Loss Profiles*

Copyright © IFAC Control in Power Electronics and Electri cal Drives, Lausanne , Switzerland . 1983

PULSE WIDTH MODULATED (PWM) INVERTERS FOR EFFICIENCY OPTIMAL CONTROL OF AC DRIVES SWITCHING ANGLES AND EFFICIENCY ILOSS PROFILES F. C. Zach and F. A. Thiel Institut fur AUg . Elektrotechnik und Elektronik - Power Electronics, University of Technology, A-1040 Vienna, Austria·

Abstract. Solutions are given for efficiency optimal PWM (pulse width modulation) inverter operation for AC motor drives. Exact motor models are used. There, e. g., temperature and rotor skin effects can be considered. The exact solutions are compared to solutions found based on simpl ified motor models and to the selected harmonic el imination (SHE) method. It is shown that the PWM switching pattern of the efficiency optimal control (EOC) is nearly insensitive to motor parameters. (Absolute loss val ues are, of course, very much dependent on motor parameters. ) Therefore this control method is equally easy to apply as, e. g., SHE. Also, further investigations based on other literature indicate that also motor dynamics can be improved in comparison to el imination of only specific harmonics with SHE. Keywords. Electrical drives; inverters; pulse width modulation; optimal control; efficiency optimization.

INTRODUCTION

NM PM p R,S,T SHE

Various forms of PMW techniques for inverters (Fig. I) have been proposed. El imination of selected harmonics (SHE) [I, 2] and the subharmonic oscillation method have been treated for many years. Only recently more interest has been on efficiency optimization of inverter drives [10, 12 - 14].

s* ~d UROl U(=Urel) un (Xi

Itl

negative modulation positive modulation number of pole pairs motor phases selected harmonic el imination sign (+ 1 for NM, -I for PM) DC supply voltage peak value of uRO-fundamental normalized OROI normalized voltage harmonics switching angles overall drive efficiency

LIST OF VARIABLES AND ABBREVIATIONS A, B, C, D, E EOC f in L MEC

VOL T AGE PATTERNS

denomination of different minima of drive loss efficiency optimal control

The basic voltage pattern is shown in Fig. 2. The number of switching angles o(i should not be too low in order to give enough degrees of freedom for optimization purposes.

inverter output (=motor supply vol tage) frequency current harmonics (amplitudes) motor representation by inductance only motor equivalent circuit

*

The authors are very much indebted to the Austrian "Fonds zur Foerderung der wissenschaftl ichen Forschung" for the generous support of this project. 231

One degree of freedom is needed for adjusting the voltage fundamental. Three [)(i are treated in this paper where two o(i are discussed as special case. More switching angles will increase inverter switching losses. E. g., let us consider a 50 Hz base frequency for a thyristor inverter in an 11 kW drive. There the losses of a pat-

F. C. Zach and F. A. Thiel

232

tern according to Fig. 2 might be in the same order of magni tude as the motor losses. Then increasing the number of switching angles wi II considerably increase the inver-

ing frequency. However, at higher frequencies the voltage harmonics un will result in lower current harmonics due to the motor reactance. There voltage control ac-

ter losses. How an increase of the !Xi can decrease the losses will be treated in [5). (For low base frequencies a current control

cording to Fig. 2 is preferable due to switching loss I imitations.)

scheme might be of interest. There a sinusoidal current reference is followed rather

Two different modulation forms can be defined: at first - for giving an equal basis - we assume that the vol tage fundamental in both

ciosely, but at the expense of higher switch-

cases has to be positive in the first half period. As can be seen from Fig. 2, the modulation pattern can start with a posi ti ve or negative pulse at Cl( =wt=O. We define negative modulation for negative pulse at rr/ 2 (and therefore - due to three C( i - pos it ive pulse at 0(=0) and vice versa. Fourier analysis yields (4/rr )(Ui 2 )( 1-2cOS()(j+ 2cos (){2. A + 2cos C(3 ) j I=s* (4/11' )(Ui 2 )( 1-2cos 0(1+ U RS 2cosC(z-2coscx.s)' Normalization yields u=s * (1-2cos 0( + 2cos 0( -2cos 0(,).

URO I=s*

(I )

V3

Fig.

la. Basic structure of inverter drivesj Si: switches (Fig. lb)j ACM: alternating current machine b. Realization of one switch by transistors Qj OF: freewheel ing diodesj G: triggering signals

URo

8g8~.:..:.

----

"111

11 I

~h Ut

--

-- 1""_ ,

~'N

IT . --

"

(3)

A.

s* has to be sel ected such that URO I be comes positive according to the definition made. This of course is in conformity with the Fourier analysis: st" = I for NM (positive pattern at t=O), s* = -I for PM. un=s* (1-2cosnol + 2cosno
~Ir-

- - ---

2.

1

(2)

wt

RH ~ 5

URS

n

-

,

,,

--

.-

,

"

-

a

.8 0.6 kX 0.4 URS .-f"'- - -I-

"' ....

"

,

-I-

I--

"" "

-

-

'--

Fig. 2. PWM voltage patterns: a, b, c for NM, d, e for PM. Shown are the special cases of u=I/2 with
2

3

f (kHz)

J,

b

Fig. 3a. Motor equival ent ci rcui t(s) MEC I: all parameters constant MEC2: L2d=L20' kx(f), R 2 = R20. k R(f), RH=RHO. (fO/f)O. 8 sn: slip for nth harmonic b. Coefficients kx(f), kR(f) representing rotor skin effects

Pulse Width Modulated Inverters the line to line voltages all harmonics are 0 for n=3k; k=l, 2, 3, ••• If two different un are set to 0, the two degrees of freedom discussed can be used to achieve this purpose. E. g., el imination of the 5 th and 7th harmonic of the motor I ine to th line voltage will eliminate the 6 order motor momentum. However, due to the sharp increase of 11 th and 13 th order vol tages, the 12th order momentum is highly increased. A thorough analysis (4) has shown that other solutions will improve motor dynamics. This and especially the need of energy conservation, furthermore the need of battery rating improvement for electric vehicles, has made efficiency optimization of the drives interesting and important.

233

SOLUTIONS As can be seen from Figs. 4 and 5 (and 9 and 10), solutions vary with u. For almost every u five pronounced optima can be found, two for NM, three for PM. The representation of losses is chosen such that the lowest loss value for a specific u is given 0 points, the highest is represented by 100. The scale is chosen separately for PM and NM such that 0 points in the PM profiles gives anoth e r (usually higher) loss than 0 points for NM. Figs. 4 and 5 show the trajectories of equal point ratings. The trajectories ha ve been determined for different motors (e. g. one for 1.1 kW, p=2, another for 11 kW, p=8). Also, the trajectories have been calculated for MEC 1 as well as for MEC2. Differencies in trajectories between the different motors are negl igible, differences bet-

MOTOR MODEL For three O(i first attempts to find efficiency optimal control (EOC) were made based on a motor model us i ng pure inductance in (8). Also other optimization criteria for a motor model using only an R-L series circuit with constant parameters have been presented in [9], but consistency with [8] is not evident. Here, a motor model is used according to

ween use of MECl and 2 are small and will be shown in Figs. 7,8 and 11. The area of possible solutions is I imited by C(1max, s* 2= 1 from Eq. (3): Wi th s* 2= 1 from Eq. (3): cos ()(.1=-s* u/2+ 1/ 2+ cos 0(2-cOS ~3. (5) Since ()(3'" ()(2, and 0"'CX 2 , 0i.3"'Tr/ 2, we have cos ()(3~coscx.2. Therefore, ()(1 becomes a maximum for ci2= C(3: cos ()(1= 1/2-st.' u / 2. (6) For NM (s t" = 1):

Fig. 3a. Dependencies on temperature are accounted for by varying the resistances. For rotor skin effects see Fig. 3b.

O~cos 0(1~1/2 and 90 0 ",
For PM (s'~ =-1): + cos ()(1=u/2+ 1/ 2, 600~ O(lmax~Oo. For ()(20max (Fig. 6) with cxl=O:

EFFICIENCY OPTIMIZATION METHOD

cos 0<.20= s t" u / 2+ 1/ 2+ cos cx3 giving a maximum for ()(20 at 0i.3= Tr/ 2: COScX20max=s':< u / 2+ 1/ 2;

The task is now to provide the Fourier analysis for the voltage patterns of Fig. 2 according to Eq. (4) and to calculate resistances and reactances according to Figs. 2a and b for obtaining the current harmonics in the various legs of the MECs. This analysis and the calculation of the losses according to the MECs and the computer program are omitted here for brevity. The optimization of the ()(i has been performed by a search over the entire region of possible c(i and by a steepest descent method [7,11). The reason for applying the former method is that the latter method provides a multitude of local optima (as well as many inval id solutions). Therefore one has to carefully check the results either by comparing all local optima or, more exact, as done here, by searching the whol e region. For 0(3 see Eq. (3). O(j 6 0lj!6 has to be observed. On the other hand,

OS

searching all possibilities takes too much computer time, wherefore steepest descent methods have been used a I so.

(7)

0-=Oi.ZO max"'60 0 for NM 60o.:CX20max-=900 for PM. Further analysis provides for any gi ve n u: + +CX 1max= o(20max and C(lmax=()(20max. Five pronounced loss minima can be seen in Figs. 4 and 5. Comparing the losses shows that up to u=O. 8 solution E provides the absolute minimum (the notation of Fig. 6a is used). Solution D yields the optimum for 0.8"'u'"'l. Trajectories vary slightly for using MECI or L, but even much less for changing from one motor to another. AI though absol ute loss values are very sensitive to parameter changes, locations of the minima do not change much. This is especially true for the most important solutions D, E for 0"'-u"'0.9. For the effects remaining see Fig. 11. It can now be seen that choosing only two C(i (equivalent to (:(1=0; Fig. 2) yields loss values of :;'10 points above the minimum. Pure square waves correspond to 0(1=0(2. Trajectories A, B, C (for PM) in Fig. 6b are relatively sen-

234

F. C. Zach and F. A. Thiel

Fig. 4. Equal loss trajectories for a: u=O.5, b : u=O.8, calculated based on exact motor representation (MEC2). Locations of minima: t:. L, + MEC I, o MEC2, with three different local solutions A, B, C

a

b

Fig. 6. Locations of the minima (traj e c (below) tory E gives ab s olute minimum for O~u~O. 8, 0 for O. 8~u~ I) L: based on pur e inductance as MEC [8]j MECI, 2: traj e ctori e s based o n MECs according to F i g. 3j SHE / A: s e lected harmonic elimination accord· ing to [1,2, 3)j SHE / B: n e w solution for selected harmonic el imination [5] j _. _. _: boundari e s of r e gions of po s -

o

sible solutions for different Uj a: negative modulationj b: posi t i ve modulation

9Ot------

90,---~r_---~~---~

" ' / ../ I //' I 1//

.8 .7

. 5., .4.

60

--

u=O.!

t ClC.2

60 ,

()(2

u=0 .5

:1.3 _ -.1

o

MEC 2 . 9

/1 ../

A

__

/ '/

u=0.1 1

I

/

I I

___

J u, 0.9

1

I

u= 0.5

I

)

I

I

I

L ••9 MECt

/

/

I / 1/ /

(deg)

(deg)

30

30

SHE/A

/

/

./

u·0 .9

.--

/

/"

0

/

a

/

0

(Xl

20

(creQi 60

b

8'0

(deg)

30

0(1- 60

Pulse Width Modulated Inverters

235

0(10(190~O____~10_____2~0____3~0_____4rO____5,0__~-.60 9o r o ----~ 10~--~ 2r O --~3TO----~40

80

A

L

+ o

MEC 1 MEC 2

c

*

SHE/A 5 HE/B 701-----+~~hH~---4~-+~

70

50~~~~+---~~7Y7L-f

t

0(2 'Or---~~~~~~~~~-f

0(1-

10

o c

Fig. 5. As Fig. 4 but for positive modulation; 0 points corresponds to a usually higher loss (lower efficiency) value than in Fig. 4; also, 100 points here gives higher loss than 100 points in Fig. 4. a: u=O. 1; b: u=O. 5; c: u=0.9 AL, +MEC1, 0 MEC2, 0 SHE / A,

* SHE/ B

F. C. Zach and F. A. Thiel

236

sltlve to the motor model (see also Fig.

11).

90

Although A is the best solution for PM, C is much less sensitive to nonexact O(i and is almost as good as A, especially for u"'O. 3. SHE-solutions exist only for PM. Solutions SHE/A have been discussed many times in

6

the literature, e.g. [1,2,3]. A second, probably new solution (SHE/B) has been found in [5]. It comes close to relative minimum A. Figs. 7 and 8 show the solutions for the (Xi ( cx 1, D<.2 as in Figs. 6a, b). Figs. 9 and 10 show the losses for u=O. 5. Finally, Fig. 11

L IJ

I

MEC1

MEC 2 SHE!S

~

t~

a

30

shows the errors made by using nonexact motor models. E. g., if MECl is used in place of MEC2, the optimum would be expected at C<1=66. 38~ CX2=76.44~ CX3=85. Ig0for u=O. 5 with the real optimum (according to MEC2) being at (66. 16~ 76. 20~ 85.16°). This means that the loss calculated based on MEC2 for

0 0

0.5

u

the cx i gained from MECl will be 0.037% too high (see Fig. 11). The results obtained

L ,MEC1

verter AC motor drive. The resul ts are highly consistent with the ones found here. Details will be presented in [6].

b

MEC2

have al so been checked by actual measurement and on-I ine optimization of a PWM in-

-

90

SHE! A

60

,

...........

---

Cl IJ ~

t~ 30

'"

60

-,.. - ~ =-:-: ~}for

L,MEC1

90

for MEC2

-

60", '+, ,

L + MEC1 o MEC2

C

tg 30 Fig.7. Optimal Dl.i according to trajectories D and E in Fig. 6a for NM based on di fferent motor model s. The absol ute opt imum changes from one set of solut i ons Dl. i to another set at u=0.8

O~"-'""'.~T'~:. :~=-:~·-~_ I~_.·-~ _·~ _~·~;-_~ ,_- _':~ _~___-:_: -'~-: _____:- ~_o_-_._._.o_'-'---'T"-'---'T~-'--,o

0.5

u -

Fig. 8a. Optimal O(ifor PM based on different motor model s corresponding to relative minimum A of Fig. 6b. b. corresponding to relative minimum B c. corresponding to relative minimum C (this solution disappears for u~O. 3 for using L)

237

Pulse Width Modulated Inverters

loss

Fig. 9. Losses versus switching angles 0( 1,0(2 (for u=O. 5 as in F i g. 4a) for NM (see also th e Appendix)

Fig. 10. Losses versus switching angles 0(1, CX2 (for u=O. 5 as in Fig. Sb) for PM (see also the Appendix)

100 2

+ MECl 11 L

o SHE/A

t ?,. ,: 't

t: (

CONCLUSIONS

-If: 5 HE! B

p. .. . 0 .

0,

Fi g. 11. Errors due to use of nonexact motor model s. Error assumed 0 for using MEC 2. Higher values for MEC 1 and L for 0.8"'u"'"1 for considering solutions D (absolute minimum in this region). Lower values for only considering solution E. Note the relatively large values (large scale) for SHE

It has been shown that various local minima exist for the losses in AC motor drives using PWM inverters. The locations of the absolute minima (not the loss values, of course) are highly independent of the motor parameters and of the exactness of the motor models used. This means that (as for selected harmonic el imination method) the efficiency opt i mal switching angles only depend on the vo I tage I evel to be adj us ted. The sens i t i vity of the losses to errors in the switching angles is rather small. This makes the efficiency optimal switching pattern very attractive for practical appl ications. Furthermore, investigations in (4] indicate that this solution is also better for motor dynamics than SHE. Another paper will show a much larger number of switching angles per quarter period can further reduce the losses. This wi II provide a basis for making the necessary tradeoffs due to the then higher inverter switching losses.

F. C. Zach and F . A. Thie l

238

REFERENCES (1)

[2]

[3]

[4]

(5]

(6)

[7]

[8)

[91

Turnbull, F.G. (1964). S e l e ct e d h a rm o nic r e ducti on i n s ta ti c D - C A-C i n ve rt e r s . IEEE Tran s . C o mmun. & El e ctro-

(10) Zach, F. C., R. J. B e rth o l d an d K. H. Ka ise r (1982). G e n e r a l p urpo s e m i cr o pr o c es s o r m o dul a t o r f or a wi d e rang e o f

n i c s ,374-378. Patel, H. S., and R. G. H o ft (1973, 1974). General iz e d t e chniqu e s of harmonic el i minati o n ••• IEEE Tran s ., IA - 9, 310317, IA-l0,666-673. Jack son, S. P. ( 1970). Mult i ple pulse m o dulat i on in s tati c inve rt e r s ••• IEEE Tran s ., IGA - 6, 357-360. Ch i n, T.H., andH. T o m i ta(1981). Th e princ i pl es o f e l i minati o n pul s ating t o r qu e ••• IEEE T r an s . , IA- 17, 160- 166. Zach, F. C., and H. Ertl (1983). Effic i e ncy optimal c o ntro l ••• (t o appear s oon). Zach, F. C., R. J. B e rth o ld, K. H. Kais e r and E.D. T o puz o glu (1983). Aut o matic o n-I i ne o pt i mi z ati o n o f micro pro c e s s or contr o ll e d AC m o t o r dr i v es . (t o appear so o n). Lev enberg, K. (1944). A method for th e s o lution of c er tain n o n-lin e ar pro bl e m s

PWM techn i qu es for AC m o t o r co ntro l. C o nf. Rec. 1982 Annual M ee ting IEEE Ind. Appl. Soc.,44 6 -451. [11] Marquardt , D. W. (1963). An algori thm for least-squares e stimati o n o f nonlin e ar

..!...!.J£i.

parameters. J. SIAM, [12] Murphy , J.M.D., an d V.B. H o n s ing e r (1982). Effici e n cy o ptimizat io n o f i n ver ter-fed induct i on m o t o r dri ves . C o nf. R e c. IAS Annual M ee t i ng 1982, 544-552. [13) H o nsinger, V. B. ( 1980). In d uct io n m o t o r o perating fr o m inve rter s . C o nf. R e c. 1980 Annual M ee ting IEEE Ind. Appl. S o c., 1276-1285. [14) B o s e , B.K. ( 1981). A d ju s tabl e s p ee d AC drive s ys t e ms. IEEE Pre ss and Wile y .

APPENDIX As exampl e s for abs o lut e l oss va lu es f o r U d =110V w e pre s ent th e f o ll ow ing cases:

i n l e ast squares . Quart. Appl. Math.,

a. M o tor Ml llkW (Rl=0.45Q ,R~O=O. 1046 Q,

~,

R HO =100Q, L1d = 0.0007321H, ~2o'O = 0.0007321H, L H = 0.003422H, S I =5%) b. Motor M2 1.lkW (R l = 8.02Q, R20=4.49Q, R HO =800Q, Lld =0. 0204H, ~2o' 0=0. 0306H, LH=5,lH, sl=8%) Approximat e l y c o nstant flu x o p e r a ti o n i s assumed [10], f o r Ml: f=u·100H z (fo r f up t o 100Hz ), f o r M2: f=u ·37. 5Hz (for f up t o 37. 5H z ). Th e r e sults ar e gi ve n in Tabl es 1 a nd 2:

164 - 168.

Buja, G. S., and G. B. Indri (1977). Op timal pulse w idth m o dulation for feeding AC motors. IEEE Trans., IA-13, 3844. Casteel, J. B., and R. G. Hoft (1978). Optimum PWM wa ve form s o f a microproc e ssor c o ntroll e d inve rt er . Proce e d i ngs PESC, 243-250.

u

(;)/1

C(2

M1 0(3

M2

ex l

los s/kW

Ci

C)(2

3

l oss/kW

min 61 79 81.03 61. 12 O. 105 * 2.583 80.56 82.47 1--- - - -- - ---- ---0 ma x 0 35 106.491 6.034 72.89 35 72.89 66 76 85. 11 min 6.663 66. 16 76.20 ~5~6_ ~207 t;t; ! -- 70.59 - 0.5 0 31 31 ma x 19.337 0 70.59 1.352 7 min 12 87.96 7. 157 6. 18 _1~6§... ~7!:!9_ ~2~~ -- -- -0.9 - - - 18 0 64.78 0 18 ma x 8.544 64.78 0.335

O. 1

-

-

- -

- ---

-

--

-

-

Table 1: Soluti o ns for selected ca s es u s ing MEC2; e qu ival e nt numb e r s f o r SHE: t; SHE / A:O. 155; SHE / B:O. 140;*-SHE/ A:0. 323; SHE / B:O. 237;

Cl

u

O. 1 0.5 0.9

+ SHE/

A:O. 263;SHE / B : 0. 226

I

M2 loss / kW

l

min

61

79

81.03

max min

0 66

35 76

72.89 85.11

min

8

13

88.05

max

o

18

64.78

---- - --max 0 31 70. 59

5.896

Dc'1

66.38

- ;1. 853 0

Table 2: As Table 1, but use of MEC1.

()(2

()(3

10ss/kw l

76.44

85.19

0.095

18

64.78

0.257

-31 - 70:59- 0:737 -