Digital Implementation of Full-Order Flux Observers for Induction Motors

Transcription

1 Digital Implementation of Full-Order Flux Oberver for Induction Motor Marko Hinkkanen and Jorma Luomi HELSINKI UNIVERSITY OF TECHNOLOGY Power Electronic Laboratory P.O. Box 3, FIN-5 HUT, Finland Keyword Variable peed drive, induction motor, field oriented control, flux model. Abtract Thi paper deal with flux etimation for induction motor drive by uing a full-order flux oberver. A problem of full-order flux oberver i their need for computationally demanding dicretization method in order to work tably and accurately at high peed. An implementation of the full-order flux oberver uing the tator and rotor fluxe a tate variable in the tator reference frame and in the rotor reference frame, repectively, wa recently propoed. Thi paper decribe how an oberver gain can be included in thi tructure. It i hown that dicretization error of the propoed implementation are mall and that there i more freedom to chooe an oberver gain, even if the imple forward Euler dicretization i ued. Introduction Flux etimator are required in high-performance direct field orientation control of induction motor. A full-order flux oberver i a veratile choice for both peed-enored and peed-enorle drive. It offer good performance and robutne againt parameter enitivity and meaurement noie. The tator current and the rotor flux are conventionally ued a tate variable in full-order flux oberver [, ]. Some author have choen the tator and rotor fluxe a tate variable [3, 4], which enable introducing the oberver even into direct torque control or tator flux oriented control. Variou reference frame can be ued for the implementation: the tator reference frame; the rotor reference frame; or a field oriented reference frame. Propertie of the full-order oberver, e.g., it dynamic and parameter enitivity, are controllable through the oberver gain. A well-known problem of full-order flux oberver i that they require ophiticated dicretization method in order to work tably and accurately at high peed (including the nominal peed) [], [5] [7]. High-order dicrete model are undeirable ince they require plenty of real-time calculation. The imple forward Euler dicretization ẋ(t) x(t+t ) x(t) () T where T i a ampling period, can be ued intead. In thi cae, the tability ha to be guaranteed by placing the oberver pole inide the tability region correponding to the dicretization. When uing a reaonable ampling period, thi requirement i difficult to fulfill without acrificing the veratility of the full-order oberver. In other word, the deired propertie of the oberver cannot be reached due to thi limitation. Another problem of the forward Euler dicretization i the degraded accuracy due to dicretization error. Recently, an implementation of the full-order flux oberver uing the tator and rotor fluxe a tate variable in the tator and rotor reference frame wa propoed [8]. Thi natural combination of two

2 reference frame give well-behaving oberver pole in the whole peed range even with zero oberver gain. Conequently, the implementation i well uited to the forward Euler dicretization. Thi paper decribe the incluion of oberver gain in the implementation preented in [8]. If deired, very imple oberver gain, e.g., real-valued contant, can be choen. Furthermore, it i hown that the dicretization error of thi implementation i coniderably maller than that of the conventional implementation. Induction Motor Model The parameter of the dynamic Γ-equivalent circuit of an induction motor are the tator reitance R, the rotor reitance R R, the tator tranient inductance L, and the magnetizing inductance L M. The electrical angular peed of the rotor i denoted by ω m, the angular peed of the reference frame ω k, the tator current pace vector i, and the tator voltage u. When the tator flux ψ and the rotor flux ψ R are choen a tate variable, the tate-pace repreentation of the induction motor become [ ẋ = jω ] [ ] k σ r j(ω x+ u r k ω m ) (a) }{{}}{{} A B [ ] i = L L x (b) }{{} C where the tate vector i x = [ψ ψ R ] T, and the parameter expreed in term of the Γ-equivalent circuit parameter are σ = L /(L M + L ), = L /R, and r = σl M/R R. The block diagram correponding to (a) i hown in Fig. (a). 3 Full-Order Flux Oberver 3. Conventional Implementation The full-order flux oberver uing the fluxe a tate variable i defined by ˆx = Aˆx+Bu +L(i î ) î = Cˆx (3a) (3b) where the oberver tate vector i ˆx = [ ˆψ ˆψR ] T and the oberver gain L = [l l r ] T. Both the tator and rotor electrical dynamic are calculated alternatively in the tator reference frame (ω k = ), in the rotor reference frame (ω k = ω m ), or in the flux reference frame (where ω k i the angular peed of the etimated flux). All the implementation baed on one reference frame are here called conventional even though the tator reference frame i the mot uual choice. Eigenvalue λ, of the oberver (3) are obtained by olving the equation det(λi A+LC) = (4) where I i a identity matrix. The eigenvalue analyi i relevant only if ω m and ω k are aumed to be contant. Regardle of the reference frame choen, there i a peed dependent cro-coupling between the real and imaginary component of at leat one tate variable. Thi may lead to poorly damped oberver dynamic at high peed. It i to be noted that the eigenvalue are independent of the choice of tate variable. 3. Propoed Implementation In the following, the variable in the tator reference frame are denoted by upercript and thoe in the rotor reference frame by upercript m. The tate-pace repreentation () rewritten uing both the tator

3 Stator winding Rotor winding u + + ψ + σ r j ψ R j r ω k (a) + ω m ψ R e jϑm ψ m R u ψ ψ m + + σ e jϑm r ϑ m (b) Figure : Block diagram of motor winding (a) in a general reference frame, (b) in the tator (upercript ) and rotor (upercript m) reference frame. Fluxe are ued a tate variable. and rotor reference frame i [ ] [ ] ż = ejϑm σ r e jϑm z+ r }{{ }}{{} A B z [ ] i = L L ejϑm z }{{} C z u (5a) (5b) where the tate vector i z = [ψ ψ m R ]T and the angle of rotor ϑ m. The block diagram correponding to (5a) i hown in Fig. (b). Baed on (5), it i natural to form the nonlinear oberver ẑ = A z ẑ+bu ( +L z i î ) î = C zẑ where L z = [ l l r e jϑm] T. The angle of rotor ϑm can be directly meaured or it can be obtained by integrating the rotor peed ω m. If an incremental encoder i ued for meauring the rotor peedω m, ome low-pa filtering of the peed i uually needed. The delay due to thi filtering may caue dynamic error during fat acceleration if the conventional oberver implementation i ued. In the propoed implementation, the peed ω m i not needed and the correponding dynamic error during fat acceleration are thu avoided. Eigenvalue Uing Floquet Decompoition For analyi purpoe, the oberver (6) can be linearized by auming the angular peed ω m of the rotor to be contant, leading to a linear time-periodic (LTP) ytem ẑ = A z (t)ẑ+bu +L z (t)( i ) î (7a) î = C z (t)ẑ A imilar idea ha been preferred in the implementation of reduced-order oberver: the voltage model i implemented in the tator reference frame, and the current model in the rotor reference frame [9, Fig. ]. (6a) (6b) (7b)

4 where the periodic matrice are [ A z (t) = σ r e jωmt ejωmt r ] [, L z (t) = l l r e jωmt ] [, C z (t) = L L ejωmt] (7c) and the period i T = π/ω m. The eigenvalue (or the Lyapunov exponent) of LTP ytem can be determined by uing the Floquet decompoition []. Without lo of generality, the autonomou openloop ytem ż(t) = A z (t)z(t) with the initial condition z(t ) = z i conidered. The olution of the ytem iz(t) = Φ z (t,t )z, where the tate tranition matrix Φ z (t,) atifie t Φ z(t,) = A z (t)φ z (t,), Φ z (,) = I (8) For a linear T -periodic ytem, the tate tranition matrixφ z (t+t,t) i a periodic matrix and it eigenvalue are independent of t. A contant ytem matrix Ãz correponding to the periodic matrix A z (t) can be obtained baed on the Floquet decompoition by olving eãz T = Φ z (T,). The eigenvalue of à z are the Lyapunov exponent of the ytem. In practice, the tate tranition matrix Φ z (T,) can be evaluated by uing time-domain imulation of (8). Eigenvalue Uing Coordinate Tranformation Another, impler method i to tranform the eigenvalue of the conventional oberver implementation (3) to the tator and rotor reference frame. The eigenvalue of (3) correponding to the rotor dynamic calculated in the tator reference frame i tranformed to the rotor reference frame uing [] λ m (ω m ) = Re{λ (ω m )}+jim{λ (ω m )}+jω m (9) The eigenvalue correponding to the tator dynamic, λ, i not tranformed. Both the Floquet decompoition approach and the coordinate tranformation approach give the ame reult. 3.3 Comparion of Dicretization Error of Sytem Matrice Two major factor affecting the accuracy of the etimated tate are the dicretization error of the element of the dicretized ytem matrix and the frequencie of the etimated tate. In the following, the error of the ytem matrix are tudied, and the effect of the frequencie are then dicued. The open-loop ytem matrix (i.e., l = l r = ) i firt conidered. The accurately dicretized ytem matrix of the conventional implementation correpond to the tate tranition matrix Φ(T,) = exp(t A) = I+ T A! + T A! +... () The tate tranition matrix of the propoed implementationφ z (T,) can be calculated by uing (8). The ytem matrice dicretized uing the forward Euler method are A d = I+T A and A zd = I+T A z for the conventional and propoed implementation, repectively. In Fig., the normalized dicretization error of the open-loop ytem matrice of the conventional implementation in the tator reference frame and the propoed implementation are depicted when the forward Euler dicretization i ued. The parameter of a.-kw four-pole induction motor given in Table I were ued. It can be een that the dicretization error at high peed can be coniderably reduced by uing the propoed implementation. In the conventional method, the rotor reference frame or the flux reference frame could be ued intead of the tator reference frame. The change of the reference frame ha only a marginal influence on the error of ytem matrice in Fig.. In the propoed implementation, the teady-tate frequency of the tator flux etimate correpond to the tator frequency wherea the teady-tate frequency of the rotor flux etimate correpond to the lip frequency. In the conventional implementation, the teady-tate frequencie of both the flux etimate are equal. The frequency depend on the choen reference frame: the tator, rotor, and flux reference frame correpond to the tator, lip, and zero frequencie, repectively. Firtly, the input Bu doe not affect to the eigenvalue and, econdly, the cloed-loop ytem can be eaily arranged to thi form by replacinga z witha z L z C z.

5 .5 Normalized error A zd (T ) Φ z (T,) Φ z (T,) A d Φ(T,) Φ(T,) ω m (p.u.) Figure : Comparion of normalized dicretization error of the open-loop ytem matrice when the forward Euler dicretization i ued. The dahed line how the error correponding to the conventional implementation (ω k = ) and the olid line the error correponding to the propoed implementation. Table I: Parameter for the.-kw four-pole 4-V 5-Hz motor. Stator reitance R 3.67 Ω Rotor reitance R R. Ω Magnetizing inductance L M.4 H Stator tranient inductance L.9 H Moment of inertia J tot.55 kgm Rated peed 43 r/min Rated current 5. A Rated torque 4.6 Nm Due to the ignal frequency being zero in the teady tate, the actual error of the etimated tate are lightly maller if the conventional method in the ynchronou reference frame i ued. However, the poor damping of the oberver dynamic i till preent in the conventional implementation. In the propoed implementation, the damping i much better and more freedom to chooe the oberver gain i left. 3.4 Parameter Senitivity The effect of motor parameter deviation can be analyzed by the mean of teady-tate relation [9, ]. It i eay to how that the parameter enitivity analyi i equivalent for both the conventional and propoed implementation when the amel and l r are ued. 4 Oberver Gain The eigenvalue of the oberver reulting from three gain election are hortly analyzed uing the parameter of the.-kw four-pole induction motor. The gain are choen to illutrate the general difference between the conventional and propoed implementation. Furthermore, they are choen to be uable alo in the peed-enorle cae when the oberver i augmented by a peed-adaptation loop, e.g., [4]. It i, however, to be noted that the peed-adaptation loop ha an influence on the eigenvalue of the oberver. In order to get a table forward Euler dicretization uing the ampling period T, the pole of the continuou time ytem hould be located inide a circle with radiu /T centered at { /T,}. Thi tability region i plotted (dahed line) in the following root loci plot auming a ampling period of µ. 4. Open-Loop Eigenvalue The open-loop eigenvalue are obtained by chooing l = l r =. In Fig. 3(a), the root loci of the conventional implementation (3) in the rotor reference frame are plotted when the rotor peed ω m varie from to 5 p.u. A the peed ω m increae, the magnitude of the imaginary part of the eigenvalue correponding to the rotor dynamic increae. Thi lead to intability at higher peed (in thi cae at

6 .3 Im{λ} (p.u.) 3 4 Im{λ}(p.u.) 5.5 Re{λ} (p.u.) (a).3.5 Re{λ}(p.u.) Figure 3: Oberver root loci plotted, when ω m =...5 p.u., l = l r =. (a) the conventional implementation in the rotor reference frame, and (b) the propoed implementation. (b) peed higher than approximately 4. p.u.) if the forward Euler dicretization i ued. Fig. 3(b) how the root loci of the propoed implementation (6). The eigenvalue behave well in the whole peed range. The root loci of Fig. 3(a) correpond to the loci plotted in the flux reference frame with the lip angular frequency ω r being zero. Nonzero lip would jut lightly hift the loci vertically, i.e., jω r i added to the eigenvalue calculated in the rotor reference frame. When the tator reference frame i ued, the intability occur at much lower peed. In the cae of larger machine, the intability i faced at lower peed ince the real part of the eigenvalue of larger machine are maller. 4. Shifted Open-Loop Eigenvalue In order to have fater convergence of the etimation error, the open-loop eigenvalue can be hifted to the left in the complex plane. The eigenvalue of the oberver are explicitly determined a k( + r )+λ,, where λ, are the open-loop eigenvalue of the induction motor and k i a hifting factor [4]. Thi pole placement lead to the complex-valued peed-varying oberver gain l = kl l r = kl [ ( + r (k+)σ( (ω m r) +σ + r ) +ωm r +jω (k+)( + r ) m r σ) ] (a) [ ( + r (k+)σ( + r) (ω m r ) +σ ω r m r σ +jω (k+)( + r) r m σ) ] (b) The root loci correpond to the loci hown in Fig. 3 beide that they are hifted to the left. The weakne of the hifted open-loop eigenvalue in the peed-enored cae i high enitivity to parameter inaccuracie unle k i mall (for the given.-kw motor, k >. lead to high enitivity). A benefit i that higher peed can be reached compared with the zero-gain cae (in the cae of the conventional implementation). 4.3 Contant Gain A imple oberver gain i obtained by chooing contant l andl r. Depite the implicity of thi oberver gain, good parameter enitivity propertie in the motoring region can be obtained if the contant gain i uitably choen [3]. For the given motor, l = 5R and l r = were choen. The root loci of the conventional implementation in the tator reference frame and the propoed one are hown in Fig. 4(a) and 4(b), repectively. The conventional implementation uing the forward Euler dicretization i untable at peed higher than approximately.8 p.u. wherea the propoed implementation i table. An intereting pecial cae i obtained by chooing l r = R R. The propoed implementation of the full-order oberver then reduce to the well-known current model implemented in the rotor reference frame.

7 5.3 Im{λ}(p.u.) 4 3 Im{λ} (p.u.) 4 3 Re{λ}(p.u.) (a) Re{λ} (p.u.) Figure 4: Oberver root loci plotted, when ω m =...5 p.u., l = 5R, l r =. (a) the conventional implementation in the tator reference frame, and (b) the propoed implementation. (b) ψ R,ref Flux contr. i,ref u,ref ω m,ref Speed contr. Curr. contr. ˆϑ e jϑ PWM e jϑ i ˆψ R ˆψ R Flux ob. ω m IM Figure 5: The direct rotor flux oriented controller. The electrical variable hown on the left-hand ide of the coordinate tranformation are in the (etimated) rotor flux reference frame and the variable on the right-hand ide are in the tator reference frame. 5 Control Sytem The propoed and conventional oberver implementation were alo invetigated experimentally. The control wa baed on the direct rotor flux orientation and ynchronou-frame current control [3, 4]. The implified overall block diagram of the ytem i hown in Fig. 5. The peed and flux controller were conventional PI-controller. The bandwidth of the controller are given in Table II. The ampling wa ynchronized to the modulation, and both the witching frequency and the ampling frequency were 5 khz. The dc-link voltage wa meaured, and the reference tator voltage obtained from the current controller wa ued for the flux oberver. 6 Experimental Reult The experimental etup conit of the.-kw four-pole induction motor which i fed by a frequency converter controlled by a dspace DS3 PPC/DSP board. The bae value ued in the figure are: angular frequency π 5, current 5. A, and flux. Wb. In the firt experiment, the zero oberver gain wa choen. An experiment uing the conventional implementation in the rotor flux reference frame i hown in Fig. 6(a), and an experiment uing the propoed implementation i hown in Fig. 6(b). The peed reference wa tepped from zero to 5 p.u. at t =.5. No external load torque wa applied. The drive wa operating in the overmodulation region even in the teady tate due to high mechanical loe at high peed. A expected baed on the root loci of Fig. 3, the conventional implementation turned into intability wherea the propoed implementation remained table.

8 Table II: Bandwidth, bae value π 5. Current control α c = 8p.u. Speed control α =.5α c Flux control α f = α In the econd experiment, the oberver gain wa choen to bel = 5R,l r = correponding to that ued in the root loci of Fig. 4. Fig. 7(a) and 7(b) how experimental reult obtained uing the conventional implementation in the tator reference frame and the propoed implementation, repectively. The peed reference wa tepped from zero to.8 p.u. at t =.. Correpondence between the root loci and the experiment i again very good; the conventional implementation became untable wherea the propoed one remained table. The ubharmonic frequency (equal to ω m /p) een in the torque producing current component i q originate from the peed meaurement. It i to be noted that the bandwidth of the flux controller wa high. Therefore, the flux etimate trie to follow it reference, and the intability or inaccuracie of the flux etimation (due to dicretization error or inaccurate motor parameter) can be een in the flux producing current component i d. In the bae-peed region, the currenti d hould be ideally contant. However, it can be een thati d in Fig. 7(a) reduce ignificantly in the bae-peed region (t = ) along with the increaing peed. Thi i due to dicretization error of the conventional oberver implementation. In Fig. 6(a) and 6(b), the current component i d varie moderately in the bae-peed region wherea in Fig. 7(b) it remain almot contant. Thi difference i probably due to the zero-oberver-gain cae being more enitive to inaccuracie in the motor parameter ued. 7 Concluion The propoed oberver implementation ha everal advantage a compared with conventional implementation: (a) well-behaving oberver pole are obtained in the whole peed range even without oberver gain; (b) dicretization error are mall even if the imple forward Euler dicretization i ued; (c) there i more freedom to chooe the oberver gain if the forward Euler dicretization i ued; and (d) dynamic error due to the poible filtering of the meaured rotor peed are eliminated. Furthermore, the oberver gain can be elected by uing conventional method, and the reult of the parameter enitivity analyi for the conventional oberver implementation can be ued. The oberver can alo be augmented with exiting parameter (or peed) adaptation rule. Reference [] G. Verghee and S. Sander, Oberver for flux etimation in induction machine, IEEE Tranaction on Indutrial Electronic, vol. 35, pp , Feb [] H. Kubota, K. Matue, and T. Nakano, DSP-baed peed adaptive flux oberver of induction motor, IEEE Tranaction on Indutry Application, vol. 9, pp , Mar./Apr [3] B. Peteron, Induction machine peed etimation obervation on oberver. PhD thei, Department of Indutrial Electrical Engineering and Automation, Lund Univerity, Lund, Sweden, Feb [4] J. Mae and J. Melkebeek, Speed-enorle direct torque control of induction motor uing an adaptive flux oberver, IEEE Tranaction on Indutry Application, vol. 36, pp , May/June. [5] O. Vainio, S. Ovaka, and J. Paanen, A digital ignal proceing approach to real-time AC motor modeling, IEEE Tranaction on Indutrial Electronic, vol. 39, pp , Feb. 99. [6] C. Bottura, J. Silvino, and P. de Reende, A flux oberver for induction machine baed on a time-variant dicrete model, IEEE Tranaction on Indutry Application, vol. 9, pp , Mar./Apr [7] G. Griva, P. Ferrari, F. Profumo, R. Bojoi, R. Maceratini, and G. Barba, Luenberger oberver for high peed induction machine drive baed on a new pole placement method, in 9th European Conference on Power Electronic and Application (EPE ), (Graz, Autria), Aug.. [8] M. Hinkkanen and J. Luomi, Novel full-order flux oberver tructure for peed enorle induction motor, in The 7th Annual Conference of the IEEE Indutrial Electronic Society (IECON ), vol., (Denver, CO), pp , Nov./Dec..

9 ωm (p.u.) id,iq (p.u.) ˆψR (p.u.) ωm (p.u.) id,iq (p.u.) ˆψR (p.u.) t () (a) t () Figure 6: Experimental reult obtained uing (a) the conventional implementation in the rotor flux reference frame and (b) the propoed implementation. The oberver gain i l = l r = correponding to the root loci in Fig. 3. The firt ubplot how the meaured peed (olid) and the peed reference (dahed). The econd ubplot how the meaured d and q component of the tator current (olid) and their reference (dahed) in the etimated rotor flux reference frame. The third ubplot preent the magnitude of the etimated rotor flux (olid) and it reference (dahed). (b) [9] P. Janen and R. Lorenz, A phyically inightful approach to the deign and accuracy aement of flux oberver for field oriented induction machine drive, IEEE Tranaction on Indutry Application, vol. 3, pp., Jan./Feb [] W. Rugh, Linear Sytem Theory. Upper Saddle River, NJ: Prentice-Hall, nd ed., 996. [] J. Holtz, On the patial propagation of tranient magnetic field in AC machine, IEEE Tranaction on Indutry Application, vol. 3, pp , July/Aug [] M. Hinkkanen and J. Luomi, Parameter enitivity of full-order flux oberver for induction motor, in Conference Record of the IEEE Indutry Application Conference, Thirty-Seventh IAS Annual Meeting, (Pittburgh, PA), Oct.. In pre. [3] L. Harnefor and H.-P. Nee, Model-baed current control of AC machine uing the internal model control method, IEEE Tranaction on Indutry Application, vol. 34, pp. 33 4, Jan./Feb [4] F. Briz, M. Degner, and R. Lorenz, Analyi and deign of current regulator uing complex vector, IEEE Tranaction on Indutry Application, vol. 36, pp , May/June.

10 ωm (p.u.) id,iq (p.u.) ˆψR (p.u.) ωm (p.u.) id,iq (p.u.) ˆψR (p.u.) t() (a) t() Figure 7: Experimental reult obtained uing (a) the conventional implementation in the tator reference frame and (b) the propoed implementation. The oberver gain il = 5R,l r =, correponding to the root loci in Fig. 4. Explanation of the curve are given in Fig. 6. (b)