Dynamic Torque Control for Switched Reluctances Drives based on a new

Transcription

1 Dynamic Torue Control for Switche Reluctances Drives base on a new Online Machine Moel Marc Hiller Universität er Buneswehr München Leistungselektronik un Steuerungen EIT6. Werner-Heisenberg-Weg Neubiberg Phone: , Fax: hiller_marc@yahoo.e URL: Abstract A new control concept for ynamic torue control of switche reluctance rives (SRD is presente. It is base on a novel machine moel that can be calculate online without the nee of any aitional look-up tables. Due to its mathematical structure the propose moel is suitable for accurate simulations an precise online torue control up to maximum spee. All necessary control parameters can be erive by solving euations in a microcontroller, only. Furthermore the torue control using the propose machine moel is very flexible an self aapting to the variation of external parameters (e.g. the DC-link voltage. Keywors Switche reluctance rive, Reluctance Drive, Converter control, Moelling, Converter circuit Introuction In orer to open new application fiels for SR Drives (e.g. main traction rive or electrical starter/generator in an electric vehicle it is necessary to improve the rive performance. This objective can be achieve by improvements in the machine esign or by innovations in the power electronic configuration an the rive control. The authors are currently investigating the whole SR Drive system incluing the major parts namely the machine an the converter an the inherent control. In the following paper the emphasis is put on a new machine moel which can be implemente in a microcontroller for fast online control. Besies, a ynamic torue control scheme for the lower spee range base on the new online machine moel is presente. The moel an the control scheme has been evelope as a part of the work on a new converter concept for SRD which has been introuce in []. The use converter concept features eight controllable switching states for motor an generator operation moe. Since the voltages of the internal capacitors can be controlle inepenently of the DC-link voltage, the new switching states allow motor phase voltages that are either higher (positive or lower (negative compare to conventional converters [Tab. I]. Furthermore the phase current control of one phase has no impact on the control of any neighbouring phase. The new switching states provie many operational avantages, because both the magnetization an the emagnetization of the motor phases can be controlle an accelerate inepenently of the DC-link voltage. As shown in Tab. I it is always possible to apply both positive an negative voltage to a motor phase without changing the switching state of the common one-uarant chopper which is change only if the energy flow on the DC-sie is inverte. EPE 5 - Dresen ISBN : P.

2 P V C A V CA C B V CB C C V CC i A i B i C N 4 6 V A A B C Fig. : Converter concept for a 3 phase SR Drive with one aitional voltage boost capacitor per phase Table I: Switching states for phase A State No. Switches Output Voltage 7 V A sign Store Energy in C A WC A t off off on -V CA + off on on +V 3 on on on + V +V CA - 4 on off on 5 off off off -V -V CA + 6 off on off 7 on on off +V CA - 8 on off off -V motor operation generator operation In orer to be able to have full access to the avantages of this converter with its extene egrees of freeom, a new online machine moel has been evelope. The variety of ifferent switching states an the aitional energy storage evices reuire a complex an flexible control structure. Precalculating all the switching angles an current waveforms becomes almost impossible if all important values (e.g. DC-link voltage, spee, torue, temperature etc. are taken into account. In orer to be able to control the machine always inepenently of these external parameters, the new machine moel has been evelope. It comprises various elements of known approaches an is esigne for state-of-the-art controller harware. Therefore the emphasis is put on higher accuracy while the calculating spee can be kept low. Furthermore the moel can also be use for the simulation of the SR Drive. By using the same algorithms for simulation an online control, the behaviour of the rive can be preicte precisely. New online machine moel The high performance control of a switche reluctance machine (SRM reuires a precise moel of the machine characteristics. Due to the highly nonlinear relationship between the flux-linkage, the phase current an the rotor position, such an accurate moel is always of a complex mathematical structure. EPE 5 - Dresen ISBN : P.

3 Besies the high precision, a SRM moel for online computation has to be implemente in a real-time controller. Although the current evelopment in controller esign is continuously increasing the calculating power, this task still reuires a compromise between accuracy an complexity. A common approach to calculate the behaviour of a SR Drive is to store the magnetization curves of flux-linkage versus current in a look-up table. These magnetization curves represent the complete set of reuire ata for the calculation of the magnetic circuit. Since these curves are obtaine by measurement or FEM-simulation at evenly-space rotor positions an then store in look-up tables the exact ata is only available at iscrete points. In orer to have access to every point in the i-ψ-plane the intermeiate values have to be etermine by either linear or uaratic interpolation or by cubic splines [4]. This interpolation has to ensure the sufficient ifferentiability of the curves in orer to guarantee that there are no iscontinuities, e.g. in the calculate instantaneous torue or its rate of change. Severe isavantages of this metho are the extensive amount of measurement ata an the high sensitivity to small measurement errors. For having access to other system values epening on various accessible input variables (e.g. i(ψ,ϕ, T(i,ϕ, i(t,ϕ, Ψ(i, ϕ a lot of ifferent look-up tables with interpolations have to be create. For a precise moel this reuires a consierable amount of storage capacity. Furthermore this approach is very inflexible regaring the consieration of aitional system parameters which have impact on the system performance (e.g. the DC-link voltage. Another possibility is to use a moel that is base mainly on the magnetic characteristics in the aligne an unaligne position. The values for intermeiate rotor positions are etermine by simple gage curves that epen on the geometry of the machine [6,7]. On the one han these kin of moels are rather simple an suitable for online computation. On the other han the precision of those moels is not sufficient for the control of high performance SR Drives. The same is true for moels which are base on the magnetic euivalent circuit of the machine an use the results of FEM calculations [8]. In orer to meet the reuirements of a high performance SRM control (e.g. for main traction rives in automotive applications the moel has to be as precise as possible. The evelopment of computing power in moern DSPs, microcontrollers an programmable logic evices (PLDs enables the eveloper to focus on accurate moels that can be implemente in real-time systems. This paper presents an improve machine moel with high accuracy which is still suitable for online computation. A machine moel that works with of a set of analytical euations instea of look-up tables can offer significant improvements. By choosing an appropriate approach, it is possible to avoi any problems with the ifferentiability of the esire values. This approach can be finally use in a very flexible way for the simulation of rive performance as well as for online rive control. Its principle is base on the work of Miller et al. [3]. Their moel was originally esigne for rapi simulation, only. It is base on a set of normalize gage curves of flux-linkage Ψ versus rotor position ϕ at constant current. A gage curve value represents the ratio of the istance g (ϕ between the actual flux Ψ(i,ϕ an the unaligne flux Ψ (i at a given current i versus the istance between the aligne flux Ψ (i an the unaligne flux Ψ (i at this certain current i (: Ψ( ϕ,i Ψ (i g( ϕ = ( Ψ (i Ψ (i The magnetization curve for the unaligne position (-axis can be expresse by the linear function Ψ (i. In the aligne position (-axis the magnetization curve is ivie into an unsaturate an linear interval Ψ,unsat (i (<i<i s an a saturate an strongly non-linear interval Ψ,sat (i (i s <i<i max. Ψ (i = f i ( Ψs Ψ,unsat (i = i (3 i s EPE 5 - Dresen ISBN : P.3

4 i + b i + c Ψ,sat (i = a (4 i + e This function for Ψ,sat (i has been chosen because it fulfils very well both the reuirements for fast online computation an ifferentiability. It has proven its suitability for the application in the moelling of various SR machines of ifferent power ratings an imensions..4.. Ψ (i Ψ,sat i + b i + c (i = a i + e.8 ϕ.6 Flux [Vs].4.. Ψ s Ψ Ψ, unsat ( i = i s s i Ψ (ϕ,i Ψ( ϕ,i Ψ (i g( ϕ = Ψ (i Ψ (i.8.6 Ψ (i Ψ ( i = f i i s i phase current [A] Fig. : Magnetization curves with the plane of a gage curve Flux (norm g (ϕ n i Torue (norm ϕ phase angle ϕ n (norm. unaligne aligne Fig. 3: gage curves (top an their first-orer erivatives (bottom for ifferent phase currents I EPE 5 - Dresen ISBN : P.4

5 By applying this scheme to the whole set of magnetization curves, a set of gage curves (normalize flux over normalize phase angle for ifferent phase currents is obtaine (Fig. 3 top. The variation of the gage curve parameters with respect to the phase current allows a very goo conformity between the measure (or simulate ata an the moel for the entire current range. The gage curves can be ajuste to the ifferent current values by ajusting four parameters, only. The structure of the gage curves remains unchange up to maximum phase current. In contrast to Miller et al. [3] the mathematical structure of the gage curves has been improve allowing a more accurate match with the measure magnetization ata of SR machines. The basic structure of the normalize phase angle sections, j, c, k an y (from unaligne to aligne position propose in [3] remaine unchange. The same is true for the gage curve functions (g, g c an g y in sections, c an y. Therefore the first-orer erivatives in sections (starting at the unaligne position an y (ening at the aligne position are still zero an the gage curve function g c in section c is still a secon-orer parabolic function: c n g ( ϕ n = (5 g ( ϕ (6 y n = n n g ( ϕ = a + a ϕ + a ϕ (7 For the blue line (referring to phase current i in Fig. 3 (top an bottom the with of sections an y is zero, i.e. that there are no rotor positions near the unaligne or aligne rotor position where the machine prouces no output torue. The gage curve in section j ( ϕ n.375 is a trigonometric function which is more complex than the function propose in [3]: with f sin π j ϕn e( n f j g j ( n m P ( f j n m P f ϕ ϕ = ϕ j (8 π e( ϕ j n f j e( ϕ n = j ϕn. (9 f This allows a much more accurate match with the measure (or calculate magnetization ata of the machine. At the en of section j the first-orer erivative of the gage curve reaches its maximum value, i.e. as well the maximum value of the output torue. At the transition from section j to section c (.375 < ϕ n.7 the moel still ensures that no iscontinuity occurs in the output torue course. This is also true for the transition between section c an section k (.7 < ϕ n, where the gage curve function is given by the complex trigonometric expression: with π g k ( ϕn = + f k k m Q sin ϕn ( 4f ( ϕn f ( f k π f k k ϕ n = + n. ( π ϕ Fig. 3 (bottom shows the first-orer erivatives of the gage curves for selecte phase currents which are a measure for the output torue. There are no iscontinuities in these curves or their rate of change resulting in a smooth calculate torue. EPE 5 - Dresen ISBN : P.5

6 .5. 8 aligne 4 Flux [Vs] unaligne phase current [A] Fig. 4: Comparison of measure flux-linkage characteristic an the ata use in the online machine moel for ifferent rotor positions (bol lines refer to calculate curves of the analytical machine moel Furthermore a strong emphasis was put on the feasibility of calculating the esire values with a reasonable short computation time. For the torue control algorithm escribe in the following paragraph a turnaroun time of 6µs coul be achieve. This value is sufficient for the control of the instantaneous torue in the lower spee range up to nominal spee. Fig. 4 shows the goo agreement between the magnetization ata obtaine in measurements an the magnetization curves which are calculate by the propose analytical machine moel. Besies the propose moel is very flexible regaring its use various control algorithms. Furthermore, it has proven its suitability for various ifferent machine esigns. 3 Measurement results First of all the new machine moel was use for simulating the rive performance. It showe very goo conformity with the results of extensive measurements []. After that the moel was implemente in the test bench consisting of a power converter accoring to the new converter scheme in Fig. an a 3-phase 3kW/9rpm SR machine which has alreay been subject of profoun investigations [4]. For the lower spee range up to nominal spee (here: n nom =6rpm the new machine moel can be use for a real-time instantaneous torue control. In contrast to many other propose control schemes (e.g. presente in [], it oes not reuire any aitional look-up tables. Fig. 5 shows the block iagram of the torue control of one machine phase implemente in a SPACE real-time environment. The reuire average instantaneous phase current i avg is calculate by using the new machine moel (here in terms of i(t,ϕ with the ajuste reference output torue T ref an the actual phase angle ϕ act on the input sie. For calculating the phase current i avg epening on T ref an ϕ act the inverse function T ref (i,ϕ act has to be use. Due to the complexity (cause by the high accuracy of the machine moel of the cubic function i avg (T ref,ϕ act the current cannot be erive irectly from the torue comman T ref. Nevertheless, by implementing an iterative solving algorithm the reuire EPE 5 - Dresen ISBN : P.6

7 phase current [A] phas e curre nt [A] phase current [A] Dynamic Torue Control for Switche Reluctances Drives base on a new Online Machine Moel phase current can be etermine. Fortunately, this algorithm oes not nee many iterations an is therefore very fast because the current value can be estimate uite accurate in avance. Any resiual errors in the machine moel can be compensate with an integrator ajusting the reference value of the torue. The average torue can be calculate either by the machine moel itself (i.e. T act (i ph,ϕ act or by using aitional harware ([5]. In this case the phase current an voltage is use for etermining the average torue []. When using the online machine moel the actual torue can be erive from the magnetic co-energy W mag *. The co-energy W mag * (i ph,ϕ act can be calculate by integrating the magnetization curve Ψ(i ph,ϕ act : W * mag (i ph iph, ϕ act = Ψ(i, ϕact i with Ψ ( i, ϕ act = Ψ (i + [ Ψ (i Ψ (i] p i ( ϕ act ( ϕ = const. By ifferentiating the co-energy, the actual torue can be obtaine: * * Wmag (i ph, ϕact Wmag (i ph, ϕact + ϕ Wmag (i ph, ϕact ϕ Tact (i ph, ϕact = (3 ϕ ϕ Here the torue is assume to be linear aroun the point of operation. For small values of ϕ this solution provies sufficient accuracy, while the more precise calculation using the erivative of * W mag woul be more time-consuming an only slightly more accurate. * ϕ act, V, V Cx, f Sref, i ph Flux [Vs] i(ϕ act,v, V Cx,f Sref,i ph ϕ act i hys T ref Flux [Vs] i(t,ϕ i avg.5 - i max i min hysteresis current controller converter SRM - T act Flux [Vs] i ph T(i,ϕ Fig. 5: Block iagram of the ynamic torue control The switching signals for the converter are prouce by a current hysteresis controller. The with of the hysteresis ban i hys can either be constant or calculate in orer to achieve a constant switching freuency f Sref. In this case the with of the hysteresis ban i hys is erive from the actual rotor position ϕ act, the DC-link voltage V, the actual phase capacitor voltage V Cx (see Fig., the reference value of the switching freuency f Sref an the referring phase current i ph. The propose machine moel is capable of calculating the value of i hys without increasing the calculation time by a consierable ϕ act EPE 5 - Dresen ISBN : P.7

8 amount. Choosing a fixe switching freuency has a very positive impact on the acoustic noise of the machine because known resonance freuencies can be avoie. Fig. 6 shows the principle phase current waveform an the applie switching scheme in motor operation. The phase magnetization starts at α an the phase current is kept in the hysteresis ban by applying either switching state (+V or 4 (V to the phase wining. For α <ϕ act < β, switching state 3 (+V +V Cx is use instea of switching state (+V for increasing the current. The energy store in the internal capacitor C x has to be reuce in the chopping section because the capacitor has to be able to store the magnetic energy of the corresponing phase wining again uring emagnetization (ϕ act > β. For β <ϕ act < β, the phase current is ecrease by applying switching state 4 (V again. The phase wining is emagnetize slowly. This feature again reuces the acoustic noise which is - in case of conventionally esigne switche reluctance machines - very sensitive to fast phase magnetization or emagnetization in the area aroun the aligne position. For the complete emagnetization switching state (-V Cx is applie where the capacitor is recharge to its initial energy level. i ph el i hys i max i avg i hys v ph V+V Cmax ϕ act V α α β β ϕ act -V Cmax Fig. 6: Principle phase current waveform with varying hysteresis with (top an applie switching voltage scheme (bottom in the lower spee region up to nominal spee (rpm < n < 6rpm During phase commutation the torue prouce by the outgoing phase can be calculate by the machine moel (using T act (i ph,ϕ act an subtracte from the reference value of the ongoing phase. This ensures a constant output torue for all phase angles incluing phase commutations (Fig. 7. Due to the nonlinear torue characteristic of the switche reluctance machine the phase current reaches its maximum at ϕ act = β. For β <ϕ act < β the outgoing phase still prouces a consierable amount of torue. Therefore the ongoing phase can be magnetize slowly. In orer to keep the calculation time of the online machine moel short, only one phase current is calculate at the same time. For α <ϕ act < β the moel is assigne to the corresponing phase. The istance between α an β is exactly one thir (or electrical egrees for a 3-phase machine of the electrical perio. Furthermore the switching angles can remain unchange for the entire spee range up to nominal spee without affecting the uality of the output torue uring phase commutation. EPE 5 - Dresen ISBN : P.8

9 iphase [A] Tact [Nm] t [ms] Fig. 7: Measurement results: motor operation, n=rpm, V =5V, T ref =6Nm The experimental results given in Fig. 7 were obtaine at rpm an the nominal DC-link voltage V =5V. The switching freuency is about.8khz. The output torue is constant even uring phase commutation. By increasing the switching freuency the torue ripple can be ecrease significantly. Due to the large gap between the rotor teeth of the use machine the maximum reuire phase current is relatively high compare to the average value. The maximum phase current can be reuce by improving the esign of the magnetic circuit or by increasing the number of motor phases. iph [A] Vph, VC [V] aligne V =9V unaligne V =5V V ph (V =9V aligne V C (V =9V unaligne t [ms] Fig. 8: Measurement results: motor operation, n=765rpm, T ref =9Nm; phase currents (top for V =9V an V =5V; corresponing switching states acc. to Tab. I; phase voltage V ph an capacitor voltage V C (bottom for V =9V Furthermore the new online machine moel can also be use for torue control in the upper spee range above nominal spee (i.e. n nom <n<n max with n nom =6rpm, n max =9rpm, where the maximum value of the back e.m.f. is bigger than the nominal DC-link voltage V nom (here: EPE 5 - Dresen ISBN : P.9

10 V nom =5V. Using a conventional asymmetric half-brige converter it is not possible to increase or ecrease the phase current at every rotor position. Therefore the current is usually controlle by ajusting one turn-on an one turn-off angle, only. This single pulse operation moe leas to high torue ripple an the achievable output torue is very sensitive to a weak DC-supply. In conjunction with the converter concept in Fig. the number of basic switching angles is increase to a minimum of four. This allows at least one boost-perio to be ae where the current can always be controlle inepenently of the rotor position an the DC-link voltage. During this boost-perio the aitional capacitor voltage - which can be controlle inepenently of V - is applie to the corresponing phase. In Fig. 8 a point of operation at 6% an % of the nominal DC-link voltage is shown. The current waveform is improve compare to the single pulse operation of conventional converters. The same is true for the torue ripple reucing the acoustic noise significantly. Here, the new online machine moel is use to calculate the phase current value that is reuire to achieve the reference torue. Afterwars the switching angles for the following electrical perio are etermine. These angles guarantee the esire pre-calculate phase current course. Thus, both the current waveform, the torue an the capacitor voltage can be controlle leaing to an optimize performance of the SR rive. 4 Conclusion A new machine moel for the fast ynamic online torue control in SR Drives is presente. The propose moel allows a very precise simulation of the rive performance as well as an accurate online control. Due to its basic structure it is suitable for online computation an allows for all environmental parameters (e.g. a varying DC-link voltage to be consiere in the calculations. The moel has been implemente an teste in a complete 3kW-SR-Drive system incluing the converter concept in Fig.. A torue control algorithm for the lower spee range using the propose moel is escribe an experimental results are shown. It is important to emphasize that the machine moel is not limite to the lower spee range but is also very useful for the online computation of the switching angles in the upper spee range (single pulse operation above nominal spee. This is especially true in conjunction with the converter concept in Fig. with its extene egrees of freeom in phase current control. Here the number of varying switching angles is increase compare to the asymmetric half-brige converters where there are only two switching angles that have to be etermine in the upper spee range. 5 References [] M. Hiller, R. Maruart, Performance Enhancement of Switche Reluctance Drives by a novel Converter Concept, Conference Proceeings EPE 3, Toulouse [] R. B. Inerka, Direkte Drehmomentregelung geschalteter Reluktanzantriebe, Dissertation, RWTH Aachen, 3 [3] T.J.E. Miller, M. Glinka, M. McGilp, C. Cossar, G. Gallegos-Lopez, D. Ionel, M. Olaru, Ultra-fast Moel of the Switche Reluctance Motor, Conference Recors IEEE-IAS Annual Meeting, 998, St. Louis, pp [4] A. Greif, Untersuchungen an Geschalteten Reluktanzmaschinen für Elektrofahrzeuge, Dissertation, Universität er Buneswehr München, [5] J. Hilinger, M. Hiller, R. Maruart, Sensorless Position Measurement of Switche Reluctance Drives at lower spee, Conference Proceeings EPE 3, Toulouse [6] S. Hossain, I. Husain, A Geometry Base Simplifie Analytical Moel of Switche Reluctance Machines for Real-Time Controller Implementation, IEEE Transactions on Power Electronics, Vol. 8, No. 6, November 3, pp [7] M. Stiebler, K. Liu, An Analytical Moel of Switche Reluctance Machines, IEEE Transactions on Energy Conversion, Vol. 4, No. 4, December 999, pp. -7 [8] N. Raimov, N. Ben-Hail, R. Rabinovici, Simple Moel of Switche-Reluctance Machine Base Only on Aligne an Unaligne Position Data, IEEE Transactions on Magnetics, Vol. 4, No.3, May 4, pp EPE 5 - Dresen ISBN : P.