Electrical parameter estimation considering the ...

Electrical Parameter Estimation Considering the Saturation Effects in Induction Machines Zelia, Myriam Assis Peixoto Paulo Fernando Seixas Department. of Electronic Eiig. aid Telecoiiiiiiunica,t.ioni Department of Electronic Engineering Pontifical Catholic University of Minas Gerais Federal Universit,y of hlinas; Gerais assisznipapucminas.br paulosQcpdee.ufmg br Abstrut 1This paper presents two inethods to estimate the electrical parameters of induction rnuturs t o be applied at the self-commissioniug phasc. Thc first method considcrs t h e operation a t lincar conditions. From Iincar modcl all parameters of t h e machine are estimated close to the rated operating point. After this, the saturation esects are incltided in the riiodel and the inagrietizirig curve of the machiric is obtained. In both casos the cstiniatiou proccdures are based on it& coutinuuus-time modal and is performed using t h e recursive least-squares algoritiirii (RLS). The use of’t h e continuous time models requires t h e calculation of t h e derivatives of rurrent and voltage measured sigiials, which are determined using polynoxriial interpolntion (spliiic-type) tcchniques. To obtain ail the electrical parameters tho equivalent machine coiicept will be defined. In order t o avoid any machine intervention, only no-torque signals are used in the standstill tests. The conditions to produce these uo-torque voltages with tlic PWhl inverter, avoiding ripplc torque genoration, are analyzed. Thc paper includes experimentai resiilts which confirm t h e guod performance of the proposed procedures.

I. I A T R ~ D L C T I O \

iising different niagnetizing currents a i d conhidering the cffccts of rriagnetic saturatioii UII tho machine modcl, the rnagrietizing inductance is estimated. To do this automatically, the induction machine should be fed directly )I:- a P I i~ive~ter. The best coriditioris to excite the machine with voltagv signals that do riot produce clcctroinagiictic torque will be demonstrated. The parameters estimation is performed by the application of t hP induction madiiiie voltagecurrent continuous models a r i d by thc rccur sivc least syuiuc:, algorit hiri 171. In all pioceclures the calculation of the dcrivati\es of current and voltage i n c a w cd signals are obtaiiied using a polynomial interpolation technique[8]. Experiniental rehults demolistrated the good performance of the proposed t3tinmtors.

11. IsDucTlos I f AClIISL X O U E L S A. Lmear Model If the usual approximations are considered. the electrical mttchiric model caii bc supposed Jiiicur ;lud is defiricd by the

Nowadays, the higher industrial qualit! standards requirrs improved contiol systems perfor1iuuic.r that led to followiiig equations: the dcsigri of ucl\.rtnccd drive systems using induction machine.; due t o its advaritages as robust coristruction and lo\\ cost. Hon er. coinplex structures of control are required UIIW the iricluctioii iriadiirie is rcprcsentecl by a ~ i o ~ i - l i ~ i ~ t t i rrioclel a d depeiideiit of its time dependent electrical parumeters. In this context. the knowledge of the iuduction motor parameters is an important task in the design and irriI.’lerr!~Iitationo€ otiservers arid controllers that compose these diivc systems [l]. where subscripts s, T refer to the stator aid rotor rir51ariy papers have beexi published on induction rriachiiie cuits. z arid z’ arc the dq0 curlerit aid voltage wctor r e yaiaziieter estixriatioii [2][3] 151[e]. Soixie of these re- spectively, R ia resistance aiid w T is the lotor speed. L,, is searches are concerned oii the parametel estimation that the static inductance (chord slope), neglecting satumtion occurs before the motor start-up mcl are used for ini- effects. From expression (1).the rotor current vector may tial tuning of the control system, called self-commihsioning be elirriiriated and the voltage-current model is ohtainied a b phase. I41: Uhually. the induction machine inoclel is considered inmliaut with tcrnpcratulc?, excitatioii levels. ctc... Nevertheless, the resistances of stator and rotor windings depend on temperature .iyhile inductaiccs clepeiid ou niagiietic satwith: umtion [6]. This paper deals with an autoniatic identification proceduie that enables the determination of all electrical parameters during the S e ~ f - ~ U i I ~ r i i s s iphase. o r i ~ Firstly, the tests i*le performed close to the nominal excitation coriditioii arid neglecting all iriodel non-linearities. Suhseyueritly,

[4

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where pm is the angle behveeri the magnetizing current and d axis, p = d d t mid. ~ ~ = ,~ dd c o p, s ~ i L,,, sin2p m Lm, = Ld sin2 p17,+ L, COS' p,,, Ldq = ( L d - L,,,) sin pill cos pm. cross-coupling iuductance between dq axes; L,d = Lls L,,a ancl L,, = L1, -I-Lr,lq;self-iuductiuiccs of the stator windiiig alorig the dq axes: L7d = 4,. Lllld and L,, = LL, -I- L,,,,,; self-incluctarices of the rotor in the dq axes, respectively. It can be noted that as a result of saturation effects Lsd # Lq. L r d # L,q aid that Appears a Ilew Col!ll.'oIlellt of inductance that represents the coupling between d arid y axes of the stator arid rotor in the statioriary refererice fr ane. The expressions for the electromagnetic torque are the same as in the case of the unsaturated macliiiie, since saturation docs iiot intruduw IICW tcxms ill the expressioii for the elcctrornagnetic toxquc and the itssuniptions in the linear model hold.

-+ +

where a and b are defined as a = l / m L , . b = l/vL,. with L,/ R r aid 0 ( L J , - L:n) /L,L, . The electromagiietic torque. iiivariarit with the reference frauic, is given by: T, =

where P is tlie slumber of poles. Iln iriclicates the complex --f imaginar? part and 2: is the complex conjugate of rotor current vecto~.

B. Inclndmg the E8ect.s of Main Flux Saturataon As a iesult of saturation. the linear equations are niodificd and the inagiletizing inductance. stator ancl rotor induct axices arc iio more constaxit. The lealrrtge inductances caii he considered coilstarit while L,, varies xith saturatioii. It is assuniecl that the magictizing inductance is a non-lineal hiniction of the magnetizing CLKI erit I,,,: thus Lin = Lm ( ( ( m1). It caii bc dcinosistrtited that[Y][lU]:

where

Ld

=

C. Equivaleirl: R;lachi,nes The determination of all t.he physical paranieters present, in equations (1)(5)is not possible frorn the coefticierits estimated with eyi.iation (2). The iiiodel is over-parameterized. Neilertheless, it is possible to ovexcorne this drawback by defining electrical equivalent machines. Ths inethod is valid for both models m d mill be illustrated to the first case. X!Iultiplying both sides of equation (1) by an arbitrary constarit h,mnand replacing the rotor current by i:. = i,/m: e q u a t h i ( 6 ) is obtained.

& lpnl/ is a dyiiztmic inductmice aiid

- L n = !dis the btatic (chord slope) inductance. Undei liiicar magiittic coiiditiow the dyiiruilic. iiiductaiice is equal to the static niagric~tizinginductance ( L n z .) Figure (1) presents the space phasor dagrani showing magnetizing curreiit in relation to the statioIiary reference frarne. The equations in tlic stator rcfereiice franne caii be obtained its:

$-77LL,n

(dz-

1, - .?W;)

The expressioiis giveii by (6) define exactly tlie same electrical machine ciescribed by (I) for auy d u e of 772. nz # 0 arid # x' [Ill. The stator voltage a i d curelit, the stator pararneters, the clectroxnagiietic torque arid rotor losses ale invariant t o the trailsfonnation of the rotor variables. So, a machine with parameters R,. L,, R, , L,ard L, is electiicall? equivalent to the rnachine with paraIrieteis R,.

L,. ui2R,. rdL,.. tuicl mL,,,. The choice of 711 coircsporicls exactly to xtbc an arbitrary value for the relation between rotor arid stator ledage inductances like in the classical tests, i.e. the standard noload arid the locked-rutor tests. For example, if m is set q u a l to . the stator self-indudaiice will be equal = 1. A relatioiito the rotor self-inductance arid k = ship between Lb arid LL, must he a m m e d . Wheu desigli details me riot available the ratio k must be assumed ac cordiiig to the design type [E]:

,/m

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e

TABLE I COMMA3D SIGNALS. OCTPCT VOLTAGES A N D VECTORS IN FWIf INVERTERS CONNECTED T O

l o l o l 0 l o

Y

LOAD

I o I o I o I o IC1

In all regular PWRl methods, the reference voltage vector is sampled at each modulation period, T; and the pulse widths are then calculated to make the mean voltage vector at the inverter output equal to this reference value. Computing the mean wltage vector v d q s (I;) =

3 (G,,

(k)

+ a ;bs

(I;)

+ cz*

ohtairi: 7dqa

G ~ ,( E ; ) )

from equation (9) we

T Cdqs ( k ) E

(A) = -

4

(10)

wl-hcre 7-dq,q( k ) = (7-1 ( k )$. up2 ( k ) -t- u273 ( k ) ) is defined as the pulse width vector. It c m be noted that while the voltage vector hab null zero sequence coniporieelit. the pulse width wrtur must have, always:

k = 2 dt-3ign A. D a i d wouiid rotor. k = 0.67 dcsigri B. k = 0.43 design C. In the est iinatiori process to be present4 the following yroceduie will he adopted: Firstly. the X: ratio is assumed equal to oiic that 111caiisL, = Lp. With this itssurnption. all the physical parameters call be calculated fi.0~11the estimated coefficients of equation (1). The parameters of debigii B and C motors can be found by iui appropriate tiaiisfoimation. The value of rri for this tiailsforination is gi\rn by relation (7):

( k - I) L;,?& in =

-

1)L p 4- 4k ( L y

"kLf

Thv elect~icalparameters are then deterrriined by: L: = m2L;. Ltn = Lt,,.Hp = r1L2R;

Applying the inverse transformation to the pulse width vector. ivc finally obtain:

{

71

(k)=

( k ) -t- 70

2% +

72 ( k )= (k) 70 7 3 ( h ) = F V C S ( k ) 4-7 0

(12)

where 0 7 L( k ) T . z = 1.2.3. . . . This cxpicssion is general. valid for all iegula sampled PW31 methods. A particular PIVN method cmi he obtained l1?7 the appioyriate choice of 7 0 ( k ) . We ciui KKXV concludc in our study of nutorque voltage signals. From table (1) the condition 'I'd, = 0 is fulfilled o d y + -2 ( 7 ) by the applicatiori of zero voltage vectors V 0 and T/ 7 . So. thih condition can not bo achievecl with a wve connected diarge iii tu1 instaiitaneous scnsc. -Uevexthclcss, this coxiditiou call be accomplished in a mean sense by making:

(8)

In these equations the superscript means the actual value of k.

Ill. No- 1C)IKjL-L.

\'C)LIAC:L SI

The machine ~iiodelat standstill is obtained b> iriakirig = 0 at (2) arid (5) that implies in the natural decouplirig In a regular sampled P\YlI method as described by (12). of the d and q circuits. By inspection of expression (3). the .(I ( k ) is tuiy desired voltttgc wavcfurni. sufficieiit conditioris to achieve zero torque are uds = U for The output mean torque will he eqiiaf to zero but some any uys.or z9$= a for any ?'&. hi this section, the s? ntliesis torque oscillations will exist. From table (I) the con& of such no-torque signals with a thrtwphasc PWM voltage tion wuqs = U is fulfilled by the applicatiou of thc vcctors --t iiirw ter is rtnal-vmd. Table I shows the co:olniriaud signals, - + + e 1'0. V 1, t 4 and V 7 . The condition zjqs = U implies: output phase voltages and thc respcctivc d, q components of each ioltage vector of the inverter feeding a wye coimected load. Titblc (I) ail expression relatirig the cwrirriarid s i g d s pulse widths 7 1 ( k ) , 7-2 ( k ) , 7 3 ( k ) mid the meal1 value of U),.

-

-

the phase voltages Tab (k), ubs (k) , LI,, (A) . dwing the I t h PTYM period is found as:

-1

-1

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By substituting tkiesc plwe voitagc \.'itlu~ujat expression (12) we obtaiii 7-2 ( k ) = 7-3 ( k ) , arid for a centered pulse PWM this also means c2 ( k ) = C j ( k ) . Looking for this coriditioxi at table (I) w e observe that the o d y vcctors that 3 4 3 uill be used by the PWLl converter are exactly V 0. 17 1, V 4

156

4

arid V 7 . So. tlie PWII modulator assweb iristtuitarieously the condition uqs = U with a AYC coriiiectecl chaigc. Tlic same arialysis call be carried out to a delta coniiected load, showing that in this case the condition wds = 0 is assured instantanmrusly by the PWM converter.

Iv.

EbTlAIATIOl PROVEDL RES

Fiom this equation, it is possible to deterrrlirie a current\-oltage model similar to (2). The current/voltage model under saturation is presented to the q axis. The expression to the d axis ha\ equuvdmt coefficients provided tlie change of tlw subscripts y by d. So. coiisideriiig = Ll, = Li the current/\ oltagc model includiiig saturation effcctb will be @ m i l bj :

A . D a w t Estmation frons Itnear model The linear regression inoclel of the induction machine at standstill is obtained taking w7=0 iri(2) ancl building the inodcl in the parairictcr liiicm form. MTith uds= 0, it ciui he writteii as: Y

=

r

=

8

=

(15)

%IS

[ [

61

[

1 (T7,Lr

Llqs

GqS

e,

-& ]

-tqS

#,

84

1

]T = 1

aL,

UT,T,

1'

Considering the thrwry prcseiitcd in the subsection (11-C) arid taking the estimated coeffkieiits from (15). dl the pa1 ameteis of the induction machiric. cmi be deteirrrirled as: f, =

82 -

(16)

61 r

e,

R, = 7 01

., e,e, is= Lf =

-e,& B:

1, = L1 (Ll . I ZL,,):

(20)

Applying (7) and (8) the values can he adapted to specific desigri class rnacliixic.

(24)

ai1

B. Paramete?. Estiriidioari.frduding Saturation Eflwts It follows from figure (1)and equation (5) that if pTn= 0 or ,uVn= 5, the cross-saturation coupli~igdocs riot cxist (L,a, = 0). This corresponds to the situations wliere only

(25)

the direct or cpadratrue t i x i s winding is excited, respect,ively [IO]. In addition, if the motor is operating at standst.ill (z+ = 0)! the inductioii nmhine moclul (5) will he given 1JJ' [13]:

(26)

(27) The expression, 1, :call be determined by:

The dyrianlic hductaiice is then calculated as: 0-7803-5692-6/00/S10.00(c) 2000 IEEE

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TABLE I1 ESTlblATED PARAMETERS 14 1TH 31ACHINE AT ST.4NDSTILL

Parameter

K,. -(!a \ ,

The dyriarriic inductance cmi be directly calculated from equation (27) if the leakage iiidiictame is luio~vn[14]. Then, tlie rriagrictizing flu iuid the static iiiductaiice crui he calculated by:

R,. (i2j L, ( H ) Lr ( H )

I I

Classical tests 3.416 5.643 0.302 0.307

I Eq(15) I

3.898 2.379 (1.316 0.32s

(30)

Ym =

V. DERIVATIVEE511iUATIOh

I'KOCEDU~~L:

Usually, the clerixatives a ~ estimated e using filters or classical diffeieiitial approaches. This M oik uses a polynomial iriteipolatiori mczthod [SI.This method fits a polwomiitl to the data windows of In points usirig staiidard least square teclmiques. Such data correspond to the time signals of which one should estimate sticcessive derivatives. Tlie indcpcridcrit vitriablc of the fittccl polynouiial is tirnc. After fitting the polyniornial fuiiction. oiie call detciirliiic the dcrivati-\e5. Assuming that the delivatives of the signal y(t) are ~ e quired at saniple k. the following window of data is choseri:

where subscripts indicate the sarriples. Subscqucntly a polyl~lilialof the type: The iiiachiric riiodcl (1)was shridatcd using both parui(33) eters sets presented in table (11) aiid were compared with actual measured stator cunent. Figure (2) shows the reis fittccl to thc data window. Findy. the derivatives at sults obtaiiied b57 rriodcl (15) wlicrc a diffcrciit excitation point y ( k ) are determined by differentiation of this poly- level was used but closc to the rated conditions. mmial. This method was applied to derivative calculation It can bc observed that tlic traiisiciit aiid steady state of the voltagcs ruid currcsits. responses using the estiiiiated parameters are closer to the actual currents. VI. EXPERIAIESTAL RE~L-LTS

+X

g ( t )= X c n t f *

n - p

+ ..'+

)clt

-tX"

The experiinextal set-up is composed by a delta coilnected cage indtiction motor. a tlirw-phase IGBT inverter tuid a Pcritium/300 microcomputcr. The test voltages n'erc produced by a regular sampling PWlI with switching f r e picncy equal to IOkHh. The sampling pcriod was loops. Two currents and two voltages were measured using 12 bits A/D converterb.

A. Estzrnataon from Lziaear Model The voltage signals used wcrc step type and set closc to the rated excitation condition. The experinimt was repeated ten times. The estiinated final vdues are the inem of all thcsc cstimatioiis. Table (11) shows tlic ohtnirid results. 111 the secoiid colu m n arc the values obtained with the standard no-load and lucked-roto1 tests while the third colmnn presents the estimated results from the direct estimation model (15). The classical results were obtained uxider specific conditions in accordance with [E?]. 0-7803-5692-6/00/'S10.00(c) 2000 IEEE

B. Estzrnatton zncli~dinySaturatzon Ejfects From the expressions (24) it is possible to determine the machine inductances mider saturation conditions. For this purpose, the motor was fed with different cuurent lexcls rarigirig fivrri zero to 2.5 times thc rated xilagnetidrig current. The estimated dynamic iriductarice using (29) is shown in the figure(3). Tlie static inductance obtained frurii nu-load tests is Iircwiited. too. Figue (4) preberits the validation results w~klchw e r ~perforiricd using the estimated parairictcrs rind thc d u e s obtained froin classical ric-load and locked iotor tests. It can be observed that there are no dif3erenc.e hetween the CUTrcrits obtained usirig estimated pararnctcrs arid the actual value. VII. coscLLJlovs The application of the continuous time xniodel simplified the estimation of the induction motor parameters a i d the

1567

0

4

1

.

.

,

,

.

,

recovering of the physical paraK1leteI'S. The preSeKlt.Cd~ I G cedues arc adequate to the seli-commissioning phase siiicc no adclitional hardware or machine intervention are needed, enabling its iiiclusioii into tlie drive system control software. The conditions to feed the rnacline without torque production were analyzed. It was demonstrated that the vdtage vector compoucrit to be usccl is. a fuicticxi of tlic loitd coimection. With this choice it is possible to minimize torque oscillatioiis during standstill tests. Through the clefinitiori of eyuideiit. rnachines it was possible to recoi'er all the machine parameters in all proposed pocedures. The dynamic inductance knowledge as a function of the irmgiietizing current c m be used into adjustable flux systems to iinprove rriachinc efficiency.

REFEREXES C h / ~ . k odf Elect?-/c~d D.ril!r.s. 19%. (11 W.Lcoii1ii1~1~ [2] .I. Gudlicrscii: P.Tliup,crscn, iiud R1. 'I'oniics. "A pructic;il itloiiti-

fiiwtioii sc:lir?rrie h ~ iiitlucticiii r inotors i i t statidstill itsiiig oiil: a

LS

tiiiltoi-," iu EPB Y 1 - ,7?1r Ewop U 11 PO I l ' t f ' Ell?f 7'0 jC.5 Id71 d A plJ'IbC:lL/d 0 / I S , ( ' ~ - ~ U l l d h ])I). 3370 %%j.i, St'Jlt, 7-9 l ! ) Y i . [3] hi. Bcrtoluzzo, G . S.Buja, ;uid R.. Meiiis, "Iuvcrtar voltiago dropfree recursive least squares p"i(!t m identificaticin of a 1)wni

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