A New Reduced Order Model Of Im

A New Reduced Order Model of Induction Motors Pekik Argo Dahono

Qamaruzzaman

and

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 proposed in this paper. Instead of neglecting the stator transients as in the case of standard reduced order model, the new model is derived by neglecting the transients in the leakage inductances. It is found that the new model produces better results than the standard reduced order model and, in some cases, the results are almost the same as that of the full order model.

1.

Introduction

A significant proportion of power system load is made of induction motors and, therefore, accurate representation of it is important in power system dynamic analysis. Although the full order model (i.e. fifth order) of induction motors has been considered to be the most accurate model, the required computation time is very large if we dealt with a multimachine system. In order to reduce the computation time, the order of the model should be reduced. The standard method to reduce the order of induction motor model is by neglecting the electrical transients in the stator[l-51. Since the transients in transmission lines are usually neglected in power system analysis programs, reduction of induction motor model by neglecting the electrical transients in the stator is very useful. The fact that the stator variables changed much faster than the rotor variables is the basic argument of neglecting the electrical transients in the stator. In the computation of the rotor variables, therefore, the changes of the stator variables can be neglected. In the reality, however, the rotor variables begin to change before the stator transients completely subside. Thus, the predicted induction motor behaviors will be different from that of the full order model. Several methods to improve the standard reduced order model were poposed in the literature[6-7]. Unfortunately, these methods are either complicated or need a switching from the full order into reduced order models. In this paper, a new reduced order model of induction motors is proposed. The new model is derived by neglecting the transients in the leakage inductances. With

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the transients in the leakage inductances are neglected, the mathematical model of the induction motor can be formulated in terms of either the magnetizing currents or mutual flux linkages. In most cases, the transients in the leakage inductances can be neglected because &heleakage inductances are much smaller than the magnetizing inductance, In this paper, the proposed model is used to predict the behavior of induction motors during transient conditions (e.g., start-up, short-circuit, and sudden changes of load). The accuracy of the proposed model is compared to the standard reduced order model and the full order model. The influence of the motor size on the acuracy of the proposed model is investigated. In some cases, it is found that the proposed model can give accurate results as accurate as the full order model.

Full Order Model

2.

The full order model of induction machines in synchronous reference frame can be written in either of currents or flux linkages as state variables. This model can be visualized by the T-form equivalent circuits as shown in Fig. 1. In per unit form and by using flux linkages as state variables, the full order model can be written as follows:

P

=

+ aZ*qr vds - a 1 q d s + weuqqs + aZ*dr

(2)

=

a3*qs

- a4*qr - Walu*dr

(3)

=

a 3 9 d s +wsiu*gr

-*qa

=

P -*ds P

wb

wb

-*qr

wb

P

-*dr wb

vqs

- a l * q a - Wcu*ds

- aeqdr

where

65 1

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(1)

(4)

+v,sin

where 6=

(B + 3 1

I,'

wedt

+ e(0)

(18)

Eqns. (1)-(5) can be solved numerically to obtain the transient behaviors of induction motors. After the flux linkages have been solved, the direct and quadrature components of the stator currents can be obtained as

=

i,,

1

5( X r r q q S - Xm*,r)

(19)

The results can be used to determine the phase currents,

Fig. 1. T-form equivalent circuits.

ib,

=

i,,

= i,,

i,, cos (wet COS ( u e t

$) +sin (wet -): (22) + $) +sin (wet + $) (23) -

The rms value of the phase currents can also be obtained as Iph = (24)

d-'

3.

Standard Reduced Order Model

The standard method to reduce the order of the model of induction machines is by neglecting the transients in the stator. The basic argument of this method is the transients in the stator are much faster than that of the rotor. By neglecting the transients in the stator, eqns. (1)-(2) become algebraic equations and the stator flux linkages can be calculated as in which r, and r, are the resistances and XI, and Xr, are the leakage reactances of the stator and rotor windings, respectively. X , is the mutual reactance between the stator and rotor windings. All rotor quantities are referred to the stator side. wb, w e , and w, are the base, supply, and rotor angular frequencies, respectively, and H is the total inertia constant of the motor and the connected load. If the phase voltages are assumed as a balanced set of sinusoidal voltages having the angular frequency of we, then the direct and quadrature components of the stator voltages in synchronous reference frame can be obtained

Qqa

=

alvqs - weuvds

+ a1a2Qqr - a2weuqdr (25) +w;,

a:

The results can be used to solve eqns. (3)-(5). The accuracy of this model was thoroughly investigated in the literature. This model usually gives a good result for small signal disturbances. For large signal disturbances (e.g. start-up and short circuit), however, the results are quite different to that of the full order model.

as

4. +U,

Vds

=

cos

(e

+

3 1

23 [v, sin. 6 -t Vb sin (6 -

F)

The New Reduced Order Model

In the standard reduced order model, the transients in the stator flux linkages have been neglected. These flux linkages consist of flux leakage of the stator winding and

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I

I D = xasxM-X& Xsa = X ~ + X M

Fig. 2. The modified equivalent circuits.

mutual flux linkage. Because the mutual inductance is much larger than the leakage inductance, the changes of flux linkages in the former will be slower than that of in the later. Thus, the transients in the mutual inductance can not be neglected. If we want to reduce the order of the induction motor model without neglecting the transient in the mutual inductance, we have to neglect the transients in the leakage inductances of both stator and rotor windings. We know, however, the rotor quantities are changing slower than the stator and the results, therefore, may not be acceptable. In order to solve this problem, equivalent circuits of induction machines as shown in Fig. 2 will be used to derive the new model. The parameters of these equivalent circuits can be derived from that of the Fig. 1 by using the following relations:

Lr = L1a + d l r LM = ~ L M l& = a%, where a=

Lm Lm +.Llr

(40)

(41)

The stator flux linkages can be separated into the leakage and mutual components, that is, *qa

*da

= =

(42) (43)

*qr+*qiu *dl

+ *dM

If the changes of leakage components of stator flux linkages are neglected, eqns. (31)-(32) can be rewritten as

P

-@qM

=

vqa

- ai*qa

-weuqda

=

z]da

- a{*ds

+

Wb

P

-*dM Wb

Weuqqa

+ +

a:*qM

(44)

a;*dM

(45)

Based on eqns. (33)-(34) and (44)-(45), the stator flux linkages can be obtained algebraically, that is,

(27) (28) (29) (30)

Though these equivalent circuits have less parameters than that of Fig. 1, both types of equivalent circuits produce the same results[8]. These equivalent circuits are commonly used to analyze induction motor drives. By using these circuits, we can neglect the transients in the leakage inductance without neglecting transient in the rotor.

where

+

+ ~3

Dm = (ai (48) The above results can be used to solve eqns. (33)-(35).

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5.

Simulated Results

In this section, simulated results on a small induction motor (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, and H = 0.45. Fig. 3 shows the stator rms current, torque, and speed during start-up. The motor is started with a load having a torque characteristic which is proportional to the square of speed. The torque at full speed is one per unit. The new reduced order model can produce almost the same results as that of the full order model, especially on the torque response. Fig. 4 shows the stator rms current, torque, and speed during short-cixuit. The motor initially running full-load and a three-phase bolted short-circuit is suddenly applied and after six cycles, the short-circuit is cleared and a full voltage is reapplied. Once again, a significant improve ment over the standard reduced order model can be appreciated from this figure.

6.

I

100

200

Angular time [rad] (4

Coiiclusion

In this paper, a simple but useful method to reduce the order of the model of induction motors has been presented. It has been shown by the simulated results that the proposed new model gives a significant improvement over the standard reduced order model.' Small signal analysis and extensions of the proposed method to other types of ac machines are currently under investigation.

I 0~ " " " "100 " " '

References

200

Angular time [rad]

[l] K. P. R. Sastry and R. E. Burridge, Investigation of A Reduced Order Model for Induction Machine Dynamic Studies, IEEE Trans. Power App. Sys., Vol. PAS-95, No. 3, pp- 962-969, May/June 1976.

(b)

FM 1-

[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 A Reduced Order Model of Induction Machines in Dynamic Stability Studies, idem, No. 4,pp. 1192-1197, J uly/August 1979. [4] N. Gunaratman and D. W. Navotny, T h esice'E of Neglecting Stator Transients in Induction Machine Modeling, idem, Vol. PAS-99, No. 6, pp. 2050-2059, Nov./Dec. 1980. [5] G. G. Richards and 0. T. Tan, Simplified Models f o r Induction Machine Transients under Balanced and Unbalanced Conditions, IEEE Trans. Ind. Appl., Vol. IA-17, No. 1, pp. 15-21, Jan./Feb. 1981.

Angular _ _ time [rad] (4 Fig. 3. The simulated results of start-up of induction motor. (a) Current. (b) Torque. (c) Speed.

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[6] F. D.Rodriguez and 0. Wasynczuk,A Refined Method of Deriving Reduced Order Models of Induction Machines, IEEE Trans. Energy Conv., Vol. EC-2,No. l, pp. 31-37, M u c h 1987. [7] S. Ertem and Y. Baghzouz, Simulation of Induction Machinery for Power System Studies, idem, Vol. E G 4, No. 1, pp. 88-94, Mar& 1988.

1

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

Angular time [lo rad/div]

(4

Angular time [lo rad/div] (b) I C

---

-

0.61

FM SRM NRM

L....'......'. Angular time Cl0 rad/div]

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

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