proposed fem-based optimization method for economical design of long permanent-magnet guideways with...

$: Proposed FEM-Based Optimization Method for Economical Design of Long Permanent-Magnet Guideways With $\hbox{HT}_{\rm c}$ Superconductors$
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 2, APRIL 2014 3600609

Proposed FEM-Based Optimization Method forEconomical Design of Long Permanent-Magnet

Guideways With HTc SuperconductorsArsalan Hekmati

Abstract—The levitation force between a superconductor diskand a permanent magnet (PM) is the origin for possible industrialapplications. The PMs form the most costly part of the permanent-magnet guideways (PMGs), particularly for longer distances. Aheuristic optimization method based on simulated annealing hasbeen proposed for the determination of the optimum arrangementand dimensions of PMs in several PMG structures. This optimumdesign is the most economical PMG design that guarantees thesatisfactory levitation performance of the PMG. As part of theoptimization process, a FEM-based method has been used, basedon the estimation of penetration depth and self-inductance of thesuperconductor disk, for the calculation of the levitation force. Theoptimization results contain the PMG structure with minimumPM consumption for a specified levitation height and supercon-ductor disk characteristics and the variation of the most importantPMG features, such as its cost and width, versus those distinctivesystem parameters. These results will provide basic analysis for thedesign of HTS-PMG levitation systems. Several guideways havebeen fabricated based on the optimization process outputs andhave shown satisfactory results.

Index Terms—Guideway, high temperature, optimization, per-manent magnet (PM), superconductor.

I. INTRODUCTION

THE LEVITATION between high-temperature supercon-ductors (HTS) and permanent magnets (PMs) has re-

sulted in a wide range of applications, such as the fields ofbearings [1]–[3] and transportation systems [4]–[8], with themain advantages of high or super-high speed, low maintenance,and low energy consumption. Among these devices, there isa large interest in Maglev systems where the levitation ofvehicles above PM tracks is attained by the bulk HTS, suchas transportation and launching systems, such that, due to theirgreat potential and vast applications, improving the levitationperformance of these Maglev systems is particularly important.As the levitation and guidance are the result of the interactionbetween permanent-magnet guideways (PMGs) and onboardHTS bulk, the PMG plays an outstanding role in Maglevsystems, and its optimization is a direct and influential approach

Manuscript received July 28, 2013; revised October 17, 2013; acceptedDecember 11, 2013. Date of publication December 23, 2013; date of currentversion January 15, 2014. This paper was recommended by Associate EditorP. J. Masson.

The author is with the Department of Electrical and Computer En-gineering, Shahid Beheshti University, Tehran 1983963113, Iran (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TASC.2013.2295840

to improving the levitation performance of the vehicle. Somearrays of PMs such as the Halbach array have been proposed[9]–[11] and applied in electrical machines, magnet bearings,particle accelerators, and Maglev designs [12]–[15]. Workshave been done on the modeling of the levitated superconductor[16]–[18], analysis of the applied magnetic field’s impact on thelevitation and guidance force [19], [20], and the experimentaland numerical evaluations of the levitation force [3], [21]–[28].In recent years, studies have been performed on the design op-timization of different types of PMGs, changing the orientationor size of the magnets [3], [8], [10], [14], [15], [20], [25]–[27], [29]–[31], modifying the dimensions or arrangements ofthe superconducting bulks [20], [27], [29], [32]–[35], studyingthe influence of the cooling process of the superconductor [3],[20], [25], [31], [36]–[41], or analyzing the superconductinglevitation and its stability [3], [11], [28], [42]–[50].

In this paper, the application of a heuristic method for theoptimization of the PMG design has been studied to makebetter use of the PMs and the superconductor. In this paper,Bean’s critical-state model has been utilized for the distributionof the current density inside the superconductor disk, and thepenetration depth of the magnetic flux is estimated by themethod in [51]. The levitation force of the YBaCuO bulk abovea NdFeB guideway has been studied in zero-field cooling [32],[52] and is calculated by Ampere’s force density [53], throughthe calculation of the self-inductance of the superconductorbulk by the formula of Lorenz [54]. The magnetic flux densityof the PMG has been calculated via the 3-D simulation of thePMG, which has been implemented using FLUX3D.

Considering the influence of the magnet arrangement onthe levitation force, a method for the determination of theoptimum PMG arrangement is presented, which, being the mosteconomical design, guarantees the satisfactory levitation oper-ation. With the results of this work, we want to establish someguidelines on what the characteristics of the superconductor andmagnets of the system should be to achieve a desired levitationforce with minimum PM consumption, which is particularlyimportant in lengthy PMGs for long-distance transportation.In the proposed optimization algorithm, the levitation forcesare calculated in different PMG structures, and the dimensionsof the PMG with minimum PM consumption are determinedfor specific levitation heights and superconductor disk dimen-sions, for each PMG structure. Based on these optimizationoutputs, the influence of the distinctive parameters of the sys-tem, such as the levitation height and the superconductor diskcharacteristics, on the important PMG features, such as its cost

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3600609 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 2, APRIL 2014

Fig. 1. Total levitation force in the antisymmetric PMG structure is zero.

and width, are investigated. Prototypes are fabricated based onthe optimization results that show satisfactory levitation andguidance operation.

II. PMG ARRANGEMENTS

In this paper, the HTS levitation system consists of a disk-shaped HTS and rectangular PMs. The HTS bulk disk in thisstudy has the diameter of 8 mm and the height of 2 mm. Themeasurement of the critical current density, i.e., Jc, for theHTS bulk with the field trap method [55], shows the value of10 A/mm2.

Regarding the longitudinal line crossing the center of thePMG, the possible PMG structures may be symmetric (asN–S–N, N–S–S–N, N–S–N–S–N), antisymmetric (as N–S,N–S–N–S, N–S–N–S–N–S), or asymmetric. It is discussed inpart III that the levitation force is the cross-product of themagnetic flux density and the screening current density insidethe superconductor disk, and therefore, the parallel componentof the magnetic field to the PMG surface is responsible forthe levitation force. Due to this fact, the levitation forcesin the antisymmetric structures would cancel out, as shownin Fig. 1. Furthermore, as stable motion is expected for thesuperconductor disk, the asymmetric PMGs—because of theunbalanced magnetic field patterns—are not considered in thisstudy. Therefore, the possible structures to be considered for themagnet arrangement would be as shown in Fig. 2(a)–(c). Thespecial pattern considered in these structures may be extendedto more complicated PMGs; however, regarding the HTS diskdimensions utilized in this study, the first three structures havebeen considered. The final results would be extendable to all ofthe structures. The two adjacent magnet rows in the PMG struc-ture may have similar magnetization directions; however, thesestructures—being considered as a subcategory of the previousmain structures—are not included in this study. At structure I,a sole magnet forms a longitudinal sample of the PMG, and thevariable parameters are the width w, the height h, and the lengthl of each magnet, as well as the distance of each magnet fromthe adjacent magnet d, as shown in Fig. 2(a). At structure II,the longitudinal sample of the PMG consists of three magnets:two side magnets with the length, width, and height of ls, ws,and hs, respectively, and the longitudinal distance do betweentwo adjacent side magnets, and a central magnet with thelength, width, and height of lc, wc, and hc, respectively, andthe longitudinal and the latitudinal displacements of dm and di,respectively, relative to the side magnets, as shown in Fig. 2(b).The longitudinal sample of the PMG at structure III is formedof five magnets: two side magnets, two middle magnets, and

Fig. 2. PMG structures considered. (a) Structure I. (b) Structure II.(c) Structure III.

one central magnet with the characteristic parameters as inFig. 2(c).

The rectangular magnets are available with a high variety ofdimensions. It has reasonably been assumed at this study thatthe magnets have constant magnetization density, and the costof the magnets is proportional to their volume, which is a goodassumption according to the market.

First, the levitation force has to be calculated for the PMGstructures.

III. CALCULATION OF LEVITATION FORCE

The shielding current is induced in the superconductor bulk,in the process of imposing the external field, to cancel thevariation of the field in the bulk. It is necessary to determine thesupercurrent distribution in the YBCO disk in order to calculatethe levitation force by the specific method presented.

A. Shielding Current Distribution

The shielding current density is assumed to be constant at Jc,which is the critical current density of the HTS disk, based onBean’s critical-state model [56]–[58]. This model has shownsatisfactory results in levitation force calculation as in [59]and [60]. To specify the distribution of the shielding currentin the superconductor disk, the flux penetration depth should bedetermined. Generally, the dimensions of the superconductordisk may affect the pattern of the current density distribution. In[61], it has been shown that for relatively long HTS cylinders,

HEKMATI: OPTIMIZATION METHOD FOR ECONOMICAL DESIGN OF LONG PMGs WITH HTc SUPERCONDUCTORS 3600609

Fig. 3. (a) Simulation of PMG structures in flux 3-D. (b) The radial and axialcomponents of the magnetic flux density in a typical height above the PMGstructures.

the assumption of constant Jc over a constant penetration depthis quite acceptable but for very thin disks; in addition to theconstant Jc over a penetration depth, there are current densitiesof low magnitude for smaller radii. The current density sharplydecays from Jc to zero in this area inside the superconductordisk [60], which has been ignored in this study. The bulksuperconductor disk with the radius of rd and with the invasiondepth of δ is assumed to be a coil of outer diameter 2rd,inner diameter 2(rd − δ), height hd, and self-inductance L(δ).Using the shielding current distribution Jc, the total current thatflows in the bulk would be as Jcδhd, and at the same time,using the HTS disk self-inductance L(δ), the current is equalto φ/L(δ), where ϕ is the total magnetic flux generated in

Fig. 4. PM volume per unit length of the PMG for known levitation height andHTS disk characteristics, it is assumed: ws = wc = w, hs = hc = h, ls =lc = l, di = dm = do = 0.

the superconductor bulk. Thus, the penetration depth may bedetermined by solving (1) numerically [51], i.e.,

Jcδhd =φ

L(δ). (1)

The total magnetic flux generated in bulk—to cancel out the ex-ternal magnetic flux—is calculated from the following equation:

ϕ =

∮B.dS =

2π∫0

rd−δ∫0

Bexz(r, ϕ)r dr dϕ (2)

where Bexz is the axial direction ingredient of the externalmagnetic field (perpendicular to the PMG surface) and may becalculated everywhere in the space from the FEM simulationresult. In addition, L(δ) was calculated by the exact formula ofLorenz [54]. There, it has been proved that the inductance is as

L =32

3

πr3mh2d

{2k2 − 1

k3E(k) +

1− k2

k3K(k)− 1

}(3)

for an integrated coil or disk of rectangular cross section, whererm is the mean radius: rd − δ/2, and k is calculated from thefollowing equation:

k2 =4r2m

4r2m + h2d

(4)

where K(k) and E(k) are, respectively, the complete ellipticintegrals of the first and second kind, of modulus k [54].

B. Levitation Force

The levitation force between a PM and an HTS disk isthe Lorentz force between the external magnetic field and theshielding current and can be calculated based on Ampere’sforce density, i.e., f , which is the cross-product of the currentdensity and the magnetic flux density. Thus, levitation force Fz

is given by [53]

Fz=

∫∫∫VSC

f.dV

=

∫∫∫VSC

(JSC ×B).dV

=

∫∫∫VSC

(JSC×BPM ).dV +

∫∫∫VSC

(JSC ×BSC).dV (5)


Fig. 5. Optimization flowchart.

where V is the volume, Vsc is the HTS volume, Jsc is thesupercurrent density, B is the total magnetic flux density, BPM

is the magnetic flux density generated by the PM, and Bsc isthe magnetic flux density generated by the HTS.

Since the cross-product of the current density and the mag-netic flux density is zero for a superconductor, (6) is obtainedfrom (5) [53], i.e.,

Fz =

∫∫∫VSC

(JSC ×BPM ).dV. (6)

To solve the given equation, it is necessary to know the appliedfield BPM and the supercurrent distribution JSC , wherein thefirst is obtained from FEM simulation results, and the latteras discussed in part A. Regarding the superconductor diskdimensions, the levitation force would be

Fz =

h0+hd∫h0

2π∫0

rd∫rd−δ

JcBexr(r, ϕ, z)r dr dϕ dz (7)

where h0 is the levitation height, and Bexr is the radial directioningredient of the external magnetic field (parallel to the PMGsurface).

Generally, Jc may change with the temperature and theapplied magnetic fields. The dependence of Jc to temperatureis not considered in the proposed optimization method. Thevalue of Jc in different applied magnetic fields may be chosenfrom the empirical curves of Jc variation versus the perpen-dicular magnetic fields or from the well-known Kim model[62], considering the maximum perpendicular ingredient of theexternal magnetic field, occurring at the outer radius of thesuperconductor disk, as the criterion for the Jc determination.

The levitation force is equal to the disk weight modified toinclude the buoyancy of liquid nitrogen. Thus, (8) yields therequired levitation force [63]. Thus

FZ = (Md − VdρLN2)g (8)

where Md and Vd are, respectively, the disk’s mass and volume,and ρLN2

is the density of liquid nitrogen.


IV. SIMULATION OF PMG STRUCTURES

The discussed PMG structures have been simulated in flux3-D through which the radial and axial components of themagnetic flux density may be determined at every point oftypical cross sections along the PMG structures. As an example,the 3-D simulation and the components of the magnetic fluxdensity for a cross section in PMG structure II and also in atypical height from the PMG surface are shown in Fig. 3.

V. OPTIMIZATION PROCESS

There is an optimized PMG design with minimum PMvolume and, therefore, minimum cost, for every proposedstructure. There is a simple explanation for this. Assume, forexample, the PMG structure II with the same width and heightfor all the magnets and all the interdistances equal to zero. Thesimulation results show that changing the width and the heightof the PMs, for an HTS disk with parameters as mentioned inpart II and for a levitation height of h0 = 1 cm, the PMs volumeper unit length of the PMG would be as in Fig. 4.

As it is evident in Fig. 4, there is a PMG design with min-imum PM volume (w = 11.3 mm, h = 4.7 mm). Generally,finding the optimum design is not easy, and heuristic optimiza-tion methods should be used. The simulated annealing methodhas been utilized to determine the optimum arrangement anddimensions of the magnets.

The overall proposed optimization process is presented inFig. 5. In this process, the target function is chosen to bethe volume of the PMs per unit length of the PMG, which iscalculated as the volume of the PMs in the longitudinal sampleof the PMG structure divided by the length of the sample andis a measure of the PMG cost. It may be calculated as (9),(10), and (11) for the three structures I, II, and III, respectively.Thus

f(w, l, h, d) =w×l×h

l+d(9)

f(ws, ls, hs, wc, lc, hc, do)

=2×ws×ls×hs+wc×lc×hc

ls+do(10)

f(ws, ls, hs, wm, lm, hm, wc, lc, hc, do)

=2×ws×ls×hs+2×wm×lm×hm+wc×lc×hc

ls+do. (11)

During the optimization process, the levitation force for aspecific sample of PMG dimensions is calculated, as discussedin part III, and is compared with the required force for thelevitation. To consider the entire PMG longitudinal sample,the levitation force is calculated in different positions of thesuperconductor disk along the sample. To do this, a step of αis defined as being dependent on FEM modeling accuracy andalso the dimensions of the superconductor disk and the PMGlongitudinal sample. Typically, it may be chosen about 1 mm.The process continues until an acceptable level of accuracy isobtained.

Fig. 6. PM volume per unit length of PMG versus the levitation height, radiusand height of the HTS disk: rd = 8 mm, hd = 2 mm.

VI. RESULTS AND DISCUSSION

The PMs’ volume per unit length of the PMG has beenobtained for different levitation heights at constant dimensionsof the superconductor disk. The result is plotted in Fig. 6.

The important result is at different levitation heights; theminimum volume of the PMs is obtained via different PMGstructures. For example, for the assumed dimensions of thesuperconductor disk, the radius of rd = 8 mm and the heightof hd = 2 mm, for the levitation height beyond approximately14 mm, PMG structure I seems the most economical, for thelevitation height between 6 and 14 mm and below 6 mm, PMGstructure II and PMG structure III are the most economicalstructures. This is mainly due to the fact that the factor forinducing the screening currents at the superconductor disk is theaxial component of the magnetic flux density—perpendicular tothe PMG surface—at the superconductor disk position, but theeffective factor on the levitation force is the radial componentof the magnetic flux density—parallel to the PMG surface—atthe superconductor disk position [51]. Relation (7) for thecalculation of the levitation force shows the dependence of thelevitation force on the supercurrent density and its distributionin the superconductor disk and also on the radial component ofthe magnetic flux density at the superconductor disk position.Different PMG structures may differ in these two factors,and a tradeoff is always present between the axial and radialcomponents of the magnetic flux density at the superconductordisk position. Based on the levitation height needed, one ofthese factors becomes eminent, making the related structure themost economically suitable PMG. At first glance, the differencebetween the curves in Fig. 6 may seem not that much, but itshould be considered that the volume of the PMs is presentedfor a length of 1 mm on the PMG, and for the long PMGs, thedifference in PM volume could be quite considerable.

The variation of the PM volume per unit length of PMG ver-sus the radius of the HTS disk is shown in Fig. 7, for different


Fig. 7. PM volume per unit length of PMG versus the radius of the HTS disk,height of the HTS disk: hd = 2 mm, and levitation height of (a) h0 = 4 mm,(b) h0 = 20 mm.

levitation heights. As may be seen, increasing the radius of theHTS disk, the cost of the PMG rapidly increases, and whichPMG structure is more economical is directly dependent on therelated levitation height. The order of the PMG structures fromthe economical viewpoint is quite changed when the levitationheight is varied from 4 to 20 mm, as may be seen in Fig. 7(a)and (b). Similar curves are obtained for the variation of PMGcost versus the height of the HTS disk.

Another factor that is important in the PMG structures isthe width of the guideway, its importance being due to theprovision of the motion band for the HTS disk. Experimentsshow that wider PMG structures provide a more stable motionfor the HTS disk particularly at twisty guideways. PMG widthis extracted from the calculated optimum PMG dimensions

Fig. 8. PMG width versus (a) height of HTS disk, (b) radius of HTS disk, and(c) levitation height.

obtained from the optimization process. It is calculated as (w)for PMG structure I, as (2ws + 2di + wc) for structure II,and as (2ws + 2wm + wc + 2di + 2dc) for structure III. Theresults for the variation of the PMG width versus the heightand radius of the superconductor disk and the levitation heightare shown in Fig. 8(a)–(c), respectively. It is evident in the


Fig. 9. Typical PMG fabricated.

Fig. 10. Fabricated PMG for the 1-cm levitation height of an HTS disk withradius, height, and critical current density of 8 mm, 2 mm, and 107 A/mm2,respectively.

figures that PMG structure I has always the least width and thatstructure III is always the widest guideway. Furthermore, theradius of the HTS disk has the most severe impact on the PMGwidth relative to the levitation height and the height of the HTSdisk, as shown in Fig. 8.

Prototype guideways with the optimum dimensions proposedby the optimization process have been fabricated and tested.A typical fabricated PMG is shown in Fig. 9. For the HTSdisk with the characteristics as in part II and a levitation heightof h0 = 10 mm, the most economical structure is structure II,according to Fig. 6, which its dimensions are calculated as:ws = 9.3 mm, wc=11.8 mm, ls=19.3 mm, lc=24.3 mm,hs=3.6 mm, hc=4.2 mm, do=6.1 mm, di=3.4 mm, dm=10.1 mm. The fabricated prototype with these calculated di-mensions, shown in Fig. 10, has shown satisfactory levitatingoperation.

VII. SUMMERY AND CONCLUSION

In this paper, a process has been proposed to determinethe optimum design of the PMGs, which yields the minimumvolume of the PMs. Therefore, this design is the most econom-ical design guaranteeing the satisfactory performance of thePMG for specific superconductor disk dimensions and levita-tion heights. The influence of varying the superconductor diskdimensions and the levitation heights on the PMG design and

characteristics have also been presented. The levitation forcebetween HTS bulk and PM is analyzed by using 3-D FEM.

It is concluded that at different levitation heights, the mini-mum volume of the PMs is obtained via different PMG struc-tures. This is due to the tradeoff between the axial and radialcomponentsc of the magnetic flux density at the superconductordisk position at different levitation heights. It was also shownthat increasing the radius of the HTS disk, the cost of the PMGrapidly increases, and the economical PMG structure is directlydependent on the related levitation height.

The width of the PMG, as an important characteristic of thePMG, has also been compared in different structures. It hasbeen shown that structure I has the least width, structure IIIis always the widest guideway, and the radius of the HTS diskhas the most severe impact on the PMG width relative to otherparameters.

These results may provide basic analysis for the design ofHTS-PMG levitation systems. Some guideways have also beenfabricated and tested with satisfying results. In this paper, threesample PMG structures have been studied; however, more com-plex structures of higher orders may be studied and comparedwith other known arrays as the Halbach array, and multiobjec-tive optimization problems may be defined with target functionsas the motion stability, the cost, and the width of the guideway.

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Arsalan Hekmati was born in Iran in 1982. He received the B.S., M.S.,and Ph.D. degrees in electrical power engineering from Sharif University ofTechnology, Tehran, Iran, in 2005, 2007, and 2011, respectively.

He has been a member of the Superconductive Electronics Research Labo-ratory since 2007 and was a Postdoctoral Research Fellow from 2011 to 2012with Sharif University of Technology, where he conducted projects in appliedsuperconductivity, such as the fabrication and characterization of superconduc-tor YBCO cylinders and the fabrication and test of inductive SFCLs for the firsttime in Iran. From 2006 to 2011, he was the Head of the Research DevelopmentDepartment, Iran Transformer Research Institute and a Project Leader with IranTransfo. Company, Ofogh Consulting Engineers Company, and the DistributionNetwork Companies. Since 2012, he has been with the Department of Electricaland Computer Engineering, Shahid Beheshti University, Tehran. His researchinterests include superconducting power devices, insulation and high-voltagesystems, and electric machinery, particularly transformers.

proposed fem-based optimization method for economical design of long permanent-magnet guideways with...

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