A Unified Method for Modeling and Simulation of Three Phase Induction Motor Drives

8/19/2019 A Unified Method for Modeling and Simulation of Three Phase Induction Motor Drives

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A UNIFIED METHOD FOR MODELING AND SIMU LATION

OF THREE PHASE INDUCTION MOTOR

DRIVES

H. Maqbahi,A. Ba-razzouk,J Xu, A. Cheriti and V. Rajagopalan

Chaire de recherche industrielle Hydro-QuCbecNQTR

Dtpartement de genie Clectrique et gtnie informatique

Universitk du QuCbec

2

Trois-Rivi res

C.P. 500, Trois-Rivikres,PQ, G9A 5H7, CANADA

E-mail: [email protected]

ABSTRACT

: This paper presents a new approach

for modeling and simulating the dynamic behavior of

a large, interconnected, nonlinear, time-varying

physical system consisting of a three-phase induction

motor drive. An example of a vector-controlled

induction motor shows that the proposed method is

very accurate and very fast. The proposed simulation

method and algorithm are expected to overcome

some of the disadvantages of the existing methods.

Simulation results are presented to illustrate the

capabilities and flexibility of the simulation

technique. A comparative evaluation with PSIM

results is also provided to confirm the simplicity and

the accuracy of the proposed unified method.

1.

INTRODUCTION

The design and performance analysis of many

power systems require computer simulation since

prototype testing has become almost impossible due

to high costs. If the circuit incorporates a number of

semi-conductor switches, machines and feedback

control, computer simulation is the only viable

solution for testing a design.

In

the literature

[l],

numerous digital computer simulation studies

of

induction motors fed from power electronic

converters have been reported. Majority of these

studies neglect the power electronic topology, and the

converter is modeled as an ideal source

wt

zero

impedance.

Current research is directed towards developing

simulation methods, which can efficiently and

accurately describe the behavior of the circuit without

complexity in problem formulation or long

computation. An alternative way for formulating the

dynamic drive analysis was suggested [2], using: first

a

simplified dq transformation

so

that the stator

variables are in the actual abc phase quantities,

second the semiconductor switches are modeled as

binary inductors, a low value during the conduction,

and infinite otherwise. Additionally, this analysis has

been based on a state space formulation of the circuit

equations. This, in it self, poses several difficulties,

when considering the development of a general-

purpose software

[3].

Despite the fact that a variety

of state-space based formulations have been

described in the literature, it is not particularly well

suited for implementation within a general simulation

package.

An alternative way for formulating the dynamic

drive analysis is suggested [4] using Transmission-

Line Modeling (TLM) technique. The induction

motor is presented in dq frame. Furthermore, the

authors have decoupled the system by introducing

small inductors between the inverter and the motor

section. The disadvantage of this approach is the

introduction of additional elements, i.e., the number

of nodes and branches increases. Furthermore, the

link of transmission line must be large in order to

reduce the errors when performing the decoupling of

the system [ 5 ] . However, from a simulation and

control perspective, only the actual phase variables

are needed. Consequently, in control schemes and in

simulating real power networks we are forced to

transform back current and voltage variables to the

phase reference frame.

The main goal of this paper is to present a new

dynamic model for three-phase induction motors

using a new unified method. This method is based on

the discrete-time machine model. In our approach,

we will directly use the phase variables to simulate

electrical machine dynamics. The solution presented

in this paper

will

be used for implementation on a

parallel computer and will be given in a next paper.

This paper

will

demonstrate that it is feasible to

simulate a vector-controlled induction motor drive

with the proposed method used along with the

commercially available software: MATLAB-

SIMULINK [6].

2.

BASICS

OF

DISCRETE-TIME MODELING

Before considering the simultaneous solution of

the whole network as presented in next section, the

basic numerical equations for individual power

systems components shall be first reviewed. From

the numerical integration point of view, the

differential equation characterizing a capacitor or

inductor can be approximated by a resistive circuit in

series or in parallel with

the

integration algorithm

[7].

This formulation can be called Discrete Time

Modeling @TM)

.

0-7803-5957-7/00/ 10.00000 EEE

345



(b) (d)

(0

Fig.1: Discrete time circuit model (b,d,f),

For an inductance, the voltage-current relationship

v L L

dt

can be discretized by using the trapezoidal rule :

of linear lumped elements (a,c,e)

di ' (1)

at each time step, expression 2 can be regarded as a

conductance

in

parallel with a history current source

(Figure 1). Similarly, the voltage-current relationship

of a capacitor can be derived. The table of Appendix

1, Table summarizes the expressions used for

three

different solution algorithms: Backward Euler,

Trapezoidal and Second order Gear algorithm

171.

The nodal equation can be written as :

A x = b (3)

where A is the system matrix,

xrepresents the

unknown

vector in discrete-time modeling and

includes all nodal voltages and branch currents.

Vector b is a function of network sources, including

current sources associated with discrete time model

of switches and energy-storage components, as well

as

netvat

. ;x

independent current and voltage sources.

3.

MODELING

OF INDUCTION

MOTORS

The equations of three-phase symmetrical

induction motor in abc reference frame

are

given in

[SI. The model used to represent the machine

assumes sinusoidal distribution of the m.m.f, linear

magnetic circuit and the absence of skin effect.

Moreover, magnetic frames of the stator and the rotor

are both taken cylindrical, smooth and separated by a

constant

air

gap.

In order to obtain a dynamic model as accurate as

possible, we have to work in the (a, b, c reference

frame) without any dq transformation. This

representation takes into account physical parameters

of each of the stator and the rotor phase windings.

The electrical stator equations of the

written as follows:

machine can be

where the stator and rotor resistances are:

5 )

The rotor quantities have siniilar expressions. Also,

the mutual coupling matrices are:

cos @, ) cos @,+@

cos @,- @

cos @, @

cos @, ) cos @,+ @

cos @,+

cos @,

-a)

c o s @ , )

Where 4 =

2x13

7)

The electromagnetic torque developed by the motor

is given by equation 5 , as follows:

and the mechanical equation take the following form:

dw

dt

T,

- T ,

= J i +

w

9)

Where J is the moment of inertia, f, is the viscous

friction, TL is the load torque, T,,

is

the

electromagnetic torque and a, represents the rotor

velocity. Despite these simplifications, a computer

simulation of the system still remains complex.

So,

the

goal of our work is to find a new model easier to

implement and more accurate to model the transient

and the steady-state operation of the motor.

The stator phase (a) equation is given by equation

lo),

as

follows:

di 2 d

dt 3 dt

v

= R , i ,

+ L , - + - L L , - J J ,

where

J ,

=

(imcos@,+ cos @, +a)

i

If we assume a non-linear current source J a as

follows:

3 L,

-a))

J

=Z L i,

cos

6, + i cos 8, +D im cos @, -

(1

1)

and a new variable:

we can deduce the following differential equation:

v

=

R ,

i ,

+

M,

where M,

1.4,

3 Ls

The new state formulation becomes:

=

+

J s n

12)

dt

346



where

-

2

Lm

3

L,

The time-varying current sources are as follow:

.T,~ = i

cos @,) , cos @,+

@

+

,

COS @,

-@

J ,

=

, cos @,) , cos @,

+ @

+ cos @,

-@

J ,

= , cos @, ) , cos @,+ @ + cos @,-0

J , =

i

cos @,)

,c cos @,+ @ +

i b

COS @,

-

@

J ,

= i

cos @,

+

,

cos @,

+

@

+

,=

cos @,

-

@

J ,

=

,<c o s @ , ) ,

+

@

+

cos @,

-

@

The electromagnetic orque is given by:

(16)

i m i m

+ i

+ imiSc)sinOr

i ~ , ~i , b i s c +i,i,a)sin(Q,

-a)+

i,i,c

+ , i , +

,i,b)sin @,

+

0

Equations (11) to 15) lead to a state-space system

easy to implement and to compute. This model is

illustrated in Figure

2

Fig.2:

Induction motor model

This model can be transformed using DTM

technique. Contrary to the conventional state-space

approach, the DTM technique derives a discrete

model directly from the physical system to be

simulated without setting up sets of differential

equations. The DTM algorithm provides an exact

solution to the discrete model. Reference [7],

presents the principles of a discrete time modeling.

The DTM model of the motor was derived by

replacing all self inductances by their DTM

equivalents and the current sources of the stator

(controlled

by

rotor currents) and the rotor

(controlled bv stator currents).

0

Fig.3:

DTM induction motor model

4. VALIDATION OF MOTOR MODEL

In order to illustrate the validity of the proposed

DTM motor model, the direct on-line motor starting

was simulated.. The computer simulation was carried

out using the Backward-Euler algorithm for a 2 kW

induction motor with two pairs of poles and fed by a

balanced three-phase sinusoidal source. Figures (4.a,

4.b, and 4.c) shows DTM simulations results for rotor

speed, stator current and electromagnetic torque

during a no-load start-up and steady state,

respectively.

These plots were obtained from a

digital computer simulation using this unified

method.

. . . . . . . . . .

'

0.3 0 . 2

0.3 0.4

0 . 5 0 . 6

T l m a I

.e5

Fig.4.a:

Rotor speed using DTM

1 0 0

I

I

0 0.1 0.2 0.3 0 .4

0.5 0.6

Tlin.

I

s e

- 8 0 I

Fig.4.b:

Stator current using DTM

I

0 0 1

0 2

0 3

0 4

0 5 0 6

Tlm. I

* e 5

-20

Fig.4.c:

Electromagnetic torque using DTM

347



Fig.5.a: Rotor speed using PSIM

Fig.5.b: Stator current using PSIM

~-

1 -

7Z - _

Fig.5.c: Electromagnetic torque using PSIM

Figures 5.a, b and c show the commercial package

PSIM [9] numerical results for the same induction

motor. The no-load motor start-up is simulated

during 600 ms. Comparing the above figures, good

agreement between PSIM results and DTM results is

observed. However, some remarks must be taken

into consideration. The model of induction motor

implemented

in

a PSIM is based on a dq model

whereas the model presented is an in-phase (a,b,c)

model. The results are both for a no-loaded motor

fed by a balanced three-phase voltage source.

5. SIMULATION EXAMPLES

In the following example, the filed-oriented

control of induction motor [lo] shown in Fig6 is

used to demonstrate the efficiency of the proposed

simulation methodology. The effects of

the

drive

control dynamics

are

also included in the simulation.

This drive is implemented with vector control circuit

in order to compare the actual stator currents with the

generated three reference currents. This control

scheme performs a control of flux and torque using

vector current control. In addition, the inverter

supply is included in the simulation formulation.

This supply is composed by a three phase diode

rectifier and a dc-link filter LC (inductor and

capacitor) forming a dc voltage source. This filter is

used to filter out the harmonics in the rectifier output

voltage and also to provide a smooth current. The

machine model is used to interface directly with the

inverter, which consists of six switches. Simulation

was carried out by using the parameters given in

reference [lo] A more detailed information about

this control scheme is presented in reference [lo].

C

e phas c w m AC-Dc mnvwion DC-ACconversion

In ction

m m

Fig.6: Induction motor drive

Fig.7:

Variation

of

flux

Fig.8:

Variation

of torque

ref. From

0.55

to 0 4Wb

ref. From 5 to 10 N.m

The flux and electromagnetic torque waveforms are

shown in figures 7 and 8 respectively. The transient

effects are also

shown

in simulation traces. It was

found that the simulation results obtained from the

present method are matching well with the simulation

results of reference [lo].

MATLAB-SIMULINK IMPLEMENTATION

There is a great number of universities and

researchers who use

the

MATLAEVSIMULINK

software

[6]

in the filed of electrical machines. Some

of its advantages are: a user friendly visual oriented

programming concept, a variety of given line: tnd

non-linear standard blocks, a large numbc- of

toolboxes for special applications. The new model of

induction motor drive was coded s a m-file but in

future

it

will

be coded,

as

a mex-file to accelerate the

computing time. The methodology is given in Fig 9.

The S-function will be masked to produce a custom

block with user-definable parameters for induction

motor, inverter, rectifier, input voltage and controls.

348



MATLAB

SIMULINK

Algorithm Capacitor Inductor

ik+l

=

yc

Vk+l

-

c

Vk

4 + l

=

YL vk+l+

4

ackward Euler

Trapezoidal

i k + l = K v k + l - y C v k + i k )

ik+l=yLvk+l+( vk+ik)

S~cond-orderGear

i,,, =

Y,.V,+,

- --.Vk --.Vk-,)

i

=Y‘.vk+l -.ik - i k & l )

c c 4 1

h 2 h

3 3

/Computation

of

inverse SMNA-matrix1 j

Impedance

Y“’;;

C

Y L = i

y ,

= =; =

h 2 L

3c 2 h

Y, = -; Y =

2 h

‘

3 L

Switch test for hlockin

<x

Yes

Updating switches,

voltage and current

Update time

w

= ?

. - - - - - -- - -

.._-_

.... --s _.___- - - - ----- -

Fig. 9

:

Schematic of the computational procedure

CONCLUSION

The new dynamic model of induction motors

presented in this paper has shown accurate simulation

results. The simulation of the behavior of an

induction machine has been canied-out with the help

of MATLAB-SIMULINK. For comparison, the

commercial package PSIM was used. Results show a

good agreement and similar simulation time.

Furthermore, the solution presented here can be

implemented concurrently in the parallel computing

systems and

will

be presented in a next paper.

ACKNOWLEDGMENT

This work is supported by funds from Hydro-

QuebecNQTR industrial Research Chair at the

University of Quebec at Trois-Rivihes.

[31

[41

[61

[71

r.81

[91

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&

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IEEE Transactions On Power Electronics

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349

A Unified Method for Modeling and Simulation of Three Phase Induction Motor Drives

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