SYNCHRONOUS MACHINE MODELS – 1

ByProf. C. Radhakrishna

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CONTENTSSYNCHRONOUS MACHINE MODELS – 1

Synchronous Machine Theory and Modelling

Direct and Quadrature Axes

Mathematical Description of a Synchronous Machine

Representation in System Studies

Typical values of standard parameters

Simplifications essential for large-scale studies

Neglect of Stator pψ Terms

Neglecting the Effect of Speed Variations on Stator Voltages

Simplified model with Amortisseurs NeglectedConstant Flux Linkage ModelClassical ModelConstant Flux Linkage Model Including the Effects of Subtransient CircuitsSummary of Simple Models for Different Time Frames

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SYNCHRONOUS MACHINE MODELS – 1Synchronous Machine Theory and Modelling

• The power system stability problem is largely one of keepinginterconnected synchronous machines in synchronism.

• An understanding of their characteristics and accuratemodeling of their dynamic performance are of fundamentalimportance to the study of power system stability.

Direct and Quadrature Axes• Magnetic circuits and all rotor windings are symmetrical with

respect to both polar axis and the inter-polar axis.• The direct (d) axis, centred magnetically in the centre of

the north pole;• The quadrature (q) axis, 90 electrical degrees ahead of the

d-axis.• The position of the rotor relative to the stator is measured

by the angle θ between the d-axis and the magnetic axis ofphase 'a' winding.

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In developing equations of a synchronous machine,assumptions are made

Representation in System Studies

Mathematical Description of a Synchronous Machine

The swing equation, expressed as two first order differentialequations, becomes

1 ( )2

rm e D r

d T T Kdt H

0 rddt

In the above equations, time t is in seconds, rotor angle δ is inelectrical radians, and ω0 is equal to 2πf. We will assume thevariables ∆ωr, Tm and Te to be in per unit. However, t will beexpressed in seconds and ω0 in electrical radians per second.

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Typical values of standard parameters

Table 1

d q q d q dX X X X X X

0 0d d d d kdT T T T T

0 0q q q qT T T T

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Table 1

Parameter Hydraulic Units Thermal Units

Synchronous ReactanceXd 0.6 – 1.5 1.0 – 2.3

Xq 0.4 – 1.0 1.0 – 2.3

Transient Reactance

0.2 - 0.5 0.15 – 0.4

- 0.3 – 1.0

Subtransient Reactance

0.15 – 0.35 0.12 – 0.25

0.2 – 0.45 0.12 – 0.25

Transient OC Time Constant

1.5 – 9.0 s 3.0 – 10.0 s

- 0.5 – 2.0 s

Subtransient OC Time Constant

0.01 – 0.05 s 0.02 – 0.05 s

0.01 – 0.09 s 0.02 – 0.05 s

Stator Leakage Inductance X1 0.1 – 0.2 0.1 – 0.2

Stator Resistance Ra 0.002 – 0.02 0.0015 – 0.005

dX

qX

dX

qX

0dT

0qT

0dT

0qT

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Simplifications essential for large-scale studiesFor stability analysis of large systems, it is necessary to

neglect the following for stator voltage:• The transformer voltage terms, pψd and pψq• The effect of speed variations.

Neglect of Stator pψ Terms• The pψd and pψq terms represent the stator transients.• With these terms neglected, the stator quantities contain

only fundamental frequency components and the statorvoltage equations appear as algebraic equations.

• This allows the use of steady-state relationships forrepresenting the interconnecting transmission network.

Neglecting the Effect of Speed Variations on Stator Voltages• The assumption of per unit ωr = 1.0 (i.e., ωr = ω0 rad/s) in the

stator voltage equations does not contribute to computationalsimplicity in itself.

• The primary reason for making this assumption is that itcounterbalances the effect of neglecting pψd, pψq terms so far asthe low-frequency rotor oscillations are concerned.

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• The first order of simplification to the synchronous machinemodel is to neglect the amortisseur effects.• It may contribute to reduction in computational effort byreducing the order of the model and allowing larger integrationsteps in time-domain simulations.

Simplified model with Amortisseurs Neglected

For studies in which the period of analysis is small incomparison to , the machine model is often simplified byassuming (or ψfd) constant throughout the study period.This assumption eliminates the only differential equationassociated with the electrical characteristics of the machine.A further approximation, which simplifies the machine modelsignificantly, is to ignore transient saliency byassuming , and to assume that the flux linkage ψ1qalso remains constant. With these assumptions, the voltagebehind the transient impendencehas a constant magnitude.

Constant Flux Linkage Model:Classical Model

0dT

qE

d qX X

a dR jX

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This model offers considerable computational simplicity; it allowsthe transient electrical performance of the machine to berepresented by a simple voltage source of fixed magnitudebehind an effective reactance. It is commonly referred to as theclassical model, since it was used extensively in early stabilitystudies.

Constant Flux Linkage Model Including the Effects of Subtransient Circuits

Figure 1 Simplified transient model

This model is used in short-circuit programs for computing theinitial value of the fundamental frequency component of short-circuit currents. As the rotor flux linkages cannot changeinstantaneously, the value is equal to its prefault value. Such aconstant flux linkage model would not be generally acceptable forstability studies.

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Summary of Simple Models for Different Time Frames

• Simple models of the synchronous machine applicable to the three timeframes; subtransient, transient, and steady state.

• The subtransient and transient models assume constant rotor fluxlinkages, and the steady-state model assumes constant field current.

• These models neglect saliency effects and stator resistance and offerconsiderable structural and computational simplicity.

Figure 2 Simplified model for subtransient period

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is the predisturbance value of internal voltage given by

(a) Subtransient model

0E

0 0 0t tE E jX I

is the internal voltage

(b) Transient model

0E

0 0 0t tE E jX I

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(c) Steady-state model

Figure 3: Simple Synchronous Machine Models

0 0q t s tE E jX I

q ad fd IE X i E

REFERENCES :[ 1 ] P.M. Anderson & A.A. Fouad : “Power System Control and Stability” , 2nd edition, IEEE Press

Power Engineering Series, Wiley-Interscience, 2003.[ 2 ] K.R. Padiyar : “Power System Dynamics : Stability and Control” , 2nd edition, BS

Publications, 2002.

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CONCLUSIONS

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THANK YOU