Parameter identification of induction motor model using genetic algorithms

F.Alonge F. D’lppolito G.Ferrante F.M. Raimondi

Indexing term: Induction motors, Genetic algorithms

Abstract: The paper deals with methods of identification of the parameters of an induction motor model using genetic algorithms. It is supposed that the inverter supplying the motor is directly accessible for control of the conduction sequences of its power switches. This makes it possible to carry out a test consisting of a transient from standstill to steady-state operation at a given frequency and successive free motion to standstill. During this test, data are acquired referring to stator voltages, and currents and speed. Then, a genetic algorithm is employed with the aim of determining the mechanical and electrical parameters of the model, so as to reproduce the input-output behaviour of a real open-loop system.

List of symbols

R,(R,.) = stator (rotor) resistance, Q L,(Lr) = stator (rotor) inductance, H Lm = mutual inductance, H

1 Introduction

As is well known, the first issue arising when model- based synthesis of a controller has to be performed for motion control systems with an induction motor is to determine the mathematical model of the motor and its parameters. Obviously, such models are also very use- ful for designers of electrical machines and for simula- tion purposes.

With regard to the model, for control purposes, it is sufficient to consider the approximate mathematical description obtained, neglecting anisotropy of the magnetic structure, iron losses and saturation. With reference to determination of the parameters, the oldest method is that based on the traditional no-load and locked rotor tests, but the resulting values are inaccurate and, consequently, not suitable for the 0 IEE, 1998 IEE Proceedings online no. 19982408 Paper first received 21st November 1997 and in revised form 18th June 1998 The authors are with the Instituto di Automatica e Sistemistica, Engheer- ing Faculty, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

IEE Proc-Control Theory Appl.. Vol. 145, No. 6, November 1998

synthesis of high dynamic performance systems [l]. Recently, some methods of parameter identification

have appeared in the literature. Some of them use clas- sical statistical approaches [2-51 or model reference adaptive techniques [I] to estimate the electromagnetic parameters of the mathematical model of the motor, forcing the motor to operate at standstill; in this case, the model is linear and time invariant. To carry out the experimental test, it is necessary conveniently to supply the motor so as to generate a zero driving torque and currents of small values, because at standstill there is no ventilation of the motor windings. However, more importantly, mechanical parameters cannot be identi- fied.

Other methods require data acquired during normal operation of the motor using suitable signals to excite the dynamics of the system. In [6], a method is described based on both local linearisation of the model at various steady-state operating points and, for each point, the approximate decoupling of the model into three models whose parameters are, finally, identi- fied separately. In [7] , using suitable digital filters, line- arly independent signals are generated, and a relationship between them is determined whose coeffi- cients are the same as those of the non-linear model of the motor. Then, a total least-squares algorithm is used to identify these coefficients, from which the physical electromagnetic parameters are computed.

In this paper, a new method is described for off-line identification of the electromagnetic and mechanical parameters of a mathematical model of an induction motor, including dynamics not usually modelled, such as Coulomb’s friction. Identification is performed using data acquired during a test consisting of a transient from standstill to a certain speed for a certain period of time and successive free motion to standstill. These data treat stator voltages as input variables and stator currents and speed as output variables.

The method in question is based on the idea of deter- mining the unknown electromagnetic and mechanical parameters of a model of the motor, so that speed and stator currents, computed by means of implementation of this model on a PC, match those generated by the motor supplied by the same input voltages. Obviously, to employ the motor as a variable speed actuator, it is necessary to supply it by means of an inverter, and, consequently, for identification it is convenient to acquire data from the inverter-motor system. In this case, it is necessary to use identification methods that do not require computation of derivatives of voltages

587

or currents, such as least squares-based methods, To attain this objective, a genetic algorithm (GA) is imple- mented to minimise a cost function consisting of a weighted sum either of square or absolute differences of experimental and computed output variables consid- ered at the same instant of time.

( a ) a mathematical model having the desired complex- ity can be considered without difficulty, including non- linear terms such as Coulomb’s friction (b) both electromagnetic and mechanical parameters can be simultaneously identified (c) computation of derivatives is not necessary, which allows identification to be performed using the con ven- tional supply of the motor obtained by means 01- an inverter.

The advantages of the method are

2 Proposed identification method

The idea on which the method is based is that of deter- mining simultaneously all the parameters of a mathe- matical model of the motor with the desired structure, so that the model is able to match the input-oui.put behaviour of the motor. This can be realised using the following approach: Step 1: An experimental test is carried out consisting of a transient from standstill to steady-state operation, at a desired supply frequency, and successive free mo:ion to standstill, producing three stator voltages, two stator currents and speed. Step 2: A mathematical model with the desired struc- ture is implemented with the aim of simulation of the experimental test of step 1. Step 3: A cost function is computed as a weighted sum either of square or absolute differences of the output variables acquired experimentally and those computed by simulation at the same instants. Step 4: The unknown parameters of the motor are iter- atively updated so as to minimise the above cost fmc- tion. Obviously, steps 3 and 4 have to be performed using suitable algorithms. Fortunately, today, powerful numerical combinatory optimisation algorithms exist and powerful PCs for their implementation. The algo- rithm we use in this paper is a genetic algorithm (GA).

2.1 Mathematical model of the motor The mathematical model of the motor. referred to ab-

where

588

Re = R, + (L$/Lz)Rr Le = L, - L”,LT a11 = Re/Le a33 Rr/Lr c1 = 1/Le

~2 L,/L,L, a13 = ~ 3 3 ~ 2 ~ l 3 l = RTLm/Lr

7, = J / , f u bm = P / J i,, (&,) and i,, are the components of the stator cur- rent (rotor flux); w is the electrical angular speed; t,n, t , and t,. are motor torque, load torque and Coulomb’s friction torque, respectively; J , f, and p are the inertia, friction coefficient and pole pairs.

As is well known, to obtain components along ab- axes po and PI,, from phase quantities pR, ps and pT, which can represent currents, fluxes or voltages, the following equations have to be used:

Pa = P R - (Pus + P d / 2

P b = fi /2(PS - LLT) ( 8 )

2.2 Experimental test With reference to the experimental test, we assume that the conduction sequences of the power switches of the inverter can be easily imposed. In this case, it is possi- ble to design and carry out the desired experimental test. Otherwise, a test can be carried out supplying the motor with three-phase line sinusoidal voltages.

- Fig. 1 Busic scheme of inverter-motor system

Fig. 2 Vectorid representation of supply voltages

As is well known, the inverter supplying the motor can be schematically represented as in Fig. 1, where Ti (i, ..., 6) denotes the ith power switch. The inverter is able to produce supply voltages that, in the ab-plane fixed with the stator, can be schematised by means of eight vectors, as illustrated in Fig. 2. Six of these vec- tors have amplitude and phase equal to (3/2)Ed and (E/ 3)i ( i = 0, ..., 5) , respectively, whereas the other two vectors are null vectors corresponding to the two short- circuit configurations, i.e. T, , T,, T3 ON and T,, T,, T6 ON. A tern of bits, C X R C X ~ C X T , is associated with each vector and defines the conduction status of the switches of phases R, S and T, respectively. The value of a generic bit is 1 or 0, depending on whether the corre- sponding phase is connected to the negative or positive edge of Ed. Consequently, the two short-circuit configu- rations are 000 and 1 11.

Now, the phase voltages supplying the motor can either be computed or acquired experimentally. In the

IEE Pvoc -Control Theory 4 p p l . Vol. 145, No. 6, Noveruber 1998

first case, terns C ( ~ C ( , ~ C ( ~ , describing the conduction con- figurations of the inverter, have to be memorised, and the stator voltages can be computed by means of the following equations:

UR = ( E d / 3 ) ( 2 ~ ~ - ais - C X T )

VS = (Ed/3)(-aR + 2Q.5' - Q T )

U T = (Ed/3)(-aR - QS + 2QT) (9) requiring only the measurement of Ed. This choice can be considered a good alternative, because it causes a slight modification in the estimated parameters of the model. However, in this paper, voltage measurements are also performed.

Let us choose a sampling period T,, asynchronous speed w, = 601(6NcTg), where N, is an integer, and a sequence of supply voltages such as, for example, { 100, 110, 010, 01 1, 001, lOl}. Then, a suitable experimental test is performed choosing a supply method to the motor that consists in forcing the inverter (a) to maintain each conduction configuration of the sequence for a time interval equal to N,T, (6) to give N sequences of supply voltages for a time interval T I = 6N,T,N (c) to give a null voltage vector, for example that corre- sponding to 11 1, after T I .

2.3 Identification of parameters b y means of GAS As is well known (see, for example, [SI), GAS work according to the mechanism of natural selection, in which only stronger individuals survive, and so species evolve. In practical applications, each individual is cod- ified into a chromosome consisting of genes, each rep- resenting a characteristic of the individual itself, and the species evolution is evaluated by assigning a posi- tive number to each chromosome, named fitness value, representing its degree of goodness.

For parameter identification of a model whose struc- ture is known, the individual is the set of unknown parameters, and, consequently, each gene coincides with a parameter.

Now, the problem is to define the genes. With regard to this, we note that we have eight unknown physical parameters, five electromagnetic and three mechanical, but these parameters can be reduced to seven by assuming L, = LA. Concerning this, as the rotor flux vector is unknown, the electromagnetic parameters of the considered model cannot all be identified. In fact, as is easy to verify, any set of electromagnetic parame- ters with identical values of

- - " ", R,, L , R,' L,

produces the same input-output behaviour (see, for example, [9]). The same situation occurs if we use steady-state current against speed and torque against speed characteristics [lo]. For simulation and control, it is usual to choose L, = L, (see, for example, [7, 91). It is preferable not to constrain L,, because the model con- sidered is able to describe the operation of the motor during saturation, in an approximate manner, assum- ing L, as a time-varying parameter [l 11.

Now, nine coefficients appear in the model of the motor, and, consequently, two relationships exist between them. The examination of the mathematical model shows that the dynamics of the stator currents

depends on a,, and, consequently, on the equivalent inductance L,. Therefore it is convenient to assume L, as a gene, assuming the mutual inductance L,, to be a dependent quantity that can be computed by the equation

L, = J L J L , - L,) (10) Regarding the mechanical parameters, it is convenient, first of all, to verify the influence of Coulomb's fric- tion. This can be done by considering the stop tran- sient, which consists of two parts: the first part, of very short duration, depends on electric transients owing to the short-circuit configuration forced in the inverter; the second part, of much longer duration, depends on the mechanical transient and consists of free motion. It is convenient to restrict our attention to the second part of the above transient. To this end, there is taken as the time origin an instant in which free mechanical motion has definitely taken place, and the correspond- ing speed 00 is taken as the initial speed. The following two cases can be considered:

Case 1: Coulomb's friction is neglected. Examination of eqn. 5 shows that the mechanical

transient can be described as follows:

w ( t ) = exp(-t/.,)wo (11) provided that load torque is zero. To compute z i71 , it is sufficient to determine the instant in which the speed value is equal to (wo/e).

With regard to the choice of genes, two different approaches can be followed, depending on whether the inertia coefficient is known (several manufacturers give the value of the inertia) or not. If the inertia is known, the viscous friction coefficient can be computed as f , , = J/z,,, and the genes total four. if the inertia is unknown, it is convenient to include J in the genes, which total five, computingf,. as above.

Case 2: Coulomb's friction is taken into account. In this case, the mechanical time constant cannot be

determined as before, because the mechanical free motion depends also on Coulomb's friction, which acts as a forcing term. The free motion is described by eqn. 5 , in which t,, = t , = 0, whose solution over time t is

The identification procedure can be performed in two successive steps. In the first, z, and V;/J) are obtained, so that eqn. 5, with t , = t , = 0, matches the mechani- cal free motion using a least-squares approach. Then, the parameters are obtained, so that the model is able to match the transient from standstill to steady state, solving a genetic optimisation problem in which chro- mosomes have four or five genes, depending on whether J is known or not, respectively. In both cases, parameters j;, and f, are computed from z,, J andf,lJ.

Now, the genetic optimisation problem consists in minimising a cost function. With regard to this, two functions are considered, with the following structure:

IEE Proc.-Control Theory Appl . , Vol. 145, No. 6, Noveinher 1998 589

or

where iUsj ibAj and wJj are data acquired experiment;illy, iaj, ibj and wj are data computed using the mathematical model, and k,, kh and k , are weights. The fitness can be computed as the inverse of function eqn. 13.

I sync. 1 :' r-1 I I A/D PCL 1800

- 1 I I I I I

Fig. 3 Basic scheme of experimental equ@ment

3 Practical application

3. I Data acquisition The identification method described in the preceding Section is applied to a 1 kW two-pole induction motor supplied by devices consisting of an AC/DC converter, a capacitive filter and an IGBT-based inverter. The block scheme of the experimental equipment is given in Fig. 3. The 'sync' block generates a 12kHz synchroni- sation signal that acts on a digital input from an ADVANTECH A/D converter and constitutes the time base for external PC operations. More precisely, only when the above signal commutates from low to high level does the PC generate command signals for the inverter at a frequency of (50 x 6) Hz and acquires :sta- tor voltages, stator currents and speed.

Voltage signals in the range [-lo V, +1OV] are acquired by means of three voltage dividers (PI, P2 and P3), realised with resistors having accuracy equal to 0.1%, which create an artificial neutral point and, con- sequently, are proportional to phase stator voltages. Two stator currents are acquired by means of two Hall transducers that generate two voltage signals in the range [-5V, +5V]. The DC tachometer T generates a voltage proportional to the speed that, by means of a calibrated voltage divider (P4), is converted into a :sig- nal in the range [-IOV, +lOV] and then acquired.

400 T

-400 1 Fig. 4 axis

Waveform acquired during starting: voltage component along a-

and currents i, and ib are computed using eqn. 8. Then, an input file can be generated containing pairs of data relative to v, and vb, the number of these pairs and the sampling period. Moreover, two output files are gener- ated: the first contains data acquired during the start- ing transient, and the second contains data relative to the stop transient. Figs. 4-9 show the acquired wave- forms.

200

8

-200

Fig. 5 Waveform acquired during starting: voltage component along b- ax1s

T

Fig. 6 Waveform acquired during starting: current component along a- axis

T

Fig. 7 axis

Waveforni acquired during starting: current component along b-

T

The values of voltages, currents and speed are cam- puted via software using the gains of the relative trans- ducers and voltage dividers. Then, voltages v, and v b

590

0.2 0.4 0.6 0.8

t, s -50 - l O

Fig. 8 Waveform acquired during starting: angular speed

IEE Proc.-Control Theory AppL. Vol. 145, No. 6. November 1998

Note that anti-aliasing filters are not used for data acquisition, because negligible components of the sig- nals involved are expected to be beyond the Nyquist frequency.

300 t

loo I 0

0 5 10 15 20

t, s Fig. 9 Speed acquired during stop transient

3.2 Data conditioning Examination of Figs. 8 and 9 shows that both wave- forms contain quantisation noise. Moreover, the first part of the stop transient corresponding to electrical transients is not recognisable. To carry out the proce- dure of parameter identification, it is convenient to remove noise from the speed data and eliminate the first part of the stop transient. The last objective is achieved by cancelling data corresponding to the first 0.2s. The convenience of noise removal is due to the fact that the model to be identified produces noise-free waveforms; in particular, the true speed is not affected by quantisation noise. Consequently, it is convenient to cancel the effects of noisy speed data from the cost function of eqn. 10. Moreover, in view of the applica- tion of least-squares approaches for estimation of z, and (@J) (see case 2 of Section 2), it is necessary to cancel quantisation noise from the stop transient speed to obtain good results.

To avoid amplitude and phase distortion and delays, it is necessary to use a filter having a frequency response with a sufficiently flat amplitude and phase equal to zero in the pass-band, and sharp attenuation in the stop-band. A filter satisfying these requirements is the digital anticausal filter. This filter has a transfer function given by

w(ej"Tc) = IF(e3wTc)l2

where F(dwT<) is the transfer function of a generic filter of order n. The anticausal filter is of order 2n. In this paper, we use an anticausal filter of order 10, based on a Chebyshev low-pass filter of fifth order, with ripple and bandwidth equal to 0.5dB and 50Hz, respectively. Figs. 10-13 show the filtered speed data and the noise superimposed on them, computed as the difference between acquired and filtered data.

Now, the data useful for identification by means of the GA are at our disposal. These data are

two input vectors whose elements are the components of the stator voltages along the ab-axes and are dis- played in Figs. 4 and 5

three output vectors containing the stator current components along the ab-axes (see Figs. 6 and 7) and the filtered speed displayed in Fig. 10

a vector containing the filtered speed data corre- sponding to the free motion of the stop transient (see Fig. 12).

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7 300 f - 2001 / m 3

I I 0 0.2 0.4 0.6 0.8

Filtered speed corresponding to datu acquired from starting up t, s

Fig. 10 to steady state

8I I

4 r

v? U

- !O (U

3

-4

-8 Fig. 11 up to steady state

Quantisation noise superimposed on data acquiredfrom starting

300 T

0 5 10 15 20 t, s

Fig. 12 motion of stop transient

Filtered speed corresponding to dccta acquired during free

4 T I 2

- U)

p o B

-2

I

Fig. 13 transient

Quantisation noise superimposed on data acquired during stop

3.3 Computation of mechanical quantities To test the influence of Coulomb's friction, it is assumed that the free motion of the stop transient is described by eqn. 11, where wo = 250.664 rad ssl and

59 1

z,,, = 10.455s are computed using experimental data. Fig. 14 shows that the computed and experimental waveforms are different; this can be avoided by taking into account Coulomb’s friction.

300 T -k

0 4 -I 0 5 10 15 20

t, s Wavefornis ofspeed rekutive to stop transient Fig. 14

( i ) Experimental ( i i ) Estimated using eqn. I I , with coo = 250.664 rad s I, s,,, = 10.455s

Using least-squares techniques applied to the mechanical eqn. 5 , with t , = 0, and considering filtered data relative to the stop transient (see Fig. 12), the Yob lowing values are obtained:

r, = 19.417 f c / J = 6.6928 The error relative to the stop transient is given in Fig. 15.

2’oI 1.5

1 .o

- 0.5

s o a3

U) U

-0.5

-1 .o

-1.5 1 Fig. 15 $xed error relative to stop trutisient

3.4 Parameter identification by means of GA Assuming J is unknown, the genes are R,, L,, L,, R, and J. The cost function is eqn. 13 with the option of eqn. 14 or eqn. 15. The genetic algorithm employed is Genesis, with the following values for the principal parameters: total trials = 20000; population size = 100; structure length = 55; crossover rate = 0.6; mutation rate = 0.001; generation gap = 1.0; scaling window = 5; genes evaluation = 2; dump interval = 3000; dumps saved = 1; options = acefgl; random seed = 123456789; rank min = 0.75. The initial ranges chosen for the parameters are sufficiently large to allow con- vergence towards a global minimum. The initial values of the parameters are randomly chosen.

Table 1: Results of simulation t’est

The computation of the cost function requires that of the output variables corresponding to the input volt- ages of Figs. 4 and 5, which is carried out by digital simulation based on the model of eqns. 1-7. This simu- lation is effected by means of the fourth-order Runge- Kutta method.

To test the capabilities of the GA and to gain insight into the weights appearing in the cost function, a pre- liminary test was carried out considering the cost func- tion of eqn. 13, with the option eqn. 14 and k , = k h = 50 and k , = 100. The following parameter values were obtained:

R, = 4.96434 s1 L , = L, = 0.1075 H

R, = 1.55618 s1 Le = 0.01098 H

J = 0.00642 Nms2 to which correspond L, = 0.1019H f u = 0.00033Nms f c = 0.04297Xm

Then, a test was carried out considering the speed and the stator currents obtained by simulation with the esti- mated parameters and assuming the same input volt- ages of Figs. 4 and 5 as the true speed and stator currents. Then, a new identification procedure was car- ried out, assuming k,, = kl, = 1 and k , = {0.5,1,2}. The best results, given in Table I , were obtained after 60000 trials, choosing option eqn. 15 in eqn. 13 and assuming k , = 0.5. The results show the validity of the approach and give some idea of the cost function and the weights that can be used.

Choosing the same cost function and the same values of the weights that give the above minimum, the fol- lowing parameters are obtained:

R, = 4.85442 s1 L, = L, = 0.13332H

R, = 1.68149 s1 Le = 0.01251 H

J = 0.00649 n”s2 to which correspond f, = 0.0003342 Nms and A = 0.043436 Nm.

The waveforms of stator current components and speed computed by simulation are then compared with those acquired experimentally. Both waveforms are dis- played in Figs. 16-20,

T 40

20

a i o

3 ._

-20

-40 Fig. 16 sient: current along a-axis

Conipitrd and experimental wavrfortns relutiiv to starting trm-

Computed - - - ~ Acquired

L,= L, Le Rr J f” fc

True 4.96434 0.1075 0.01098 1.55618 0.00642 0.00033 0.04297

Identified 4.94379 0.10806 0.01084 1.55132 0.00646 0.000332 0.04279

592 IEE Proc.-Control Tlieory Appl , Vol. 145, No. 6 , Noveinher 1998

2o I 10

U .- k 0

-1 0

-20 I Fig. 17 sient: current a i n g a-axis ~ Computed ~ ~ ~ ~ Acquired

Com uted and experimental waveforms relative to starting tran-

quantities [12]. Examination of Figs. 16-19 shows that computed and experimental currents are practically superimposed. Fig. 20 shows that, at the start and dur- ing the initial phase of the transient, the speed acquired is greater than the computed one, owing probably, to the fact that the mathematical model is able to describe single cage induction motors, whose characteristic torque speed increases continuously from starting torque up to maximum torque and then decreases to zero, whereas conventional induction motors have a characteristic that decreases from starting torque down to minimum torque, then increases to maximum torque and finally decreases to zero. After about 100 ms, experimental and computed speed waveforms are prac- tically superimposed.

4 Conclusions T

40

20

Q .- 5 0

-20

t, s Fig. 18 sient: current a i n g b-axis ~ Computed - - - - Acquired

Com uted and experimental waveforms relative to starting tran-

20T

-20 1 Fig. 19 sient: current u l n g b-axis ~ Computed - - - - Acquired

Com uted and experimental waveform relative to starting trun-

T

0.2 0.4 0.6 0.8 t, s

Fig. 20 ~ Computed - - - - Acquired

Expermental and computed speed: starting transient

The validation of the model is carried out by means of simulation, comparing experimental and computed

IEE Proc -Control Theory Appl , Vol. 145. No. 6, November 1998

The paper deals with the problem of identification of the parameters of an induction motor for control pur- poses. The principal peculiarities of the method are (a) it allows identification of the parameters of the motor that permit a good reproduction of the input- output behaviour of the motor itself (6) as a consequence of (a), the goodness of the identi- fied parameters depends on the excitation of the dynamics of the system (c) it allows us to estimate Coulomb’s friction, making it possible to compensate for it for control purposes (6) it is not based on the traditional no-load and locked rotor tests, and, consequently, tests do not require par- ticular attention, and identification is performed using transients instead of steady state.

Moreover, the identification method cannot be applied on-line, because real-time implementation of the GA is, currently, not practicable.

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References

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