a new reduced order model of im

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A New Reduced Order Model of InductionMotors

Pekik Argo Dahono and Qamaruzzaman

Electrical Energy Conversion Research Laboratory,Department of Electrical Engineering, Bandung Institute of Technology

J1. Ganesa No. 10, Bandung 40132, INDONESIA

Abstract

A new reduced order model of induction motors is propo-sed in this paper. Instead of neglecting the st ato r tran-sients as in the case of standard reduced order model,the new model is derived by neglecting the transients inthe leakage inductances. It is found tha t the new model

produces better results than the standard reduced ordermodel and, in some cases, the results are almost the sameas that of the full order model.

1. Introduction

A significant proportion of power system load is madeof induction motors and, therefore, accurate representa-tion of it is impo rtan t in power system dynamic analysis.Although the full order model (i.e. fifth order) of induc-tion motors has been considered to be the most accuratemodel, the required computation tim e is very large if wedealt with a multimachine system. In order to reduce the

computation time, the order of the model should be redu-ced. The sta nda rd method to reduce the order of induc-tion motor model is by neglecting the electrical transientsin the stator[l-51. Since the transients in transmissionlines are usually neglected in power system analysis pro-grams, reduction of induction motor model by neglectingthe electrical transients in the s tat or is very useful. T hefact that the stator variables changed much faster thanthe rotor variables is the basic argument of neglecting theelectrical transients in the stator. In the computationof the rotor variables, therefore, the changes of the s tat orvariables can be neglected. I n th e reality, however, the ro-tor variables begin to change before the stator transientscompletely subside. Thu s, the predicted induction mo-tor behaviors will be different from tha t of the full ordermodel. Several methods t o improve the standa rd reducedorder model were poposed in the literature[6-7]. Unfor-tunately, these methods are either complicated or need aswitching from the full order into reduced order models.

In this paper, a new reduced order model of inductionmotors is proposed. The new model is derived by ne-glecting the transients in the leakage inductances. Wit h

the transients in the leakage inductances are neglected,the mathematical model of the induction motor can beformulated in terms of either the magnetizing currentsor mutual flux linkages. In most cases, the tran sients inthe leakage inductances can be neglected because &he ea-kage inductances are much smaller than the magnetizinginductance, In this paper, the proposed model is used

to predict the behavior of induction motors during tran-sient conditions (e.g., start-u p, short-circuit, and suddenchanges of load). The accuracy of the proposed model iscompared to th e sta ndard reduced order model and thefull order model. The influence of the motor size on theacuracy of the proposed model is investigated. In somecases, it is found tha t the proposed model can give accu-rate results as accurate as the full order model.

2. Full Order Model

The full order model of induction machines in synchro-nous reference frame can be written in either of currentsor flux linkages as sta te variables. Thi s model can be visu-alized by th e T-form equivalent circuits as shown in Fig.1. In per unit form and by using flux linkages as statevariables, the full order model can be written as follows:

(1)

(2)

(3)

(4)

P- * q a = vqs - l * q a - Wcu*ds + aZ*qrwbP

-*ds = vds - 1 q d s +w e u q q s +aZ*drwbP

-*qr = a3*qs - 4*qr - Walu*drwbP

-*dr = a 3 9 d s + w si u* gr - e q d rwb

where

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+v,sin (B+31

Fig. 1. T-form equivalent circuits.

in which r, and r , are the resistances and XI, and Xr, arethe leakage reactances of the stator and rotor windings,respectively. X , is the mutual reactance between thestator and rotor windings. All rotor quan tities are referredto the s tator side. w b , w e , and w, are the base, supply, androtor angular frequencies, respectively, and H is the totalinertia constant of the motor and the connected load.

If the phase voltages are assumed as a balanced set ofsinusoidal voltages having th e angular frequency of we,then the direct and quadrature components of the stator

voltages in synchronous reference frame can be obtainedas

+U, cos (e + 31Vd s = 2 [, sin 6 -t Vb sin (6 - F)

3 .

where

6 = I,' wedt + e(0) (18)

Eqns. (1)-(5) can be solved numerically to obtain thetransient behaviors of induction motors. After the fluxlinkages have been solved, the direct and quadrature com-ponents of the stator currents can be obtained as

(19)1

i,, = 5 X r r q q S - Xm*,r)

The results can be used t o determine the phase currents,

ib , = i,, cos ( w e t - $) +sin ( w e t - : (22)

i,, = i,, COS ( u e t +$) +sin (wet +$) (23)

The rms value of the phase currents can also be obtainedas

Ip h = d- (24)

3. Standard Reduced Order Mo-del

The standa rd method t o reduce the order of the model ofinduction machines is by neglecting the transients in thestator. The basic argument of this method is the tran-

sients in the stato r are much faster than t ha t of the rotor.By neglecting the transients in the stator, eqns. (1)-(2)become algebraic equations and the stator flux linkagescan be calculated as

(25)lvqs - e u v d s +a1a2Qqr - 2weuqdr

a +w; ,qa =

The results can be used to solve eqns. (3)-(5). The accu-racy of this model was thoroughly investigated in the li-terature. This model usually gives a good result for smallsignal disturbances. For large signal disturbances (e.g.start-up and short circuit), however, the results are qu ite

different to that of the full order model.

4. The New Reduced Order Mo-del

In the standard reduced order model, the transients inthe s tat or flux linkages have been neglected. These fluxlinkages consist of flux leakage of the sta tor winding and

652




I I

Fig. 2. The modified equivalent circuits.

mutual flux linkage. Because the m utua l inductance ismuch larger than the leakage inductance, the changes offlux linkages in the former will be slower than that of inthe later. Thus, the transients in the mutual inductancecan not be neglected.

If we want t o reduce the order of the induction motormodel without neglecting the transient in the mutua l in-ductance, we have to neglect the transients i n the leakageinductances of both stator and rotor windings. We know,however, the rotor quantit ies are changing slower tha n thestator and the results, therefore, may not be acceptable.In order to solve this problem, equivalent circuits of in-duction machines as shown in Fig. 2 will be used to derivethe new model. The parameters of these equivalent cir-cuits can be derived from that of the Fig. 1 by using thefollowing relations:

Lr = L1a + d l r (27)L M = ~ L M (28)

l& = a%, (29)

(30)

whereLm

L m +.L l ra =

Though these equivalent circuits have less parametersthan that of Fig. 1, both types of equivalent circuits pro-duce the same results[8]. These equivalent circuits arecommonly used to analyze induction motor drives. Byusing these circuits, we can neglect the transients in theleakage inductance without neglecting transient in th e ro-

tor.

D = xasxM-X& (40)Xsa = X ~ + X M (41)

The s tato r flux linkages can be sepa rated into the leakageand mutual components, that is,

* q a = * q r + * q i u (42)*da = *d l + *dM (43)

If the changes of leakage components of sta tor flux linka-ges are neglected, eqns. (31)-(32) can be rewritten as

v q a - i*qa - w e u q d a + a:*qM (44)

P

-@qM =Wb

-*dM = z]da - {*ds

+Weuqqa

+a;*dM (45)

W b

Based on eqns. (33)-(34) and (44)-(45), the stator fluxlinkages can be obtained algebraically, tha t is,

where

T h e above results ca n be used to solve eqns. (33)-(35).

Dm = (ai + +~3 (48)

653




5. Simulated Results

In th is section, simulated results on a small induction mo-tor (10-hp, 220 V, 6-pole) will be discussed. In per unit,the data of the motor are r , = 0.0453, T , = 0.0222, X s ,= 2.119, X,, = 2.074, X, = 2.042, an d H = 0.45.

Fig. 3 shows the stat or rms current, torque, and speedduring start-up. The motor is started with a oad havinga torque charac teristic which is proportional to the squareof speed. The torque a t full speed is one per unit. Thenew reduced order model can produce almost the sameresults as that of the full order model, especially on thetorque response.

Fig. 4 hows the stator rms current, torque, and speedduring short-cix uit. The motor initially running full-loadand a three-phase bolted short-circuit is suddenly appliedand after six cycles, the short-circuit is cleared and a fullvoltage is reapplied. Once again , a significant imp rov ement over the standard reduced order model can be ap-preciated from this figure.

6. Coiiclusion

In this paper, a simple but useful method to reduce theorder of the model of induction motors has been presen-ted. It has been shown by the simulated results that theproposed new model gives a significant improvement overthe standard reduced order model.' Small signal analysisand extensions of the proposed method to othe r types ofac machines are currently under investigation.

References

[l ] K. P. R. Sastry and R. E. Burridge, Investigation of AReduced Order Modelfor Induction Machine DynamicStudies, IEEE Trans. Power App. Sys., Vol. PAS-95,No. 3, pp- 962-969, May/June 1976.

[2] P. C. Krause, F. Nosari, T. L. Skvarenina, and D .W. Olive, The Theory of Neglecting Stator Transients,idem, Vol. PAS-98, No. 1, pp. 141-147, Jan./Feb. 1979.

[3] T. L. Skvarenina and P. C. Krause, Accuracy of AReduced Order Model of Induction Machines in Dy-namic Stability Studies, idem, No. 4, p. 1192-1197,J uly/August 1979.

[4] N . Gunaratman and D. W. Navotny, T h e E of

Neglecting Stator Transients in Induction MachineModeling, idem, Vol. PAS-99, No. 6, pp. 2050-2059,Nov./Dec. 1980.

[5] G. G. Richards and 0. T. Tan, Simplified Modelsfo r Induction Machine Transients under Balanced andUnbalanced Condition s, IEEE Trans. Ind. Appl., Vol.IA-17, No. 1, pp. 15-21, Jan. /Feb. 1981.

I

100 200

Angular time [rad]

( 4

I ~ " " " " " " '0 100 200

1 -

Angular time [rad]

(b)

FM

Angular_ time [rad]( 4

Fig. 3. The simulated results of start-up of induction motor.(a) Current. (b ) Torque. (c ) Speed.

654




1

Angular time [ lo rad/div]

(4

Angular time [ lo rad/div]

(b)

I C

0.61

FMSRM

NRM

----Angular time Cl0 rad/div]

(4

L . . . . ' . . . . . . ' .

Fig. 4. The simulated results of short-circuit of induction mo-tor. (a) Cdrrent. (b ) Torque. (c) Speed.

[6] F. D. odriguez and 0 . Wasynczuk, Refined Methodof Deriving Reduced Or der Models of Induction Ma-chines, IEEE Trans. Energy Conv., Vol. EC-2, o. ,pp. 31-37, M u c h 1987.

[7] S. Ertem and Y. Baghzouz, Simulation of InductionMachinery for Power System Studies, idem, Vol. E G4, No. 1, pp. 88-94, Mar& 1988.

181 G . R. Slemon, Circuit Models fo r Polyphase InductionMachines, Electric Machines and Power Systems, Vol.8, pp. 369-379,1983.

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a new reduced order model of im

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