32

    = 8.33 = + 8.34From Equation (8.34)

    = ;:Y:;: 8.35

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    Figure (8.2) Thed- and q-axis equivalent circuits with one rotor circuit in each axis

    Substitute in Eq. (8.32) gives, = + :;: = 7 ? + ;:;: @

    Where, 7 = :" > ;:" = Similarly, for the q-axis

    = 7 ? + Z[Z[ @Where, 7 = From Equation (8.16), the d-axis stator voltage is given by

    = = +

    Substituting for from equation (8.35) gives = + 7 ? + Z[Z[ @= + + 7 7 ?Z[Z[ @ 8.36= + J + 57

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    Where, 57 = ?Z[Z[ @ 8.37Similarly, the q-axis stator voltage is given by

    = + J + 57 8.38Where, 57 = ?;:;: @ 8.39With transient saliency neglected J7 = J7 the stator terminal voltage is

    + K = 5 +K5 + K + J K= 5 +K5 + K KJ + KUsing phasor notation, we have

    5] = 5P +KJSP 8.40Where, 5P = 5 +K5

    = ? Z[Z[ + K ;:;: @The corresponding equivalent is shown in figure (8.3)

    Figure (8.3) Simplified transient model

    With rotor flux linkages and constant, 57 and 57 are constant. Therefore the magnitudeof5 is constant. As the rotor speed changes, the d- and q-axes move with respect to any generalreference coordinate system whose R-I axes rotate at synchronous speed, as shown in figure ( ).

    Hence, the components 5

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    Figure (8.4) TheR-Iandd-q coordinate system

    This model offers considerable computational simplicity; it allows the transient electrical

    performance of the machine to be represented by a simple voltage source of fixed magnitude

    behind an effective reactance. It is commonly referred to as the classical model, since it was used

    extensively in early stability studies.

    8. SMALL SIGNAL STABILITY OF A SINGLE MACHINE INFINITE BUS SYSTEM:Small signal stability is the ability of the power system to maintain synchronism when subjected

    to small disturbance. The small perturbation continuously occurs in any power system n due to

    small changes in load and generation. For analyzing the small signal stability of any system the

    system model can be linearized around an operating point i.e the disturbances are considered tobe so small or incremental in nature so that we can develop a linear model of the system around

    an operating point. Once we develop the linea

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