odelling and Powe Facto Impovement of Switched Reluctance oto Dive X. D. XUE, K. W. E. HENG, S.. HO and Y.P.B.YEUNG Depatment of Electical Engineeing The Hong Kong Polytechnic Univesity Hung Hom, Kowloon, Hong Kong P. R. of HINA Abstact: - This pape pesents a summay of eseach wo on the modelling and development of powe facto impovement techniques fo switched eluctance moto (SR) dives. Fou models of SR ae pesented, which include an intepolation model using two-dimensional (2-D) bicubic spline, an analytical model based on 2-D least squaes technique, a hybid model based on 2-D bicubic and 2-D bilinea splines, and also a model including mutual coupling. The authos have also investigated the effects of the contol and output paametes upon the powe facto in SR dives. Futhemoe, a new contol stategy and two eal-time schemes to impove the powe facto of SR ae developed and epoted. Key-Wods: - achine odel, Powe Facto, Switched Reluctance oto (SR) Dive Intoduction SR has salient poles on both stato and oto. It has vey simple stato windings and thee ae no oto windings. The ai gap eluctance between the salient stato and oto poles ae dependent on the oto position. Usually, SR opeates in the magnetic flux satuation egion so as to poduce a high toque to mass atio. Hence, it is essential to study the degee of magnetic satuation in high pefomance SR dives. On the othe hand, the nonlineaity aising fom the high satuation of the magnetic chaacteistics tend to complicate the analysis as well as the contol of SR motos. Thus, the modelling of the nonlinea magnetic chaacteistics and the establishment of the model of SR dives ae cucial in the design, pefomance pediction, as well as contol of SR dives. Fo SR dives, the ase cuents ae pulsating by vitue of its ysical featues. Hence the powe facto of SR dive ae geneally poo and the loss in such motos ae high. Theefoe, a study to impove o coect the powe facto in SR dive has become a topical eseach aea ecently. In this pape, a wealth of infomation on the modelling and development of techniques to impove the powe facto of SR dive ae epoted in details. 2 odelling of SR 2. Intepolation model using 2-d bicubic spline This is essentially a piecewise intepolation model. 2-D bicubic spline is selected as the intepolation function. The 2-D bicubic spline function is given by (). g( i) g2 ( i) ψ ( θ, i) ( g( θ ), g2 ( θ ), g3( θ ), g4 ( θ )) A g ( ) () 3 i g4 ( i) whee ψ denotes the flux linage, θ denotes the oto position, i denotes the cuent, both g (θ) and g (i) (, 2, 3, and 4) ae polynomials and A is a 4 4 matix. The flux linage, which is taen as the state vaiable in the poposed simulation model of the SR dive, is defined as an intepolation function of both the oto position and the cuent. In Ref [], the authos descibe the computations of g (θ), g (i) (, 2, 3, and 4) and the matix A in details. The intepolation coefficients in the bicubic spline function can be computed fom a limited numbe of given data. The intepolation of the flux linage with espect to the oto position and the cuent based on this function is quite accuate, as the 2-D bicubic spline function is a thid-ode function which is eadily applicable in both scientific and engineeing poblems. The simulation model of SR dive with voltage souce can be descibed by (2) at steady state. dψ ( u Ri) (2) ω
whee ω denotes the oto speed, u denotes the voltage applied to the ase winding, and R denotes the ase esistance. Fig. ompaison between the expeimental and intepolated magnetic chaacteistics It can be seen that the intepolated cuves as shown in Fig. ae smooth and accuate. Hence, the bicubic spline intepolation can descibe exactly the nonlinea magnetic chaacteistics. In addition, the cuent wavefoms fom the simulation and expeiment in Ref [] ae also demonstating that the poposed intepolation model based on 2-D bicubic spline can poduce excellent simulation of the SR dive. The poposed simulation model equies elatively few expeimental o numeical computation data and the simulation is moe accuate and faste than pevious ones using conventional intepolation methods such as linea intepolation [2], quadant intepolation [3], Dd cubic spline intepolation [4], as well as 2-d bilinea intepolation [5]. 2.2 Analytical model based on 2-d least squaes A novel analytical model of the nonlinea magnetic chaacteistics that consists of 2-D othogonal polynomials fo N nown oto position data and nown cuent data ae given in (3). p q ψ ( θ, i) a ( θ θ ) ( i i ) (3) whee p is a positive intege (p N), q is also a positive intege (q ), a ae the coefficients that will be detemined fom 2-D least squaes technique, off line o on line, and N θ θ / N, i i / (4) It should be noted that the coefficients a have to be computed using the ecusive algoithm. The detailed desciption and mathematical deivation ae given in Ref [6]. In geneal, the accuacy of the poposed analytical model is dependent on the maximum ode numbes (i.e. p and q) of the polynomial. Howeve, that does not mean that the highe ode numbes would esult in small computation eos. Futhemoe, the highe ode numbes ae not popitious to apid computation. Hence, the detemination of both p and q should tae into account of the numbe of the nown oto positions, the numbe of the nown cuents, the computation eos of the model, and the speed of computation. In this study, it is suggested that the maximum elative eo (RE) is used to detemine the maximum ode numbes in the poposed model in ode to obtain good accuacy and fast convegence. Hee RE is defined by RE max N f ( θ, i ) ψ ψ (5) whee f(θ, i ) epesents the computed flux linage values fom the poposed model and ψ epesents the flux linage values fom the nown data. Fig. 2 ompaison between the computed and expeimental values Fig. 2 shows that the expeimental cuves of the flux linage agee vey well with the computed cuves obtained fom the poposed model. It indicates that the poposed analytical model can chaacteize the discete expeimental magnetic data accuately. Based on the above analytical model of the nonlinea magnetic chaacteistics, a new model of SR is also developed, in which the dynamic model of SR and the steady model ae given by (6) and (7), espectively. di dt u Ri ω a ( θ θ ) p q a ( θ θ ) ( i i ) p q ( u Ri) / ω a ( θ θ ) di p q a ( θ θ ) ( i i ) p q ( i i ) ( i i ) (6) (7)
Subsequently, the expession of the instantaneous toque T poduced by one ase winding can be obtained as shown in (8). T ( θ, i p q ) [( i i ) a + + ( i ) ( θ θ ) + ] (8) The simulation and expeimental esults given in Ref [6] show that the simulation cuent wavefom is highly consistent with the measued cuent wavefom. That veifies the poposed model of SR based on 2-D least squaes technique is accuate. In shot, the poposed analytical model of the nonlinea magnetic chaacteistics and the developed model of SR can significantly enhance the convegence speed and computational accuacy in the simulation of the SR dive because of use of the 2-D least squaes algoithm. Thus, the poposed model should be bette than the pevious analytical models such as models with the exponent component [7], models with fist- and second-ode functions [8], models with the compound expession consisting of the exponential and sinusoidal functions [9] and models with the compound component consisting of sinusoidal and polynomials functions []. 2.3 Hybid model based on both 2-d bicubic and 2-d bilinea splines Fist, a lage numbe of fine mesh finite elements ae geneated automatically using 2-D bicubic spline intepolation () as descibed in Section 2.. Then, in each of the element, a 2-D bilinea spline is defined as the element shape function as given in eq.(9). ψ ( θ, ) c + c θ + c i + c iθ (9) i 2 2 22 whee the coefficients c (,, 2) in (9) ae dependent on the coesponding element paametes and thei computation is discussed in [] in details. Equation (9) is used to simplify the moto voltage equation as shown in (2) into the analytical expession of the cuent, which is given by eq. (). i( θ ) Ω c + D 22 u / ω c 2 2 ( + R /( ω c c + c )) 22 (+ R /( ωc 22 )) 22θ () whee Ω epesents the -th element and D, which is the integation constant of the -th element, is computed using the poposed position stepping method by consideing the continuity condition of the cuent fom eq. () as follows: D Ω ( i u / ω c 2 ( θb ) Ω c22 ( + R /( ωc22 (+ R /( ωc 22 )) c2 + c22θb ) )) () whee θ b is the oto position value on the bounday between two adacent elements. (a) Simulation (b) Expeiment Fig. 3 uent wavefoms (the speed 8 pm) The simulation and expeimental esults illustated in Fig. 3 show that the poposed hybid model is accuate and the developed position stepping method is coect. The salient meit of the hybid model is summaized as follows. The fine elements ae geneated automatically using a 2-D bicubic spline to tae cae of the magnetic nonlineaity as descibed ealie. By intoducing a 2-D bilinea spline, the teminal voltage equation of SR dive can be simplified significantly, and the deived analytical expessions endes the computation of the cuent simple and apid. Futhemoe, the developed position stepping method ensues that the poposed model can be solved accuately. In othe wods, the developed hybid model combines the excellent featues of both intepolation and analytical models. 2.4 odel including mutual coupling In geneal, two o moe ase windings in an SR dive ae excited simultaneously when the SR dive
is unning. Hence, thee is mutual coupling between the ase windings. By intoducing the self-inductance, mutual inductance, and coupling coefficient, the model of multi-ase SR with mutual coupling can be deived as shown in (2). di u [ ] [ ω [ ] ] R [ s] [ ω [ d ][ i] ] [ i] (2) di whee [ ] is the deivative matix of the ase cuent, [s] is the switching coefficient matix, [i] is the ase cuent matix, [ ] epesents the coupling matix that is detemined fom (3). The pesented model with mutual coupling is simple and is suitable fo apid compute calculation when compaed with the taditional models based on magnetic cicuits [3] [4]. Futhemoe, the poposed model is moe accuate compaed to othe pio at models because it can accuately descibe the nonlineaity in SR using bicubic spline intepolation. (a) Simulation (the speed 785 pm) [ ] N N N N N N N N N N N N (3) whee denotes the coupling coefficient of ase- to ase-, denotes the mutual inductance of ase- to ase- ( ), and denotes the self-inductance of ase-. d [ ] is the invese matix of [ ]. [ ] is computed fom (4). d d d [ ] dn N N d d N dn N N N N N d N d N d N N N (4) The instantaneous toque T geneated by the ase- is classified into two components, one fom the self-coupling and the othe fom the mutual coupling as shown below. T W ( i, θ ) + θ N W ( i θ, θ ) (5) whee W (i,θ ) denotes the co-enegy poduced only by ase-, i epesents the cuent in the ase- at θ, and W' epesents the co-enegy fom the mutual coupling of ase- to ase-, i epesents the cuent in the ase- at θ ( ). Fig. 4 shows that the ase cuent wavefoms fom the simulation ae consistent with the measued ase cuent wavefoms. (b) Expeiment Fig.4 uent wavefoms (ase-, 2ase-2, 3ase-3, 4ase-4) The moe detailed desciption of the deivation and solution algoithm ae given in [2]. 3 Powe Facto Impovement of SR Dive 3. Effects of the contol and output paametes on the powe facto The powe facto in SR dive should be computed fom the following expession. Pa + Pb + Pc PF (6) S + S + S a whee PF epesents the powe facto of the SR dive, P a, P b and P c ae the thee-ase aveage powe, S a, S b and S c ae the thee-ase appaent powe, espectively. In [5], a detailed desciption on the computation of these powes is given. The effects of the contol paametes and the outputs on the powe facto can be summaized as follows [6] [7]. The tun-on and tun-off angles have significant effects upon the powe facto fo single-pulse, b c
voltage PW, and cuent hysteesis contols. The vaiations in the tun-on and tun-off angles inevitably esult in a change in the powe facto. Neglecting othe limitations on the tun-on and tun-off angles, the maximum powe facto can be ealized though optimizing the tun-on and tun-off angles. The feewheeling angle in single-pulse contol has some effects upon the powe facto. The powe facto may be impoved by changing the feewheeling angle. Howeve, both the tun-on and tun-off angle have a stonge effect than the feewheeling angle upon the powe facto. Fo single-pulse and voltage PW contols, the input voltage almost has no effect upon the powe facto. Howeve, the input voltage influences the powe facto stongly with cuent hysteesis contol. Highe input voltage will esult in a lowe powe facto. Fo voltage PW contol, the chopping modes of PW have a significant effect upon the powe facto. In consideation of optimizing the powe facto, soft chopping is bette than had chopping. Howeve, the chopping modes unde cuent hysteesis contol almost have no effect upon the powe facto. Fo voltage PW contol, the duty cycle of PW affects the powe facto stongly. The powe facto inceases with an incease in the duty cycle. Fo cuent hysteesis contol, the efeence cuent has a significant effect upon the powe facto. Inceasing the efeence cuent will esult in a high powe facto. The hysteesis band affects the powe facto consideably with cuent hysteesis contol. The powe facto inceases with an incease in the hysteesis band. The dc lin capacito influences the powe facto stongly. To be specific, the powe facto deceases with an incease in the value of the dc lin capacito. The moto speed has a ponounced ffect upon the powe facto in both single-pulse and cuent hysteesis contol. The powe facto deceases with an incease in speed at specific tun-on and tun-off angles fo single-pulse opeation. oeove, the powe facto will become high if the speed goes up with cuent hysteesis contol. Fo single-pulse contol, the effect of the toque upon the powe facto is vey wea at specific tun-on angle and tun-off angles fo a given speed. Howeve, the toque has a ponounced effect upon the powe facto since the powe facto inceases with an incease in the aveage toque with cuent hysteesis contol. 3.2 New stategy and two eal-time schemes to impove the powe facto Fom the achievements in Section 3., it can be found that the tun-on angle, the tun-off angle, and the speed ae the ey paametes affecting the powe facto. Thus, the novel stategy being poposed is to impove the powe facto by adusting the tun-on and tun-off angles. Two optimization methods ae utilized to optimize the tun-on as well as the tun-off angles and to veify the poposed stategy. They ae, namely, the quasi-2d seach algoithm and the Hooe-Jeeve patten seach algoithm as descibed in [8]. The simulation and expeiment epoted in [8] show that the powe facto using the two seach algoithms is highe than that ealized by simple methods that do not have a seach algoithm. Futhemoe, the meit values of the powe facto as defined in [8] using the Hooe-Jeeve patten seach algoithm ae lage than those using the quasi-2d seach algoithm. This indicates that the Hooe-Jeeve patten seach algoithm is bette than the quasi-2d seach algoithm in optimizing the switching angles. Fom the stategy as descibed, the authos have poposed two eal-time schemes fo impoving the powe facto which ae, namely, a eal-time optimization (scheme-a) based on the quasi-2d seach algoithm as illustated by Fig. 5 and the off-line optimization (scheme-b) based on the Hooe-Jeeve patten seach algoithm as depicted in Fig. 6. Fig. 5 Schematic diagam of the scheme-a based on quasi-2d seach algoithm Fig. 6 Schematic diagam of the scheme-b based on Hooe-Jeeve patten seach algoithm It can be seen that scheme-a is suitable fo both constant toque opeation and constant powe opeation. oeove, it needs simple hadwae cicuit and can be implemented eal-time. Scheme-B
is only suitable fo constant toque opeation, but it has fixed tun-on and tun-off angles and can be implemented easily using only softwae contol. In compaison with the pevious methods fo coecting the powe facto [9] [2] [2], the advantage of the poposed stategy and the two developed eal-time schemes is thei applicability ove a wide ange of moto atings. They do not need complicated contol and hadwae cicuits. The dawbac is that both schemes cannot impove the powe facto to unity, and the efficiency of the SR dives could be slightly compomised because the optimal tun-on angle is geneally nea the egion in which invese toque is poduced. 5 onclusion Fou models of SR dive have been pesented. Thee of the schemes do not tae into account of mutual couplings and these ae i) one based on an intepolation model using 2-D bicubic spline, ii) an analytical model based on the 2-D least squaes technique, and iii) a hybid model based on 2-D bicubic and bilinea splines. The fouth model includes self-inductance, mutual coupling and coupling coefficients. All fou models have been validated successfully based on the simulation and expeimental esults. ompaing those thee models without mutual coupling in tems of accuacy, the intepolation model is the best and the hybid model is the second best. Fom the view of simplicity and computation speed, the analytical model is the best and the hybid model is still the second best. lealy, the model with mutual coupling is the best among the fou pesented models in tems of accuacy. The theoetical, simulation, and expeimental studies indicate that the contol and output paametes may affect the powe facto in SR dives. A new contol stategy has also been epoted to impove the powe facto though optimizing the tun-on and tun-off angles. Two eal-time schemes to impove the powe facto ae also developed. The one based on on-line quasi-2d seach algoithm equies simple hadwae cicuit and some simple eal-time computation. The othe which is based on the off-line Hooe-Jeeve patten seach algoithm equies neithe hadwae no any eal-time computation. Howeve both schemes have been implemented easily and successfully as epoted. Refeences: [] X. D. Xue, K. W. E. heng, and S.. Ho, Simulation of Switched Reluctance oto Dives Using Two-Dimensional Bicubic Spline, IEEE TRANSATIONS ON ENERGY ONVERSION, Vol. 7, No. 4, Decembe 22, pp. 47-477. [2] P. J. awenson, J.. Steenson, P. T. Bleninsop, J. oda, and N. N. Fulton, Vaiable-speed Switched Reluctance otos, IEE PRO., Vol. 27, Pt. B, No. 4, July 98, pp. 253-265. [3] J.. Steenson and J. oda, omputation of Toque and uent in Doubly-salient Reluctance otos fom Nonlinea agnetization Data, Poc. IEE, 979,26, (5), pp. 393-396. [4] D. W. J. Pulle, New data base fo switched eluctance dive simulation, IEE PROEEDING-B, Vol. 38, No. 6, Novembe 99, pp. 33-337. [5] S.-U. Rehman and D.G. Taylo, Piecewise odeling and Optimal ommutation of Switched Reluctance otos, Poceedings of the IEEE Intenational Symposium on Industial Electonics (ISIE '95), Vol., 995, pp. 266-27. [6] X. D. Xue, K. W. E. heng, and S.. Ho, Pecise Analytical odelling agnetic haacteistics of Switched Reluctance oto Dives Using Two-Dimensional east Squaes, The 34 th Annual IEEE Powe Electonics Specialists onfeence (PES3), Acapulco, exico, June 5-9, 23, Vol., pp. 46-42. [7] D. A. Toey and J. H. ang, odelling a Nonlinea Vaiable-eluctance oto Dive, IEE PRO., Vol. 37, Pt. B, No. 5, Septembe 99, pp. 34-326. [8] T. J. E. ille and. cgilp, Nonlinea Theoy of the Switched Reluctance oto fo Rapid ompute-aided Design, IEE PRO., Vol. 37, Pt. B, No. 6, Novembe 99, pp. 337-347. [9] S. i, I. Husain, and. Elbulu, Switched Reluctance oto odeling with On-line Paamete Identification, IEEE Tansactions on Industy Applications, Vol. 34, No. 4, July/August 998, pp. 776-783. [] D. N. Essah and S. D. 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H. Rim, W. H. Kim, E. S. Kim, and K.. ee, A hoppingless onvete fo Switched Reluctance oto with Unity Powe Facto and Sinusoidal Input uent, 25 th Annual IEEE Powe Electonics Specialists onfeence, (PES 94), Vol., 994, pp. 5-57. [2] V. K. Shama, B. Singh, and S. S. uthy, Pefomance Analysis of Unity Powe Facto-onvete-Invete Fed Switched Reluctance oto Dive, IEEE Intenational Electical achines and Dives onfeence Recod, 997, pp. WAI/5.-WAI/5.3. [2] T. Gopalaathnam and H.A. Toliyat, A High Powe Facto onvete Topology fo Switched Reluctance oto Dives, 37th IAS Annual eeting, Vol. 3, 22, pp. 647-652.