Flux estimation algorithms for electric drives: a comparative study

Transcription

1 Flux etimation algorithm for electric drive: a comparative tudy Mohamad Koteich To cite thi verion: Mohamad Koteich. Flux etimation algorithm for electric drive: a comparative tudy. International Conference on Renewable Energie for Developing countrie (REDEC 26), Jul 26, Zouk Mobeh, Lebanon. REDEC 26, International Conference on Renewable Energie for Developing countrie 26. < <hal > HAL Id: hal Submitted on 3 May 26 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 Flux etimation algorithm for electric drive: a comparative tudy Mohamad Koteich Renault Group, Technocentre Avenue du Golf, Guyancourt, France mohamad.koteich@renault.com Abtract Thi paper review the tator flux etimation algorithm applied to the alternating current motor drive. The ocalled voltage model etimation, which conit of integrating the back-electromotive force ignal, i addreed. However, in practice, the pure integration i prone to drift problem due to noie, meaurement error, tator reitance uncertainty and unknown initial condition. Thi limitation become more retrictive at low peed operation. Several olution, reported in the literature, are reviewed and compared. Emphai i placed on the low-pa filter baed algorithm that how good performance in teady-tate a well a in tranient operating condition. I. INTRODUCTION High-performance motor drive, uch a field-oriented controlled and direct-torque controlled alternating current (AC) drive, require an accurate tator flux etimation. For AC machine, there exit two model for flux etimation, namely the voltage model and the current model. The current model etimation i known to be efficient in low and medium peed range, epecially when combined with high-frequency injection-baed etimation technique []. Neverthele, the accuracy of the current model i highly dependent on the knowledge of the machine inductance. On the other hand, the voltage model, which conit of integrating the tator back-electromotive force (EMF) ignal, i known for it good performance at medium and high peed. The main advantage of the voltage model i it robutne againt the machine parameter; it only require the tator reitance, which can be quite accurately known in variou application. Both voltage and current model can be combined in one tate-oberver etimation algorithm [2], [3], [4], [5]. Note that the voltage model implementation i the ame for all AC drive, wherea the current model depend on the machine tructure. In thi paper, the voltage model i addreed. The implementation of a pure integrator i prone to drift problem due to the following practical iue [6]: a) Inverter Nonlinearity: the tator voltage are not directly meaured, they are contructed uing the reference voltage of the pulewidth modulator (PWM). Thi provide a clean voltage ignal, but it doe not exactly repreent the tator voltage a the PWM inverter introduce ditortion. b) Current Meaurement: the current meaurement channel exhibit error due to unbalanced gain and DC drift. c) Stator Reitance: the accurate knowledge of the tator reitance, which may vary, i important for accurate etimation. d) Integrator initial condition: unknown initial condition, at the tarting of the drive or when evere back-emf change occur, reult in a DC-offet in the integrator output. Variou algorithm have been reported in the literature to olve the drift problem. One imple olution conit of uing a light amount of low-pa filtering in the integration of the back-emf [7]. However, thi introduce error in the etimated flux ignal epecially when the motor frequency i lower than the cutoff frequency of the low-pa filter (LPF) [8]. In view of emulating the frequency repone of a pure integrator, the author of [9] propoe a programmable cacaded LPF method of flux etimation: three cacaded programmable LPF with magnitude compenation are deigned and hown to be efficient when applied for enorle tator-flux-oriented control of induction machine. In the ame context, everal compenated LPF algorithm have been propoed for directtorque controlled [] and field-oriented controlled [8], [], [2] AC machine. Other algorithm, concentrated on the DCoffet rejection, conit of limiting the amplitude of the output flux ignal uing an adequate aturation function [3], [4], [5]. Thi paper i focued on the etimation of the tator flux of AC machine uing the voltage model. It review the etimation algorithm reported in the literature, and ummarize them in one generalized etimation algorithm, a detailed in Section II. From the generalized algorithm, three categorie of modified integrator can be derived: ) low-pa filter (LPF) baed algorithm, which are een a open-loop etimator, 2) amplitude aturation integrator and 3) adaptive flux oberver. The lat two categorie are conidered to be cloed-loop etimator ince the etimated flux i compared to a certain correction ignal in a feedback tructure. Section III i dedicated for the tudy of LPF-baed algorithm with a detailed comparative tudy uing digital imulation. Section IV review the correction-baed etimation algorithm. Concluion are drawn in Section V. II. STATOR FLUX ESTIMATION PROBLEM A tator winding of an electric motor can be een a a connection of a reitance R in erie with a coil having time-varying inductance. Throughout thi paper, the complex pace-vector notation i ued to repreent the electromagnetic quantitie (the current i, voltage v and fluxe ψ ). The voltage model equation can be written a: v = R i + dψ () dt Therefore, the tator flux vector can be etimated by integrating the back-emf (e ): ψ = (v R i ) = e (2)

3 The frequency repone of the integrator in the Laplace domain i: Ψ () e () = The magnitude and the phae lag of the integrator are: G = ω ϕ = π 2 where ω i the angular frequency of the tator flux Ψ (). To enure an accurate etimation, everal algorithm have been reported in the literature. They can be ummarized uing the following general formulation: ( ψ = e + (t) ψ cor ψ ) (6) Thi equation i illutrated graphically on Fig.. It i written imilarly to the traditional tate-oberver equation: (3) (4) (5) III. LOW-PASS FILTER One intuitive olution for DC-offet i to implement a highpa filter, with a corner frequency, in erie to the pure integrator. Thi reult in the low-pa filter approximation of the integrator: Ψ e =. + = + (8) A LPF can be een a an integrator with a negative feedback, a how the ignal flow diagram of Fig. 2. i R Fig. 2: Low-pa filter approximated integrator ˆx = f(x, u) + (t)(y ŷ) where x i the tate, u i the input, y i the output, f i the dynamical model of the ytem and (t) i the oberver gain to be tuned. However, the equation (6) i not an exact tateoberver ince y i not a meaured ignal; a correction flux, ψ cor, i ued for comparion with the etimated flux. R I Ψ V β jω Ψ Ψ i R ωc (t) ψ cor Fig. : Generalized modified integrator tructure In the frequency domain, the equation (6) become : Ψ = e + Ψ cor (7) + The generality of thi tructure lie in the choice of ψ cor ; depending on thi choice different etimation algorithm can be derived. At a firt glance, two particular algorithm prevail. Chooing ψ cor = yield the pure integrator equation, and chooing ψ cor = reult in a low-pa filter equation, with a corner frequency. The LPF algorithm i tudied in detail in the following ection. Other poible choice of the term ψ cor are dicued in ection IV. (t) i conidered to be contant. jω Ψ θ θ Fig. 3: Vector diagram of an LP Ψ Under inuoidal teady-tate condition, the voltage model reduce to: Ψ = jω (V R I ) (9) wherea the LPF approximated integrator give: Ψ = jω + (V R I ) () The vector diagram of Fig. 3 illutrate the relation between the real flux Ψ and the etimated flux Ψ. If the corner frequency i very low, the LPF i brought cloer to the pure integrator. In contrat to the pure integrator, the LPF i able to eliminate the DC-offet in the etimated flux. Fig. 4 how the time-domain repone of both the pure integrator and the LPF (with = 2 rd/) in preence of an initial etimation error of. W b. α

4 The magnitude and the phae of the LPF frequency-domain repone are, repectively: G = ω 2 + ω 2 c ϕ = arctan ( ω ) () (2) Hence, higher corner frequency enure fater DC-offet rejection, however, it introduce higher ditortion to the output ignal due to increaing attenuation and phae lag. The mot critical ituation prevail when the tator frequency ω i lower than the corner frequency. On the other hand, if i choen to be very low, which correpond to a large time-contant of the LPF, the drift problem perit. To overcome uch ituation, a compenation of the LPF gain and phae can be conidered in order to guaranty a pureintegrator-like overall frequency repone. Furthermore, the LPF pole (ituated at ) hould be placed far enough from the origin to olve the drift problem [8], [] Time (ec) Fig. 4: Real flux (in black), integrator output (dahed blue) and low-pa filter output (dahed red) with = 2 rd/. Magnitude (db) Phae (degree) Frequency (rad/) Fig. 5: Bode diagram of the pure integrator (black line) and the LPF for two cut-off frequencie rd/ (blue line) and rd/ (orange line), and the correponding compenation tranfer function (dahed line). Furthermore, for the ame reaon, the overall repone depend on the corner frequency of the LPF/Compenation. A. Compenation of the LPF Several olution are propoed in the literature for LPF compenation [8], [], [], mot of them are baed on the multiplication of the LPF block by the following invere of the HPF frequency repone []: + = + (3) Then, the compenation gain and phae lag are repectively: G = ω 2 + ω 2 c i R j ω (a) output compenation [] θ ω ϕ = arctan ( ω ) π ( ) 2 = arctan ωc ω (4) (5) Fig. 5 how the frequency repone of the pure integrator and two LPF with cut-off frequencie of rd/ and rd/, a well a the frequency repone of the correponding compenation tranfer function. The um of an LPF repone with it compenation repone reult in a pure integrator repone. Thi compenation can be applied to the output (flux) of the LPF [8], [] a hown in Fig. 6a, or it can be applied to the back-emf ignal, at the input of the LPF [] a hown in Fig. 6b. Note that both compenation level are not the ame regarding the overall repone, ince it i not a imple multiplication of two linear time-invariant ytem. i R j ω θ (b) input compenation Fig. 6: Low-pa filter compenation B. Corner frequency tuning The choice of the the corner frequency i crucial for good flux etimation, epecially in low-peed operating condition: a filter that can rapidly attenuate the DC-offet in medium/high peed operation might fail at low-peed and vice-vera (ee Fig. 7). Thi i due to the placement of the LPF pole with

5 repect to the tator frequency. Hence, an adaptive corner frequency tuning can be adopted by chooing to be dependent on the tator angular frequency ω a follow: = λ ω (6) where λ i poitive real number maller than one. At low peed, λ can be tuned to a low value, e.g.., wherea for higher peed, it can take higher value. In thi cae, the timecontant of the LPF, /(λ ω ), i decreaed with the increae of the tator frequency. i R λ ign ω (a) Output compenation [8] jλ θ Time (ec) (a) Initial condition error Time (ec) (b) Speed reveral Fig. 7: Time repone of the LPF for tow corner frequencie = rd/ (dahed blue) and = rd/ (dahed red). There exit two way to implement a compenated LPF with ω dependent corner frequency. The firt one conit of multiplying the LPF output by the following gain G and the phae lag ϕ of the compenation function (Fig. 8a): G = + λ 2 (7) ϕ = ign(ω ) arctan (λ) (8) The other way i to apply the compenation to the back-emf at the input of the LPF (Fig. 8b), which yield the following modified integrator equation []: ˆψ = ( λ ω + [ jλign(ω )] e ) dt (9) The comparion of the input-compenated and outputcompenated LPF i hown in Fig. 9 for λ =.2: the input compenated LPF how better behavior in the peed reveral operation, wherea the tranient behavior in repone to an initial etimation error eem to be the ame for both etimator. Furthermore, the tranient behavior i better if λ i maller, wherea a higher value of λ allow fater rejection of the DC-offet (Fig. ). Note that for λ = the pure integrator i achieved []. i R jλ ign ω (b) Input compenation [] Fig. 8: Compenated low-pa filter with peed-dependent corner frequency = λ ω Time (ec) (a) Initial condition error Time (ec) (b) Speed reveral Fig. 9: Input compenated LPF (dahed blue) v. output compenated LPF (dahed red) for λ =.2. C. Stator frequency etimation In addition to the choice of λ, the performance of the compenated LPF depend heavily on the accuracy of the tator flux angular frequency (ω ) etimate [6], which i often evaluated uing the following equation: ω = e ψ ψ 2 = ψ α(v β R i β ) ψ β (v α R i α ) ψ 2 α + ψ 2 β However, thi etimation i enitive to the tator reitance uncertainty, to the offet and ditortion in the flux and back- λ θ

6 Time (ec) (a) Initial condition error Time (ec) (b) Speed reveral Fig. : Input compenated LPF with λ =. (dahed blue) and λ =.9 (dahed red). A. Amplitude aturation integrator The amplitude aturation integration algorithm conit of chooing a correction flux having the ame phae of the integration output, but whoe magnitude i een a a aturation that limit the amplitude of the integrator output. A firt, primitive, algorithm wa propoed by [3] (Fig. 2a): the correction flux ψ cor i equal to the integrator output flux ˆψ (which yield a pure integrator) until the limiting level L i exceeded. When L i reached, the integrator output become: Ψ = e + Z(L) + + where Z(L) i the output of the aturation block whoe amplitude i limited to L [3]. One main limitation of the ue of thi algorithm i the tuning of L. Ideally, the limiting level L hould be equal to the actual flux amplitude in order to eliminate the DC-offet at the output. If L i greater than the flux amplitude, the output waveform will have a DC component, in addition to the AC component, in repone to an input offet. On the other hand, if L i lower than the flux amplitude, the etimated flux waveform will be ditorted. EMF ignal. In addition, the decreaing magnitude of the numerator at low peed deteriorate the etimation. To remedy to thi problem, the author of [6] propoe the PLL-baed etimation cheme illutrated on Fig.. The tator voltage i ued a a reference vector for the PLL, the voltage angle θ v i yntheized and ued a the angle of a rotational reference frame where the q axi component of the voltage vector (v q ) hould be null. Therefore, v q i ued a the error ignal i R (a) Saturable feedback of a Proportional-Integral (PI) controller that output the ω etimate. e jθ v q PLL ω θ v e j ˆθ L (b) Amplitude limiter Fig. : PLL-baed angular frequency etimation [6] The choice of the voltage vector for the PLL i motivated by the fact that thi vector ha everal advantage over the fluxe and current vector: it i the very clean vector available, epecially when contructed from the DC-link voltage and the witching tate of the inverter, it ha a coniderable magnitude and generate a conitent large-enough error ignal at the input of the PI [6]. IV. CORRECTION-BASED INTEGRATOR Although the LPF-baed modified integrator provide a certain amount of output feedback, it i een a an open-loop etimator ince it correpond to ψ cor = in the generalized modified integrator (6). Thi ection preent another cla of correction-baed modified integrator, baed on different choice of the correction flux ψ cor. Two broad categorie are preented: amplitude aturation integrator and adaptive flux oberver. e j ˆθ ψ.e ψ (c) Adaptive compenation uing quadrature detector Fig. 2: Modified integration algorithm propoed by [3] Another algorithm, hown on the Fig. 2b wa propoed in [3] to avoid poible waveform ditortion when L i lower than the actual flux amplitude: only the etimated flux amplitude i aturated to a limiting level L. Baed on thi algorithm, the author of [3] deign a third algorithm in which the value of L i determined uing an adaptive controller. The error ignal fed to thi controller i the dot product of the integrator input (back-emf) and output (etimated flux).

7 Thi error i zero when the input and the output vector are orthogonal (Fig. 3). Other paper, uch a [4], [5] propoe to take the amplitude of the tator flux reference (et-point) a a limiting level for the integrator, a hown on Fig. 3. Note that in [5] the corner frequency contain an integration term ( i time-variant) in order to have a PI controller fed with the difference between the etimated flux and the reference flux. However, the effectivene of the integration action i not proved. i R ψ ω e j ˆθ c (t) Fig. 3: Amplitude limitation modified integrator [4] B. Adaptive flux oberver The adaptive flux oberver are more complex etimation algorithm where the correction flux i the etimation of the tator flux uing the current model [2], [3]. The general tructure of the adaptive-oberver-baed flux etimator i hown on Fig. 4, where L eq = L q for ynchronou machine and L eq = L M 2 /L r for induction machine [7]. In low-peed operating condition the performance of both voltage and current model i deteriorated. A high-frequency injection-baed flux etimation can be applied to provide a more accurate current-model-baed correction flux [], [4], [8], [9]. i R L eq (t) Current Model ˆθ i Fig. 4: Oberver-baed integrator V. CONCLUDING REMARKS A review of the tator flux etimation algorithm ha been preented in thi paper. A a concluion, an input-compenated LPF with varying corner frequency can be a imple and efficient flux etimator. Further improvement of the reviewed algorithm may be realized by adapting (identifying) the tator reitance. Thi challenging topic ha been addreed in the literature [4], [5], and till need further invetigation. REFERENCES [] C. Silva, G. M. Aher, and M. Sumner, Hybrid rotor poition oberver for wide peed-range enorle pm motor drive including zero peed, IEEE Tranaction on Indutrial Electronic, vol. 53, no. 2, pp , 26. [2] C. Lacu, I. Boldea, and F. 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