Estimation of Saturation of Permanent-Magnet Synchronous Motors Through an Energy-Based Model

Transcription

1 Estimation of Saturation of Permanent-Magnet Synchronous Motors Through an Energy-Base Moel Al Kassem Jebai, François Malrait, Philippe Martin an Pierre Rouchon arxiv:110393v1 [mathoc] 15 Mar 011 Mines ParisTech, Centre Automatique et Systèmes, 60 B Saint-Michel, 757 Paris ceex 06, France {al-kassemjebai, philippemartin, pierrerouchon}@mines-paristechfr Schneier Electric, STIE, 33, rue Anré Blanchet, 710 Pacy-sur-Eure, France francoismalrait@schneier-electriccom Abstract We propose a parametric moel of the saturate Permanent-Magnet Synchronous Motor PMSM together with an estimation metho of the magnetic parameters The moel is base on an energy function which simply encompasses the saturation effects Injection of fast-varying pulsating voltages an measurements of the resulting current ripples then permit to ientify the magnetic parameters by linear least squares Experimental results on a surface-mounte PMSM an an interoir magnet PMSM illustrate the relevance of the approach Inex Terms: Permanent magnet synchronous motor, magnetic circuit moeling, magnetic saturation, energy-base moeling, cross-magnetization I INTRODUCTION Sensorless control of Permanent-Magnet Synchronous Motors PMSM at low velocity remains a challenging task Most of the existing control algorithms rely on the motor saliency, both geometric an saturation-inuce, for extracting the rotor position from the current measurements through high-frequency signal injection [1], [] However some magnetic saturation effects such as cross-coupling an permanent magnet emagnetization can introuce large errors on the rotor position estimation [3], [4] These errors ecrease the performance of the controller In some cases they may cancel the rotor total saliency an lea to instability It is thus important to correctly moel the magnetic saturation effects, which is usually one through -q magnetizing curves flux versus current These curves are usually foun either by finite element analysis FEA or experimentally by integration of the voltage equation [5], [6] This provies a goo way to characterize the saturation effects an can be use to improve the sensorless control of the PMSM [7], [8] However the FEA or the integration of the voltage equation methos are not so easy to implement an o not provie an explicit moel of the saturate PMSM In this paper a simple parametric moel of the saturate PMSM is introuce section II; it is base on an energy function [9], [10] which simply encompasses the saturation an cross-magnetization effects In section III a simple estimation metho of the magnetic parameters is propose an rigorously justifie: fast-varying pulsating voltages are impresse to the motor with rotor locke; they create current ripples from which the magnetic parameters are estimate by linear least squares In section IV experimental results on two kins of motors with surface-mounte an interior magnets illustrate the relevance of the approach II AN ENERGY-BASED MODEL FOR THE SATURATED PMSM A Energy-base moel The electrical subsystem of a two-axis PMSM expresse in the synchronous q frame reas = u Ri + θ t t φ q 1 = u q Ri q θ t t φ +φ m, where φ,φ m are the irect-axis flux linkages ue to the current excitation an to the permanent magnet, an φ q is the quarature-axis flux linkage; u,u q are the impresse voltages an i,i q are the currents; θ is the rotor electrical position an R is the stator resistance The currents can be expresse in function of the flux linkages thanks to a suitable energy function Hφ,φ q by i = 1 Hφ,φ q 3 i q = Hφ,φ q, 4 where k H enotes the partial erivative wrt thek th variable, see [9], [10]; without loss of generality H0,0 = 0 For an unsaturate PMSM this energy function reas H l φ,φ q = 1 φ + 1 φ q L L q where L an L q are the motor self-inuctances, an we recover the usual linear relations i = 1 Hφ,φ q = φ L i q = Hφ,φ q = φ q L q

2 Notice the expression for H shoul respect the symmetry of the PMSM wrt the irect axis, ie Hφ, φ q = Hφ,φ q, 5 which is obviously the case for H l Inee, 1- is left unchange by the transformation φ,u,i,φ q,u q,i q,θ := φ,u,i, φ q, u q, i q, θ; this implies ie Therefore 1 Hφ,φ q = 1 Hφ,φ q Hφ,φ q = Hφ,φ q, 1 Hφ, φ q = 1 Hφ,φ q Hφ, φ q = Hφ,φ q H φ, φ q = 1 Hφ, φ q = 1 Hφ,φ q = H φ,φ q H φ, φ q = Hφ, φ q Integrating these relations yiels = Hφ,φ q = H φ,φ q Hφ, φ q = Hφ,φ q +c φ q Hφ, φ q = Hφ,φ q +c q φ, where c,c q are functions of only one variable But this makes sense only if c φ q = c q φ = c with c constant Since H0,0 = 0, c = 0, which yiels 5 B Parametric escription of magnetic saturation Magnetic saturation can be accounte for by consiering a more complicate magnetic energy function H, having H l for quaratic part but incluing also higher-orer terms From experiments saturation effects are well capture by consiering only thir- an fourth-orer terms, hence Hφ,φ q = H l φ,φ q α 3 i,i φ 3 i φ i q + α 4 i,i φ 4 i φ i q i=0 i=0 This is a perturbative moel where the higher-orer terms appear as corrections of the ominant termh l The9coefficients α ij together with L, L q are motor epenent But 5 implies α,1 = α 0,3 = α 3,1 = α 1,3 = 0, so that the energy function eventually reas Hφ,φ q = H l φ,φ q +α 3,0 φ 3 +α 1, φ φ q +α 4,0 φ 4 +α,φ φ q +α 0,4φ 4 q 6 Fig 1 a φ i,i q = Constant b φ qi = Constant,i q Flux-current magnetization curves IPM From 3-4 an 6 the currents are then explicitly given by i = 1 Hφ,φ q = φ L +3α 3,0 φ +α 1, φ q +4α 4,0 φ 3 +α, φ φ q 7 i q = Hφ,φ q = φ q L q +α 1, φ φ q +α, φ φ q +4α 0,4 φ 3 q, 8 which are the flux-current magnetization curves Fig 1 shows examples of these curves in the more familiar presentation of fluxes wrt currents obtaine by numerically inverting 3-4; the motor is the IPM of section IV The moel of the saturate PMSM is thus given by 1- an 7-8 It is in state form with φ,φ q as state variables The magnetic saturation effects are represente by the 5 aitional parameters α 3,0,α 1,,α 4,0,α,,α 0,4

3 C Moel with i,i q as state variables The moel of the saturate PMSM is often expresse with i,i q as state variables, eg [5] Starting with flux-current magnetization curves in the form φ = Φ i,i q 9 φ q = Φ q i,i q 10 an ifferentiating wrt time, 1- then becomes L i,i q i t +L qi,i q i q t = u Ri + θ t φ q L q i,i q i t +L qqi,i q i q t = u q Ri q θ t φ +φ m, where L i,i q L q i,i q 1 Φ = i,i q Φ i,i q L q i,i q L qq i,i q 1 Φ q i,i q Φ q i,i q Though not always acknowlege L q an L q shoul be equal Inee, plugging 3-4 into 9-10 gives φ = Φ 1 Hφ,φ q, Hφ,φ q φ q = Φ q 1 Hφ,φ q, Hφ,φ q Taking the total erivative of both sies of these equations wrt φ an φ q then yiels 1 0 L = 11 H+L q 1 H L 1 H+L q H 0 1 L q 11 H+L qq 1 H L q 1 H+L qq H L L = q 11 H 1 H L qq 1 H H L q Since 1 H = 1 H the secon matrix in the last line is symmetric, hence the first; in other wors L q = L q To o that with the moel of section II-B the nonlinear equations 7-8 must be inverte Rather than oing that exactly, we take avantage of the fact the coefficients α i,j are experimentally small At first orer wrt the α i,j we obviously have φ = L i + O α i,j an φ q = L q i q + O α i,j Plugging these expressions into 7-8 we easily fin φ = L i 3α 3,0 L i α 1,L q i q 4α 4,0L 3 i3 α, L L q i i q +O αi,j 11 φ q = L q iq α 1, L L q i i q α, L L q i i q 4α 0,4 L 3 qi 3 q +O αi,j 1 Finally, L i,i q = L 1 6α3,0 L i 1α 4,0 L i α,l q i q L q i,i q = L q i,i q = L L qi q α 1, +α, L i L qq i,i q = L q 1 α1, L i α, L i 1α 0,4L q i q III A PROCEDURE FOR ESTIMATING THE MAGNETIC A Principle PARAMETERS To estimate the 7 magnetic parameters in the moel, we propose a proceure which is rather easy to implement an Fig Experimental illustration of equation 15: time response of i reliable With the rotor locke in the position θ = 0, we inject fast-varying pulsating voltages u t = ū +ũ ft 13 u q t = ū q +ũ q ft, 14 where ū,ū q,ũ,ũ q, are constant an f is a perioic function with zero mean The pulsation is chosen large enough wrt the motor electric time constant It can then be shown, see section III-C, that after an initial transient i t = ī +ĩ Ft+O 1 15 i q t = ī q +ĩ q Ft+O 1, 16 where ī = ū R,ī q = ūq R,ĩ,ĩ q are constant an F is the primitive of f with zero mean F has clearly the same perio as f; fig shows for instance the current i obtaine for the SPM of section IV when starting from i 0 = 0 an applying a square signal u with = 500Hz, ū = 3V an ũ = 30V On the other han using the saturation moel the amplitues ĩ,ĩ q of the current ripples turn out to be ĩ = 1 ũ +α, L q ī q L ī ũ q +L q ī q ũ L +1α 4,0 L ī ũ +6α 3,0 L ī ũ +α 1, L q ī q ũ q 17 ĩ q = 1 ũq +α, L ī L q ī q ũ +L ī ũ q L q +1α 0,4 L qī qũq +α 1, L ī ũ q +L q ī q ũ 18 As ĩ,ĩ q can easily be measure experimentally, these expressions provie a means to ientify the magnetic parameters from experimental ata obtaine with various values of ū,ū q,ũ,ũ q B Estimation of the parameters Since combinations of the magnetic parameters always enter linearly, they can be estimate by simple linear least squares; moreover by suitably choosing ū,ū q,ũ,ũ q, the whole least squares problem for the 7 parameters can be split into several problems involving fewer parameters:

4 with ū = ū q = 0, hence ī = ī q = 0, an ũ = 0 resp ũ q = 0 equation 17 resp equation 18 reas L = 1 ũ resp L q = 1 ũ q 19 ĩ ĩ q with ū q = 0, hence ī q = 0, an ũ q = 0, 17 reas ĩ = ũ 1 +6α 3,0 L ī +1α 4,0 L L ī 0 Notice 18 reas ĩ q = 0 hence provies no information with ū = 0, hence ī := 0, an ũ q = 0, rea ĩ = ũ 1 +α, L L q qī 1 ĩ q = ũ α 1,L q ī q with ū = 0, hence ī := 0, an ũ = 0, rea ĩ = ũ q α 1,L q ī q 3 ĩ q = ũq 1 +1α 0,4 L L q qī 4 q L resp L q is then immeiately etermine from 19; α 3,0 an α 4,0 are jointly estimate by least squares from 0; α,, α 1, an α 0,4 are separately estimate by least squares from respectively 1, -3 an 4 C Justification of section III-A The assertions of section III-A can be rigorously justifie by a straightforwar application of secon-orer averaging of ifferential equations [11, p 40] Inee the electrical θ subsystem 1- with locke rotor ie t = 0 an input voltages reas when setting τ = t τ = 1 ū +ũ fτ Ri φ,φ q 5 τ = 1 ūq +ũ q fτ Ri q φ,φ q 6 This system is in the so-calle stanar form for averaging, with a right han-sie perioic in τ an 1 as a small parameter Therefore its solution is given by φ τ = φ 0 ũ τ+ Fτ+O 1 7 φ q τ = φ 0 ũq q τ+ Fτ+O 1, 8 where φ 0,φ0 q is the solution of the system φ 0 = ū Ri φ 0 t,φ 0 q φ 0 q = ū q Ri q φ 0 t,φ 0 q obtaine by averaging the right-han sie of 5-6 After an initial transient φ 0 τ,φ0 qτ asymptotically reaches the constant value φ, φ q etermine by ū = Ri φ, φ q an ū q = Ri q φ, φ q Plugging 7-8 with t = τ into 7-8, an expaning along powers of 1 then yiels i t = ī + Ft ũ +6α 3,0 φ ũ +α 1, φq ũ q L +1α 4,0 φ ũ +α, φ φq ũ q + φ qũ +O 1 i q t = ī q + Ft ũq +α 1, φ ũ q + L φ q ũ q +α, φ φq ũ + φ ũq+1α 0,4 φ q ũ q +O 1, where ī = i φ, φ q an ī q = i q φ, φ q There remains to express φ, φ q in function of ī,ī q Rather than exactly inverting the nonlinear equations 7-8, we take avantage of the fact the coefficients α i,j are experimentally small At first orer wrt the α i,j we have φ = L i +O α i,j an φ q = L q i q + O α i,j Using this in the previous equations an neglecting O 1 an O α i,j terms we eventually fin Using irectly 11-1 yiels of course the same result A Experimental setup IV EXPERIMENTAL RESULTS The methoology of section III is teste on an interior magnet PMSM IPM an a surface-mounte PMSM SPM with rate parameters liste below The setup consists of an inustrial inverter with a 400V DC bus an a 4kHz PWM switching frequency, 3 Space boars DS1005 PPC Boar, DS003 A/D Boar, DS400 Timing an Digital I/O Boar an a host PC The measurements were sample also at 4kHz IPM SPM Pole pairs 6 Rate power 00 W 100 W Rate current 1 A 34 A Rate spee 1800 rpm 400 rpm Rate torque 106 Nm 9 Nm Resistance B Experimental results With the rotor locke in the position θ = 0, a square wave voltage with frequency = 500Hz an constant amplitue ũ or ũ q 30V for the IPM, 40V for the SPM is applie to the motor But for the etermination of L,L q where ū = ū q = 0, several runs are performe with various ū resp ū q such that ī resp ī q ranges from A to +A with a 03A increment IPM, or from 8A to 8A with a 05A increment SPM The estimate parameters are liste below; the uncertainty in the estimation stems from a ±10mA uncertainty in the current measurements

5 a IPM a IPM b SPM Fig 3 Measure values circles an fitte curve soli line for 0 IPM SPM L mh 919±5 1554±10 L q mh 458±1 586± α 3,0 AWb 770± ±011 α 1, AWb 535± ±07 α 4,0 AWb 3 194± ±08 α, AWb 3 18±80 876±103 α 0,4 AWb 3 66±04 118±017 The goo agreement between the fitte curves an the measurements is emonstrate for instance for 0 on Fig 3 an for on Fig 4 Notice 0 illustrates saturation on a single axis, while illustrates cross-saturation C Valiation The estimation proceure relies on 0 4, with either ī or, ie current vectors with angles 0,90,180,70 To check the valiity of the moel tests were conucte with current vectors with various angles an magnitues on the b SPM Fig 4 Measure values circles an fitte curve soli line for whole operating i = i +i q ranging from0a toa with a 03A increment for the IPM, an from0a to 55A with a 05A increment for the SPM Fig 5 shows for instance the results for a 60 current angle; there is a goo agreement between the measure values an those preicte by the moel As a kin of cross-valiation we also examine the currents time responses to large voltage steps Fig 6 shows the goo agreement between the measurements an the time response obtaine by simulating the moel with the estimate parameters; it also shows the ifferences with the simulate response when the saturation effects are omitte Fig 7 shows the goo agreement also between the measure flux values ie obtaine by integrating the measure currents an voltages an the flux values obtaine by simulation V CONCLUSION A simple parametric magnetic saturation moel for the PMSM with a simple ientification proceure base on high-

6 Fig 6 Time response of i to large step voltages in u IPM a Interior-magnet PMSM Fig 7 Saturation curve φ i IPM b Surface mounte PMSM Fig 5 Measure values circles compare to moel-preicte values soli line for a 60 current angle frequency voltage injection have been introuce Experimental tests on two kins of PMSM IPM an SPM emonstrate the relevance of the approach This moel can be fruitfully use to esign a sensorless control scheme at low velocity REFERENCES [1] J H Jang, S K Sul, J I Ha, K Ie, an M Sawamura, Sensorless rive of surface-mounte permanent-magnet motor by high-frequency signal injection base on magnetic saliency, IEEE T In Appl, vol 39, no 4, pp , 003 [] M J Corley an R D Lorenz, Rotor position an velocity estimation for a salient-pole permanent magnet synchronous machine at stanstill an high spees, IEEE T In Appl, vol 34, no 4, pp , 1998 [3] P Guglielmi, M Pastorelli, an A Vagati, Cross-saturation effects in IPM motors an relate impact on sensorless control, IEEE T In Appl, vol 4, no 6, pp , 006 [4] N Bianchi, S Bolognani, an A Faggion, Preicte an measure errors in estimating rotor position by signal injection for salient-pole PM synchronous motors, in IEEE Int Electric Machines an Drives Conf, 009, pp [5] B Štumberger, G Štumberger, D Dolinar, A Hamler, an M Trlep, Evaluation of saturation an cross-magnetization effects in interior permanent-magnet synchronous motor, IEEE T In Appl, vol 39, no 5, pp , 003 [6] G Štumberger, B Polajzer, B Štumberger, M Toman, an D Dolinar, Evaluation of experimental methos for etermining the magnetically nonlinear characteristics of electromagnetic evices, IEEE T Magnetics, vol 41, no 10, pp , 005 [7] Z Zhu, Y Li, D Howe, an C Bingham, Compensation for rotor position estimation error ue to cross-coupling magnetic saturation in signal injection base sensorless control of PM brushless AC motors, in IEEE Int Electric Machines Drives Conf, vol 1, 007, pp [8] P Guglielmi, M Pastorelli, G Pellegrino, an A Vagati, Positionsensorless control of permanent-magnet-assiste synchronous reluctance motor, IEEE T In Appl, vol 40, no, pp 615 6, 004 [9] D Basic, F Malrait, an P Rouchon, Euler-Lagrange moels with complex currents of three-phase electrical machines an observability issues, IEEE T Automat Contr, vol 55, no 1, pp 1 17, 010 [10] D Basic, A K Jebai, F Malrait, P Martin, an P Rouchon, Using Hamiltonians to moel saturation in space vector representations of AC electrical machines, in Avances in the theory of control, signals an systems with physical moeling, ser Lecture Notes in Control an Information Sciences, J Lévine an P Müllhaupt, Es Springer, 011, pp [11] J A Saners, F Verhulst, an J Murock, Averaging methos in nonlinear ynamical systems, n e, ser Applie Mathematical Sciences Springer, 007, no 59

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