Vector Approximation Method with Parameter Adaptation and Torque Control of CSI-fed Induction Motor

Kozo Ide, Zhi-Guo Bai, Zi-Jiang Yang and Teruo Tsuji Department of Electrical Engineering

Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu Japan 804

Abstract - The concept of vector approximation method for CSI-fed induction motor consists of the vector control and the shaping of a stator current wave. The vector approximation means that a desirable current space vector is approximated by two of realizable current vectors which have only six directions, and by DC link current control. The main purpose of vector approximation method is not only to decrease torque ripple but also to control instantaneous torque, which is different from that of general PWM method. Moreover in this paper, a parameter adaptation system containing a magnetic saturation table and a model reference adaptive system (MRAS) is newly added to the system, which may bring more precise vector approximation and torque control to induction motor drive systems. The simulation and experimental results are shown in this paper.

I. INTRODUCTION

In recent year, a voltage source inverter (VSI) may be more popular than a current source inverter (CSI) for AC drives be- cause the VSI has faster current response to supply and can apply for a PWM system in the high frequency region more easily due to the low impedance. This is why the field-oriented control (vector control) often apply to control VSI-fed AC motor drive systems [I]. The CSI, however, still has better features which are capa- bility of four-quadrant operation without any extra power circuit for regeneration, ruggedness and reliability, and no shoot-through fault. The high precision of current control is possible. Thus it may be appropriate for large capacity drive systems such as steel mills, elevator motors, and so on. However, the CSI-fed AC motor has problems in low frequency region, where occure torque pulsa- tion and the harmonic heating because of a rectangular current wave with 120" conduction. In order to cope with this problems, a vector approximation method was proposed [2], which would not only decrease torque ripple but also control instantaneous torque of the induction motor driven by the CSI even in transient state. Although the PWM-type CSI drive induction motor system has been proposed in [3][4] and the design of PWM signal was con- sidered to decrease the higher harmonic components of current waveforms, our method is different from the PWM methods. In this paper, a modified vector approximation method with DC link current control is shown. The vector approximation method approximates the desirable stator current vector by two of six re- alizable current vectors which are adjacent to the desirable one. The magnitude of the desirable current is adjusted by the con- trol of the DC link current for each instant. Thus the modified vector approximation method can produce any current vector so that a sinusoidal phase current is realizable, if necessary. On the other hand, the phase angle of the desirable stator current is de- termined by utilizing rotor flux angle which is computed by the on-line simulation of the mathematical equation of the induction motor. If the rotor flux angle is estimated correctly, the so-called magnetizing current and torque producing current can be decou- pled precisely and so controlled independently on vector control

[ 3 ] . However, the vector control is influenced by the rotor parame- ter variation, that is, the rotor resistance varies with temperature a.nd the inductance is a function of main flux saturation. To solve this problem, many identification or adaptation schemes have al- ready been proposed [6]~[13]. An off-line automated identifica- tion [6] and an adaptation schemes with using the reactive power transferred to the rotor [7'J have been evaluated as effective meth- ods. With a different approach, model reference adaptive system (MRAS) [8], [9] have been presented. The adaptive system esti- mates the error between the motor outputs and that of reference model, and tunes the adaption gains or parameter to decrease the error to zero. As a result, the degradation of torque control by the rotor resistance variation was indicated there. While, the se- lection of flux level with regard to the magnetic saturation effects was discussed in [IO]. The saturated dynamic models of induct,ion motor with vector control have been proposed in [11]+~[13].

Thus in our case, to accomplish the ideal vector approximation system for CSI-fed induction motor, both resistance and indue- tance variation problems have to be solved. In order to do that, the vector approximation system newly includes a magnetic sat- uration model which is obtained by a novel off-line identification mehtod. The on-line compensation of magnetizing inductance be- comes possible. This system also includes MRAS using the model reference error of torque to compensate rotor resistance variation. Consequently, this method may make it possible to asymptot- ically track the actual torque and estimate the accurate rotor resistance without the influence of inductance variation, which occurs at the field weakenning or the optimal efficiency control.

This paper discusses the vector approximation method with parameter adaptation in the following sections. The identification and adaptation method are shown. The expected simulation and experimental results are also shown.

11. BASIC THEORY

The basic equations of a symmetrical induction machine are given by

( 2) L, ., Lm

i,, = i , + -3, . where U,, i , , i:, and i,, are the complex numbers of stator volt- age, current, rotor current and rotor magnetizing current respec- tively in the stationary frame fixed to the stator, that is, a-p axis in Fig.1. R, and R, are stator and rotor resistance, L , and L, are stator and rotor self inductance, Lm is magnetizing inductance, U, is rotor angular velocity, and p is the differential operator. From (1) and (2), we have

(3)

0-7803-1993-1/94 $4.00 0 1994 IEEE 71 1

where Y = & / L , . (4)

Since it is difficult to detect i,,, the magnetizing current simula- tor is used.

pi,, = (-7 -.iwr)imr + +, .

pe = (-y - jw,)e .

(5)

(6)

(7)

Subtracting (3) from (5), we have

where . . e = t,, - t,, .

This indicates that the error e asymptotically decreases by e--@. The instantaneous torque is given by

T, = kt9(isrm,) = kt(i,((i,,Isina . (8) where

3 P L L 2 2 L ,

kt = . (9)

The P is the number of the poles of the induction motor, 9 means a imaginary part of a complex number, (*) means the conjugate of a complex number, and a is the angle between i , and i,,.

y / r o t o r axis

-./ '.,:, e d 1 mr

i sd ,cy Q P stator axis

Fig.1. The relation of current vectors.

111. VECTOR APPROXIMATION METHOD

Vector approximation method approximates the desirable sta- tor current vector iz in the sense of time average by the realiza- ble six current vectors. The phase angle of the desirable current vector is determined with regard to the phase angle of the ro- tor magnetizing current i,, and the magnitude of the required torque as follows. Assume that the rotor angular velocity w, does not change during a sampling period At. Then the space vector of the rotor magnetizing current is

i,, = im,e300 . (10)

where a,, is the phase angle of the magnetizing current vector in the stationary frame. Consider the rotating frame, that is, the d-q axis in Fig.1. Then from (3), we have the following equations in the discretized form.

where k means kat , wsr is slip angular velocity, and i,d is the direct-axis component of the stator current in the rotating frame. Consider, as the controlled variables, both the torque and the magnitude of the rotor magnetizing current. That is, li,l sina denoted by i,, is determined by the required torque, and lis[ cosa denoted by i s d is determined with regard to im7. In Fig.1, i sd and

is, are shown as the vector decompositions of iz. Thus the desired stator current vector iz is given as follows:

cy = tan-' + , (14) zsd while the realizable stator current vectors with the CSI drives are six directions as shown in Fig.2. Assume iz at the sector I in Fig.2. The i; is approximated by iSl and i s 2 which are applied for At1 and At2 respectively as shown in Fig.3.

p-ris

+ - axis

uv \r

TWV Fig.2. The six realizable current vectors and iz.

Fig.3. Princeple of Vector approximation.

Because of sufficiently high switching frequency, iz can be as- sumed as constant during one switching cycle At. Thus i: is expressed by the view point of the composition of vectors as fol- lows:

where sin(r/3 - a,) sin(r/3 + al) At1 = A t .

At2 = At - At1 . The content of the first parentheses in the righthand side of (15) gives the angle CY, and that of the second gives the magnitude of i:. Thus, the DC link current should be controlled so as to

In this case, the tip of the composed vector i: is on the line between the tips of the vectors iil and ii2 in Fig.3. With the DC link current control, i; is realizable. On the other hand, since it is impossible to apply is, and is2 at the same moment, the composition of vectors is done by time average sense.

712

Then the average torque during At is given as follows:

(20) 4 3

where IC: = k , { - I ~ c cos(7r/6 - al)} .

Since At is assumed to be sufficently short, the rotor magnetizing current vector i,, is also considered to be constant for At. Thus (19) is replaced with

From (21), the required torque is realized in the sense of the time average. As mentioned above, the main purpose of the vector approximation is to control the instantaneous torque arbitrarily. This can be achived by controlling the stator current arbitrarily. If required, the tip of the current vector can be controlled to have the locus of a circle. In this case, the phase currents are sinusoidal. This fact is confirmed by the simulations and the experiments in Section VI.

IV. ROTOR PARAMETER IDENTIFICATION

This section shows an identification method for the value of ICf and y off-line. The changes of kf and y make the errors the torque and the pase angle of i:, In order to control the torque precisely, it is necessary to know the characteristics of the parameter vari- ation. The method proposed as follows uses the experimental system shown in Fig.4 which is a vector approximation system for the identification.

I

a L

Fig.4. Control system of Identification.

Assume a sinusoidal phase current at the steady state condi- tion. Then, since i,, = i,d at the steady state from (l l) , from (13)

i, = Jm. (22) and the electrical magnetic torque T, and the slip frequency w,( are from (21) and (3)

T, = kfimrirq . (23)

where

- 3 P L k 3 P L& R, R, '-22 L, 221,+L, ' y = z = n . (25)

Let a, and a ~ ~ f be the reference input to the system shown in Fig.4. Then the reference value of i,, is written by

&Ref = cos ffRef . (26)

However, the real a does not coincide with ( Y R ~ J , if the value of y is not correctly set. Then i,, does not coincide with the reference value. However, if for constant i, the maximum torque, Te~,, ,~, is obtained for some a, then from (22) and (23), i,, is equal to iJq, so that the actual a should be 45" against a h f . Thus torque coefficient kt is identified as follows:

In order to measure the torque, the following relation is used as

T, = J* + TO dt

at neload and low speed condition. J is drive inertia, To is s e called the Coulomb torque. The Coulomb torque can be com- puted by the experimental measurements with the two kind of

dw, constant stator current i, and i: when J - and J - dt dt are mea-

sured. In fact, if kt is constant from (27), (28), and (29), To is given by

Thus kt can be identified. The identification of y is done by the following theory. Assume a h f gives maximum torque under the condition of constant i,, such would be confirmed experi- mentaiy by changing aflef and measuring the torques. Although cyRef is not necessarily 45", the actual cy should be 45". Thus y is identified with the nominal value, yo and C Y R ~ J as follows:

Thus the tan a R e f in (30) represents the deviation rate of actual y from the nominal value.

In the magnetic saturation region, however, since a at the max- imum torque may become bigger than 45". In order to identify k t in the magnetic saturation region, the following algorithm is needed.

i)

ii)

iii)

i v)

For each i,, T,(,,) is obtained experimentaly.

Assume for T,(,,,), a = 45". Then the relation

is obtained using

i2 2

T, = kt imrisq = kt 2 sin 2 a (32)

but since a is not necessarily 45", we can make a relation between T4ma,) and a using (31), for example Fig.5.

From Fig.5, for each is we have a which gives T,,,,,,, and again kt's are computed using (32). Then we have another relation such a type of (31).

Repeat ii), iii) until the graph of kt = f(iml) does not chang.

71 3

Phase angle (Y between ii and imr(deg) Fig.5. Relation between T, and cy.

Fig.6 shows the result by using the above algorithm, that is, starting from the initial graph (the upper dots), the algorithm converges to the final graph (the solid curve) by 5 times iteration.

If the rotor leakage inductance 1, is assumed to be constant, the magnetizing inductance L , is obtained by (25). This relation is shown in Fig.7. From Fig.7, a inverse magnetizing curve is obtained as shown in Fig.8. The relation shown in Fig.8 is used to identify the rotor resistance. In the experimental system, the look-up table is used as Fig.8.

0 10 20 Magnetizing Current imr[A]

Fig.6. Experimental identification results of IC:

0 10 20

Magnetizing Current imr[Al Fig.7. Experimental identification results of L,.

20

2 I .- e

I; 10 M .- .- e,

P 0

Rotor Flux @ r [Wbl

714

Fig.8. Inverse magnetizing curve obtained by the identification results.

V. TOR PARAMETER ADAPTATION

The phase angle of rotor fluxis computed by (l l) , (12), and (24). These equations contain rotor parameter 7, and in practical y changes by temperature and magnetic saturation. For precise control of torque, it is important to know the precise value of y in their equations since the value of y affects not only the estimation error but also the estimation of torque magnitude directrly. The parameter adaptation method discussed in this section makes the error of torque magnitude zero, and then the precise value of parameter is obtained and utilized. The details are shown as follows:

A. Design of the Adaptation Law

In this subsection, the compensation of rotor resistance varia- tion which depends on the temperature is considered. In order to do an on-line adaptation, MRAS approach by utilizing the model reference error of torque is applied, which assures stabd- ity. Now, notice two equations (3) and (5) which are the actual and the mathematical models respectively. Since (3) and (5) are nonlinear, they are linearized by the Taylor expansion around an operating point after each variable is expressed by the d-q com- ponents in the rotational reference frame and expressed with the small- signal forms. Assume that L, does not change, since it is not necessary to consider the saturation effect in the small range around the operating point. The state error equation between the actual and the mathematical models at the operating points can then be obtained as follows [14]:

Aim;;:Lrd 1 = [-W.,o -70 UdO] -70 [ Aimrdd,;imrd 1 + [ ;] ~ ( A R , - ah,) . (33)

where ( A ) denotes estimated values, (0) denotes the operating points. In the same way as (33), the torque error between the actual and the mathematical models from (8) are

(34)

From (33) and (34), the transfer function from the rotor resistance estimation error AR, - AR, to the output AT, - AT, is obtained as follows:

AT, - A?, = kt [z84(Ai,,d - &,d) - i r ~ A i m r q ] .

AT, - A?, G(s) =

AR - ak

where K t = - k t / L ,

Thus the pdaptation law for AR, must be determined so that AT, - AT, converges to zero. This means A& converges to AR,. In many cases, the adaptation law is chosen according to Popov's hyperstability theorem [15]--[16], however in our case the adaptation law can be linear as follows:

AA, = ( K p + Kr/s)(ATe - APe)V,;, . (37) where K p and K I are the adaptive p ins . From (35) and (37), the transfer function from AR, to AR, is

-- AR KtVslgnisdOisqO [s + ?o{l - ( i s q o / i s d o ) 2 ) I ( s ~ c p + K I ) AR, - 53 + as2 + bs + c

(38) where

Fig.9 shows the block diagram of the adaptation.

Ae= ATe - A'?e

L I

Fig.9. Rotor resistance adaptation block diagram

For the stability of the system, a, b, and c should be positive and ab - c > 0 by Routh-Hurwitz condition. Thus V,;,, K p , and KI are then determined. That is, K , , is the sign function as follows:

i) if isqo 2 0 and abs(i,,o) 2 isdo then 1 ii) if is@ 2 0 and &(is@) < i.dO then - 1

= iii) if id, < 0 and abs(i,@) 2 i J d O then - 1 iv) if i , , < 0 and abs(i,@) < i sdo then 1

(40)

2?0 ~ d o a b ~ ( K t i s q ~ ) '

2?0 - isdoabs(Ktis@)Kp i,doabs(Kti,@)

I In the case i) and iii) of (40), K p and Kl are determined as

0 < K p < .

0 < K I <

4 4 1 ) ? : { I + ( i d q o / i S d 0 ) 2 } - ?oi.doabs(Ktid@)KP{l - (is@/isdo)2)

2?0 - isdoabs(Ktis@)KP - ?O{l - (iaqO/isdO)2} while in the case ii) and iv) of (40), K p and K I can be deter-

mined as K p > O , K , > O . (42)

B. Adaptation with the saturation table On the adaptation mentioned above, the inductance variation

has not been considered yet. The inductance variation depends on z,,. Thus i,, depends on i s d . Assume i,d changes stepwise, then i,, can be considered to be the piecewise constant compared with the resistance variation which depends on temperature. From ( 3 ) ,

p -amr + i,, = i , d , ) ( 4 3 ) where L, and R, denote actual values. Let L,o and R,o denote the nominal value of L, and R, respectively, then we have from (43).

(44)

In order to take the saturation into consideration, (44) is used instead of ( 1 1 ) as follows. The rotor leakage inductance 1, is so small compared with Lr and L , that the following relation ca.n be assumed.

(45) Lm L r Lmo LIO . -- - -

where LmO denote the nominal value of L,. Thus we can obtain the modified rotor flux angle estimation cxP with the adaptation and the saturation table by ( 1 2 ) , (24), (37) and (44). It is shown in Fig.10, where the estimated rotor resistance is used.

i,, controller of Fig.8.

Fig.10. Flux angle estimation with the adaptation and the saturation table.

This method has only a drawback in the following case. Dur- ing the flux level changing, whereat the inductance changing, the adaptation may not work correctly because the inductance value is8 assumed as the constant in (33) and (44). However, once the flux level is steady at any operating points, the correct inductance value is informed to the adaptation system indirectly from the sat- uration table. This adaptation system can estimate rotor resis- tance correctly at each constant level of flux. In (37), the torque of the mathematical model is calculated with zhr(= &zmr) as shown in Fig. 10.

where kto is the nominal value of torque coefficient. It can be seen that ihr compensates the inductance variation. On the other hand, the actual torque can be obtained by the measurement by a torque detector [14] or the estimation with DC link power [17]. For the torque estimation at the steady state, the feature of CSI may be useful. Assuming no power loss in the inverter, the power into and out of the inverter is identical [18]. In this case, torque is expressed as follows:

Te = k t 0 i ~ , i s q . (46)

Te = 3[(2/3)vDClDC - Rslia12]/~i . (47) where V ~ C is inverter input voltage, R, is stator current and w; is stator synchronous angular velocity. The measuring DC link power seems to be much simpler than measuring the inverter output. l i J l and w; can be obtained with each reference value. However, since some power losses which may be regarded due to the drive amplifier and smoothing filters exist certainly, the power effciency of the inverter should be measured beforehand.

. .

71 5 the d-axis equation can be rewritten as

VI.

Induction motor Output 7.5 kw No. of poles 4 Voltage 200 V ac Current 27.2 A ac Speed 1740 r/min L , 0.04647 H Rr 0.335 fi M 0.04557 H

Speed controller Cumni conuoller

L iM. controller

- of Fig.10. Wr

Encoder

Fig.12. Control algorithm of Vector approximation.

DC generator 7.5 kw 6 220 V dc 34.1 A dc 1200 r/min

SIMULATION AND EXPERIMENTAL RESULTS

A . System configuration

The experimental system is shown in Fig.11 where the capaci- tances parallel to the induction motor act as smoothing filters of pulsating phase current and this is understood as the time aver- aging in (15). The DC generator is coupled with the induction motor as aload which is not appeared in Fig.11. The nominal rat- ings of the main equipments in the experimental system is listed in Table I.

THREE PHASE SOURCE

ttt

MICROCOMPUTER

INTERFACES

INDUCTION MOTER

ENCORDER

The control algorithm as shown in Fig.12 is executed by using DSP (NEC-pPD77230). The computational time is about 0.5 ms for one cycle.

B. Verification of Vector approximation

Fig.13 (a) and (b) shows computer simulation results of the steady state locus of the tip of the stator current vector ((a)) and the corresponding phase current ((b)) by the vector approxima- tion method with keeping the constant DC link current (the left) and with the instantaneous control of the DC link current (the right).

IP

- 1 @ - 1 I - 1 I

Fig.13 (a) Locus of the tip of the stator current vector.

I

Fig.13 (b) The corresponding stator phase current. Fig.13. Simulation results by Vector approximation with the

constant DC link current and with the DC link current control.

-2.0 J Time (0.2 sec/div)

Fig.14. Experimental waveform of the phase current by vector approximation with the constant DC link current.

71 6

-2.0 I Time (0.2 sec/div)

Fig.15. Experimental waveform of the phase current by vector approximation with the DC link current control.

It should be noted that the smoothing filter is assumed for the both case. The dark parts in Fig.l3(a) mean the limiting area of the vector approximation met hod to avoid such conditions that At , or At2 is less than the turn-off time of GTOs (see (16)(17)). The minimum of At , or At2 is set to the greater value than the turn-off time of GTOs, which is 50 psec in the experiments. This effect appears as the jumps of the waveform in Fig.13 (b). Thus vector approximation method may be suitable for medium and big-size drives with low speed. Fig.14 and Fig.15 show the exper- imental results of the phase current by the vector approximation method with the DC link current kept constant and with the in- stantaneous control of the DC link current respectively. These results almost coincide with the simulation results, which means the validity of our assumptions with respect to vector approxima- tion. Moreover, the phase current wave form is improved by the vector approximation with the instantaneous control of the DC link current.

C. Verification of Parameter Adaptation

adaptation START

The parameter adaptation proposed in Section V is tested in the conditon where the rotor speed is 120 r/min at the steady state, the reference magnetizing current i,, is 10 A, the load torque is 6.0 Nm (about 20 %of the rated torque) which was mea- sured by a torque detector. Fig.16 shows the estimated torque re- sponses and the estimated rotor resistance. The estimated torque converges to the actual value 6.0 Nm, and then the estimated ro- tor resistance converges to 69 % of the initial value, which would mean that the initial rotor resistance has been set to 1.45 times more than the actual value. According to (35), when the es- timated torque coincided with the actual value, the estimated rotor resistance would coincide with the actual value.

In order to confirm our theory more certainly, another condi- tion is tested, where the reference magnetizing current is 10 A initially, and after the operation becomes the steady state the reference magnetizing current is changed from 10 A to 6 A s t e p wise, the initial condition of the rotor resistance is the same as in Fig.16, and the heat change can be negligible since the experi- ment is continued for the short term. Since the control system has a magnetic saturation table to compensate inductance variation due to the change of flux level, the rotor resistance is expected to estimate the same value as shown in Fig.16. Fig.17 shows the re- sult in this condition. The esitmated rotor resistance converges to the same value of Fig.16, and the reference torque also coincides with 6.0 Nm. This result shows the capability of the rotor resis- tance estimation at the different flux levels. In the experiment, the adaptive gains are selected as l i p = 0.16e-4, IC1 = 0.771e- 4 which are relatively small to avoid that the estimated rotor resistance would be oscillatory.

Finally, the speed and DC link current responses in the case where the updated rotor resistance value is used in the control

I

( where i,, = lO.O[A],i,, = S.O[A])

20% of the rated value, 6.0 Nm

Fig.16. Estimated torque responses and Estimated rotor resistance (imr = 10 A).

fa ( where i,, = 6.O(A],i,, = 8.2[A])

20% of the rated value, 6.0 Nm

adaptation 7 START 2.ORO 2.0(sec)

Fig.17. Estimated torque responses and Estimated rotor resistance (zml = 6 A).

system are compared with that in the case where the incorrect rotor resistance value is used in the control system. Fig.18 shows the responses in the latter case, and Fig.19 shows the responses in the former case. From these results, it can be noticed clearly that the acceleration property is improved in the former case. This reason is understood as follows: The estimated torque is set to the rated value during the transient state. While the actual torque in the case of Fig.18 can not reach the rated value, t.he actual torque in the case of Fig.19 can coincide with the estimated torque.

717

REFERENCES

h

v E l

0.01 a I Time ( 2 sec/div)

I

0.0- Time ( 2 sec/div)

Fig.18. Rotor speed and DC link current responses with incorrect parameter.

300 K h

v S I a v1 4

0.0. I Time (2 sec/div)

Fig.19. Rotor speed and DC link current responses with correct parameter.

VII. CONCLUSION

We proposed the vector approximation method with parame- ter adaptation for CSI-fed induction motor. The features of this method is summarized as follows:

e The control of the instanteneous torque in the both tran- sient and steady state is possible by the vector approxima- tion with DC link current control, and the torque ripple can be decreased, which is proved by the sinusoidal phase current wave form.

The parameter adaptation with the saturation table can estimate the rotor resistance correctly at each different flux level, and the estimated torque coincides with the actual value which shows the capability of torque control.

This method is not so complicated to implement, and is adequate for the precise torque control of medium and big-size drives with

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_ _ _ _ .