Optimal Control of Voltage Source Inverters Supplying Induction Motors

OPTIMAL CONTROL OF VOLTAGE SOURCE INVERTERS SUPPLYING INDUCTION MOTORS

s. Halasz Department ofElectrical Machines, Techn. Univ. 1111 Budapest, Hungary

ABSTRACT The paper is concerned with the voltage source inverters ensuring the minimization of additional coil losses in the induction motor. Both simple and PWM inverters will be considered under non-modified control. Theoretically, the results are only valid if the skin effect in the motors is neglected. Nevertheless, the results as first approximation prove to be valid even for motors with relatively strong skin effect.

+ __

~ ~

6--~

(a)

INTRODUCTION

a

Additional coil losses in inverter supplied asynchronous motors can be important enough to justify the application of the optimum theory. These losses can be minimized by the optimal control of three-phase inverters supplying the motors. Our analyses refer to inverters of unmodified control, hence, the motor can only be connected to the dc network with identical phase arrangement in the one-sixth period of the fundamental component. The asynchronous motor parameters are supposed to be constant and the inverter voltage drops are disregarded. The parameter variation as a result of skin effect will be reckoned with separately.

Reol

J

b

c

STATING THE PROBLEM

(b)

The inverter configuration and the locus of the ~ator flux Park-vector ~ as well as of its fundamental component ~ 1 have been shown in Fig. I. Losses are expected to have a minimum whenever the locus of the flux vector ~ has a six-sided symmetry over the entire period of the fundamental component. The number of order of harmonics due to the six-sided symmetry may be: \I

=

I + 6k,

Fig. I. Inverter configuration (a) and flux Park-vector (b) il

fundamental component of vector i,

A ~ resultant vector of harmonic fluxes, L'

(I)

stator transient inductance..

The additional losses are proportional to the square of rms value of current

where k = 0, ± I, ± 2, . . . The one-sixth period drawn by heavy line in Fig. I will always b'e discussed later.

~ period. Ai is known

The time t = 0 will be at the middle of this The resultant vector of harmonic currents (Ref. 1) to be closely approximated by: -

-

-

~-~I

Ai = i - i l = -L-'- =

A ,I. '1!,

2 Be U="3 UdC' where Udc de supply voltage,

(2)

t where

7

stator current Park-vector,

379

tiMe in the angle of fundamental component.

380

Sandor Halasz

The problem can be stated as follows: Find u(t) amoIlg the possible time functions ensuring the minimum of integral

~'lt2

6"

1. J I ~~12 dt

=

(4)

1r -6"

with the constraint that the fundamental component of flux ~ has to remain constant (e.g. at rated value):

_ 'It}

)(

R«JI

" 3 6_ = 7r J 1/Ie- lt dt " -6

= oonst.

g j

(a)

(5)

SIMPLE INVERTER For the sake of our analysis, the inverter supply voltage udc ' hence also voltage u has to be assumed to be variable in time, even in time _.!..;; t".!.

6

6

in

such inverters. Unlimited de voltage

The minimum of additional losses may be reached by this way. This minimum can be fairly well approached in the low speed range. The additional losses of the motor will be minimized (see Appendix) if in each interval _.!."

6

Fig. 2. Optimal control, voltage unlimited. Park-vectors (a) and phase voltage and flux (b)

t" .!.6 (6)

and at time t =.!. (and t = _.!. as well) the flux ~ 6 6 and ~~ too is abruptly changed by

6 " -<

J

5=

as can be seen in Fig. 2 in the interval -

(7)

udt,

(I: -+

0). Be 1/1}

1r 1r 6 + I: < t <;·6 -

=

I, then

1::

u = cos t,

(1r+Y3). 6 """4 J, sin t - oos t . i, (cos t - 61r - 4Y3) i,

,T. . 'I' = srn t~} =

-

~1/1

=

while 5 = -.!!..- -

6Y3

(8)

where \) = I + 6k; k = 0, ± I, ± 2, . .. . The results have been complied in Table I, and compared with data relating to inverters of constant de voltage and of de voltages varying as

u = _1_2(Ref. 2). The relative cos t weight of harmonics for k> 0 is seen to have increased, namely the first and the second term in the last equation will have the same sign for these harmonics. In the cases of very high \) values the amplitude of the harmonics can be approximated by the second term in equation (10), Le. by (- l)k . 0,00865. The torque pulsations are obtained by: _

_

~= ~1/IX i}

_

+ 1/1}

Assuming as an example L'

~ I - 1 L' 1 L' X =

-

c,.1Ji.

0,2 and in no load

(11)

I} =

1

4' t:.m = 4,7 sin

Thus:

"

Y3 2 ~'lt2=7r ~ (oost-i-4) dt=0,001609. 3 6

_

X t:.i=

-

(9)

-6 It is 74,8% of 0,00215. The later figure is proportional to the motorlosses supplied by a simple inverter of oonstant de voltage (Ref. 1). The voltage harmonics in the case of optimal oontrol:

and in rated load i 1 t:.m =

where a

=

t(cos t =

[5 sin t -

le

(12)

al,

." .1 -/6+/

oos

(t -

ill (cos -a),

(13)

1r 13 6 + 4'

Torque pulsations also included in Table I. Limited de voltage

In the case of limited output voltage, the previously

381

Optimal control of voltage source inverters

Table 1. Control of simple inverters 1 cos t

u = -2-

u = const

-5

v

0,2

optimal

0,145

0,1202

0,1228

0,1405

7

0,143

- 0,2035

- 0,1899

- 0,1929

- 0,1969

- 11

0,091

- 0,08865

- 0,0038

- 0,0016

- 0,0045

"

U;

13

0,0769

0,1048

0,1456

0,1498

0,1453

'It (0)

0,9497

0,9532

0,9567

0,9562

0,9540

1,097

1,101

1,1046

1,1041

1,1016

0,00215

0,00186

0,00161

0,00164

0,00179

no load

0/ L:.m

0"--7

rated load

~ 0

Jt

-'I

-< 14\ \

:-\

.,

0

~"6

:\

I

0

'-/'

(14)

while in the range

= l/max = '3

= I

7r

1-2sinT T - 0,5 sin

'3 -

L:.~

a

V

max

.

(16)

2T

(cos t -

=

Vmax 7r [SIn. T + ---v("6 -

a)j, -

T )]

The fluxes in the interval -

T .;;;

t .;; Tare:

~3

u fJ.f

I I I

I

I

Relationships for VI = 0,25Vmax and VI = 0,75Vmax have been plotted in Fig. 3, while data relating to this two cases are included in Table I.

0

(17)

=

(15)

where V dcmax is the maximum voltage of the de circuit. VI and T can be related to each other on the basis, that the fundamental component of the voltage given by (14) and (IS) has to be VI:

V

).

I

Vdcmax;

IJt

.T\...7 "I"

:-

~I = sin t - cos t . j,

and

2

U

0

~ = sin t - aj,

deduced optimum conditions can only be approached so that in the range - T';; t .;; T u = VI cos t,

\

-j- '-./ \

\

L------

-.J(J-

-075

u l max Ut

'

I I I

.0.25 :

/'max I ------.J o

Fig. 3. Voltage time function for simple inverter

Sandor Halasz

382

and elsewhere in the sixth of the period: -. Umax '" = srn T + ~ (t -

+ [5 cos t - sin

.

.

T) - a/,

t

il~

= sin T -

sin t +

U

;;ax

+ (cos t -

(t - T)

-

.

U max

srn t + ~ (t 1

The torque pulsations at points t =

a)j.

t

±

(22)

T) ] .

i and

t = 0 are

independent of L' for a given current and can be influenced by control only in a small extent. At the same time the effect is always controversial: if the torque pulsation can somewhat be reduced at t = 0, then it will

Thus:

This result has been compared with the losses of obtained by the application of an inverter of constant de ~'I'2 ., Ut voltage (p = 0 00215) rn FIg. 4. For < 0,4

-u-

,

be accentuated by the same extent at t =

±

i.

Thus,

control can only reduce positive or negative areas, i.e. speed deviations due to torque pulsations. Table I shows

max

r

p

. [srn T

(18)

i)] .

(t -

u- cons!

(,0

0.8 J

!I

(a)

I

o

I

/

0,2 Fig. 4. Optimal control of simple inverter

the losses approach that of the inverter of optimal control. In general, results for limited voltage inverters can be stated to be in the midway of unlimited and optimized inverters and of inverters of constant de voltage. Now, voltage harmonics depend also on Ut: Uv = Ut

'1 1T

[Sin (v - I )T V - I •

+ sin (v + I )T] + v + I

7l"

+ _6 Umax srn 6 v _~

1T

Ut

In the range -

T";

•

Sin

(20) vt

_

v

t .:;;;

T

trol u = _I-2 is advisable. cos t PULSE WIDTH MODULATED (PWM) INVERTERS

T':;;; t':;;;

1T

6'

at

no load: I1m = 4,7[- a sin t + cos t· sin T +

(21 ) max

that optimal control to reduce the losses is seldom advantageous from the viewpoint of torque pulsations. If the reduction of speed pulsations is of importance, con-

torque pulsations essentially

equal (ll) and (12), while in the range

U + cos t ~

Fig. 5. Optimal control, voltage limited. Park-vectors (a) and phase voltage and flux (b)

(t - T) ] ,

t

and at rated load:

~m= [5sint~cos(t-i)] ·(cost-a)+

The motor supply is varied by means of the number and width of pulses, while de voltage is constant. Inverter control can be optimized by adequate distribution of pulses in time and by changing their width (Fig. 6). The implementation of the results obtained for simple inverters is at present rather difficult, but they help to optimize the PWM inverters by providing important instructions. This is of practical importance, since optimal control is within the technical possibility for PWM inverters. The followirIg conclusions can be deduced from the precedirIg chapters:

383

Optimal control of voltage source inverters

t, f4..-..- r-'I ......

t f t2 ,...:.-

u

I I

F depending on t I ' t 2' ... , tn ,A. Introducing notations:

....-

I

I

1

o

tn-{ tn

f=

I I

t

I

x 7

Fig. 6. Voltage time function for PWM inverter 1. The pulse density distribution in time has to correspond to Fig. 3 for a given VI' i.e. the pulse density is higher at the ends of the one-sixth period than around its middle section. 2. The last pulse has to satisfy inequality t n _ I < T, where T is determined by equation (16). For infinite number of pulses t n _ I -+ T. 3. In conformity with Fig. 4, the last pulse width increases with VI' hence with the speed. Therefore the locations of pulses are shifting towards the

i

middle of

'It = I

~. 21T VI

2

.:l'lt = 'It~

iI

sint.(-I)i I

-I + (2-)2(1 VI

(24) '

T 3 + 3T 2 ) +

1T

+Q(VV)2[U 2 -t l )2U 3 -t 2 )+ ... + 1T I

T2(~-tn)]'

n

T=

periods. This is especially true· for

ZI (-

I)i t .. I

The computation starts from a given VI and an assumed initial value X. First the deviation

VI> 0,8.

of the last pulse reaches ~ rather soon

4. Time t n

with increasing VI and pulse number.

is determined. Computing X +.:lx and the computation will be repeated with new values until

The pulse distributions minimizing additional losses can be computed by several methods. The best from convergency aspects, proved to be in our praxis one suggested by Dr. I. Racz. The function to be minimized in the method is consisting in minimizing the function

F = 'lt 2

-

'It~ + A('It~ - I).

fT·f
is infinitesimal (e.g. EO = 10- 23 ). The developed program takes into consideration other conditions, as well e.g. coincidence of certain pulses with the increase EO

of VI' or that t n = ~ is feasible. Fig. 7 presents loss

(23)

p

2

......

B--

.!fl

..........................

f

--

--_

...... ...............

sim le inverllJr--_

---

- - ....------__ - - .... --___ -_ ----_,. -_ -_ --::~ ~ ...

U""c= const Ul

--

_ _- - - -

~~

----------------------------Oph~um-----

--- Optimal control

--

o

O,f

0,2

Sy~trical

0,3

Q+

control

0.5

0,1

Fig. 7. Motor losses, PWM inverter

0.'

0,1

Sandor Halasz

384

P obtained by this method for some pulse numbers n. For the sake of comparison, losses for symmetrical distribution have been plotted as well. It may be seen that the losses can be reduced by about 20% by optimal control, or keeping the losses unchanged the pulse number can be decreased by 2 or 3 by applying optimal control. It is interesting to note that the losses may be even smaller than for. simple inverters of constant voltage. The time functions of certain variables are seen in Fig. 8 for pulse number n = 5. As an example, Fig. 9 shows torque pulsations under

optimal and symmetrical control. Optimal control is seen to be generally not advantageous for torque pulsations. It has to be remembered, however, that the error originating from the negligation of resistances is higher in the case of the computation of torque pulsation, than in the calculation of losses. Therefore the computed torque pulsations can only be regarded as first approximations, and mainly for higher fundamental frequencies. SKIN EFFECf The numerical analysis can not be carried out in general form in the case of the skin effect taken into account. Therefore, the computation to check the theoretical results have been made for an asynchronous motor selected asan example (Type: VZ 160M4, 14kW, 50Hz, 1460 I/min, 190 V, connection delta). The method of computation and the motor data can be found in (Ref. 2., 3.). It is necessary to know the harmonics of voltage in the case of calculation, so for the simple inverters see the equations (10) and (20). In case of optimal control, computed rotor losses are seen in Table 2. The losses included in Table 2 are related to the rotor winding losses of rated operation.

(a)

Table 2. Skin effect, 'It 1

[= 10 Hz

=1 [= 50 Hz

.skin

skin

Fig. 8. Optimal control of PWM inverter. Park-vectors (a), phase voltage and flux (b)

Symmetrical

Optimal

Am

0.2

Optimal

0,071

0,29

0,052

0,75

V= const

0,099

0,33

0,069

0,84

The losses are smaller' in optimal control, even if the skin effect is taken into account, though the advantage is somewhat less than it was in the case without skin effect. PWM inverter losses as a function of pulse location have been plotted in Fig. 10.

(a)

Fig. 10. Losses of PWM inverter (n

Fig. 9. Torque pulsations of PWM inverter (n = 5, Vl = 0,6). No load (a), rated load (b)

=

2, V l

=

0,77)

It turns out from the figure, that the place of minimum losses hardly depends on skin for n = 2. On the other hand, it is interesting to notice, that the value of losses is sensitive in a smaller extent to the pulse displacement

385

Optimal control of voltage source inverters originated from the control for optimal controlled inverter. Further analyses showed that the minimum of losses in general does not differ considerable between the cases calculated with and without skin effect.

The relationships deduced for simple inverter for the case of optimal control can be applied to PWM inverters. The winding losses of the motor may be reduced by about 20%. Unfortunately, however, optimal control is not always favourable for torque pulsations. The numerical computations taking into account the skin effect verify that the relations deduced hold approximately for motors having considerable skin effect. APPENDIX

= I = const,

Assuming '1'1 p.u. system:

f

~'I'2 = -

7ro

since Thus:

1/1 1

~'I'2 = a2

(4) can be written in

[(a - cos t)2 + (1/I x - sin t)2 dt

= sin t -

cos t • j

and

~'I'2 = a 2 - ~ a + 7r

I, a = ~ +

a

=-

1/I y

d~'I'2 = (2a - ~)

- dUI

6

= -7r3 da.

=

const

= -"1r3 da = 1T6 61 (cos T 1

-

cos T 2 )·

As a result of the first voltage impulse the flux change 61 in the interval T 1 .,; t<;;' T 2 and the square of its rms' value is: da )2 T - T ) d~'I' 2 = - 3 ( x 27r cos T - cos T (2 l' 1 2

d~1/Ix =

Its minimum is at

T1 =

0,17 and

T2

=~.

u

(AI)

~'I';

=0

=

may be

'

7r

da + 23 1r

!!. - 017 6' 2 (da)2. ( cos 0, 17 - cos "6 7r)

47r

YJ,

and

~'I'2 = 0,001609.

da

REFERENCES (I)

I. Racz, Betrachtungen zu Oberwellenproblemen an Asynchronmotoren bei Stromrichterspeisung, Periodica Polytechnika. Electrical Engineering Vol. 11. No. 1-2. (1967), 29-57.

(2)

S. Halasz - I. Schmidt - S. Wahsh, Additional Losses of Asynchronous Motors Supplied by Forced-Commutatiol) Inverter, International Conference Electrical Machines, Paper I 33- I, Vienna, (1976).

(3)

S. Halasz - I. Schmidt, Kenyszerkommutaci6s inverterrol taplalt aszinkron motoros hajtasok tobbletvesztesegei, Elektrotechnika. Hungary. No. 4 (1976),121-129.

+ d~'I';.

and that of the fundamental component of the inverter output voltage:

7r O.

(2a -~)

05+ 3V3 .

~:

""6 = d(~ J u

=

This function has a minimum if da = - 0,000143 and d~'I'2 = - 0,24' 10- 6. Since the deviation da and d~'I'2 very small, they can be neglected in relation to the value, obtained with the assumption ~1/Ix = o.

As a result of the infinitesimal change da the deviation

dUI

7r

cos -

has to be satisfied. By changing voltage u(t) with a Dirac impulses at times T 1 and T 2 (T 1 < T 2) the deviation of the fundamental component

= cos t

Differentiating equation (AI):

of the flux at t

= -67r{3 -da

In order to keep the flux '1'1' that is, the voltage U1 constant, the time function of u(t) has to be changed,

d~'I'2

~'I'2. x

Adequately modifying voltage achieved, hence:

=

1

The expression of the square of the total flux deviation is:

_.§ a + 05+ 3V3 + 47r 7r '

From '1'1

dU

but the restriction by which the new 'I' ( ~)

CONCLUSIONS

" 6 "6

After integration

costdt).