Force Optimization of a Double-Sided Tubular Linear Induction Motor

IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 12, DECEMBER 2014

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Force Optimization of a Double-Sided Tubular Linear Induction Motor Antonino Musolino, Marco Raugi, Rocco Rizzo, and Mauro Tucci Department of Energy, Systems, Territory, and Construction Engineering, University of Pisa, Pisa 56126, Italy An innovative double-sided tubular linear induction motor is presented, and its optimal design in terms of thrust force is discussed. A dedicated semi-analytical model of the device is developed allowing for fast and accurate evaluation of all the electromechanic quantities in the device, including the thrust force, back-electromotive force, distribution of the induced current, and average magnetic flux density in the teeth. This provides a basis for the design optimization, which has been performed by a novel evolutionary algorithm based on the self-organizing maps. Using the semi-analytical formulation, the characterization of the machine is greatly facilitated, thus allowing a fast evaluation of the cost function and design constraints. Finally, the obtained optimal design is validated by comparison with finite elements method analysis. Index Terms— Electromagnetic analysis, induction accelerators, induction motors, linear motors, optimization methods.

I. I NTRODUCTION

L

INEAR movements in mechanical engineering are usually obtained using rotating motors in conjunction with rotation to translation mechanism. Linear electromagnetic actuators are able to provide thrust force directly to the load without mechanical gears and transmission. As a consequence, the control bandwidth and dynamical performance are improved as well as the overall efficiency and reliability. Among the various linear machine configurations, the tubular one is very attractive because of reduced leakage fluxes, and for the characteristic high speed and high actuating force with relatively low mass. Two basic configurations are available: the synchronous permanent magnet tubular linear motor (PMTLM) [1] and the asynchronous tubular linear induction motor (TLIM) [2]. A number of PMTLMs have been developed [3], [4], and they are widely used in different contexts: 1) electrical power generation [5], [6]; 2) household appliance [7]; 3) automotive [8]; and 4) electromagnetic launch [9]. Despite of the reduced efficiency and power factor, TLIMs have gained interest due to their simple structure and ease of feeding. Applications to transportation systems have been recently proposed in [10], in which the self-centering feature of the device is able to provide the levitation and guidance forces, which is a valuable advantage with respect to the flat linear induction motor [11]. Air-cored TLIM, usually named linear induction launchers, are widely used in electromagnetic launch [12]. In fact, when PM linear synchronous motors are used in electromagnetic aircraft launch system, the switching frequency of the drive electronics must be, at a minimum, in the range of 1800–4200 Hz [15]. These frequencies may border the limits of the capabilities of high-power drives based on insulated-gate bipolar transistor or integrated

Manuscript received October 21, 2013; revised March 11, 2014 and June 11, 2014; accepted July 20, 2014. Date of publication July 25, 2014; date of current version December 12, 2014. Corresponding author: A. Musolino (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2343591

gate-commutated thyristor. Linear induction motors are able to provide the same thrust force with voltages and currents of approximately the same magnitude, but with fundamental frequencies in the range of 30–250 Hz. In general, linear induction motors may represent an alternative to the use of PMTLMs in those applications where very high speeds and thrust forces are required [13], [14]. In these systems, the long stator–short mover configuration is adopted, and the stator is subdivided in sections, which are fed in sequence. Voltages of the order of some kilovolt and currents of several of kiloampere are needed. Previous recent works on optimization of linear motors are mostly based on the use of equivalent network models, which, as known, are fast but may suffer of limited accuracy especially in presence of complex flux paths [16]–[19]. On the other hand, numerical tools are able to provide a detailed description of the distribution of all the electromechanic quantities in the device, taking into account the saturation and temperature effects [20], [21]. This approach is time consuming, and it is not effective to insight the dependence of the performance on the design parameters. Kou et al. [9] perform a design based on an equivalent network model without using any automatic optimization procedure. A similar approach is used in [7], where the authors use an analytical model to define a number of performance indexes, and the optimization is performed on a heuristic base. Shiri and Shoulaie [18] and Bazghaleh et al. [22] perform automatic optimization of a single-sided linear induction motor using evolutionary algorithms (EAs): 1) genetic algorithms (GAs) and 2) particle swarm optimization (PSO), respectively. They both use the well-known Duncan equivalent network, so that they need to define proper correction coefficients to take into account the influence of the edge effects. In this paper, we present a similar approach to [18] and [22], but we refer to the innovative double-sided tubular linear induction motor (DSTLIM), which is characterized by the presence of two concentric stators with a light mover between them. Moreover, instead of the Duncan equivalent network, we introduce a novel semi-analytical model, which

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Fig. 1.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 12, DECEMBER 2014

Schematic 3-D view of the proposed DSTLIM [13].

is fast and accurate and is able to take into account both the entry and exit effects without the need of correction coefficients [23]–[25]. An innovative evolutionary optimization algorithm is also used [26], which is as powerful as GA or PSO. The main advantage of the proposed approach with respect to [18] and [22] is that the proposed semi-analytical model provides the knowledge of the field distribution everywhere in the device with short computation times, and with excellent agreement with a numerical (finite element) model. In this way, in the optimization, we are able to set constraints on local quantities such as maximum admissible values of induced current density and of magnetic flux density. Using equivalent network models does not permit to accurately deal with constraints on local quantities. The proposed model is then perfectly suited for an optimization of the device using EAs, because of its short computation times, and constraint handling capabilities. This paper is organized as follows. Section II introduces the semi-analytical model of the proposed device. Section III describes the design optimization procedure. In Sectiion IV, the proposed optimization algorithm is applied to two motors for different applications. Section VI discusses the influence of leading parameters for a preliminary design. To confirm the validity of the design, finite elements method (FEM) is used and the results are compared.

Fig. 2. View of the cross section of the DSTLIM taken in correspondence of the teeth [13].

Fig. 3.

Example of a lamina.

An accurate analysis of the device requires a 3-D numerical model with consequent long simulation times [27], often resulting in unviable automatic design procedures. Faster analyses can be performed by approximated analytical models, which usually discard the end effects, the presence of the slots and cuts. As known, the effects of the presence of the slots are taken into account by introducing a Carter coefficient [28] and equivalent current sheets. As discussed in [13], the presence of the cuts marginally affects the thrust force per pole pitch length; this allows the use of an axisymmetric model. Finally, end effects are taken into account assuming an indefinite axial extension of the stator and of the mover, while the exciting current sheets occupy a portion of finite length of the stators. Once the solution of the analytical model has been obtained, the slot leakage flux and average distribution of flux density in the teeth are evaluated by approximated formulas [13], [29]. B. Motor Lamination

II. D OUBLE -S IDED T UBULAR L INEAR I NDUCTION M OTOR A. DSTLIM Description A 3-D view of a section of the proposed DSTLIM is shown in Fig. 1. The red and the blue slotted hollow cylinders represent the inner and outer stator, respectively. The coils in the external stator cannot be circular and have to be properly connected by the endwindings at the two sides of the cut. The presence of the cut is necessary in order to connect the load with the mover translating in the air gap between the stators. The mover is composed by two aluminum layers surrounding a massive iron core. Fig. 2 shows the cross section of the DSTLIM.

An important issue to deal with is the description of how the stator cores are laminated. For tubular machines, this is a practical problem that may affect the device performance. The easiest way to laminate the stator cores is to use comb laminas as in Fig. 3 arranging them as in Fig. 4, which shows the cross section of the machine. Since the proposed machine is characterized by an increased distance between the stator teeth and the iron layer, open slots are adopted to minimize the winding inductance. This configuration works well enough as far as the outer stator, but it is not acceptable for the inner stator. A possible alternative is the use of a combined lamination where disk shaped as laminas discussed [30] are added.

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will be considered in Section V, the distances between the outer stator teeth and iron layer are 4.9 and 9.5 mm, and these distances have to be multiplied by the Carter factor, which is always greater that one. Similarly for the inner stator. C. Semi-Analytical Model The axisymmetric semi-analytical model of the device is briefly summarized here; details are in [13], [24], and [25]. The governing equation in terms of magnetic vector potential A constrained by the Coulomb gauge (∇ · A = 0) for linear isotropic medium moving with velocity v, is written as   ∂ 2 (1) A − v × (∇ × A) ∇ A = μσ ∂t Fig. 4. Cross section of the machine. The stator windings and mover are not shown. Light gray: gap between the teeth of the stators.

Fig. 5.

Example of combined lamination.

where σ and μ, respectively, indicate the conductivity and permeability of the medium. The feeding current sheets attached to the stators at radii r2 and r7 can be written as   J int (r, ϕ, z, t) = ϕIm Jint e j (ωt −kz)     L L ×u z − − z δ (r − r2 ) u 2 2   J ext (r, ϕ, z, t) = ϕIm Jext e j (ωt −kz)     L L ×u z − − z δ (r − r7 ) (2) u 2 2 where u(z) and δ(r ) are the unit step and Dirac function, respectively, and ϕ is the unit vector in the azimuth direction. Jint and Jext are the amplitudes of the current sheets, L is the axial length of the current sheets, and k = 2 pπ/L, where p is the number of pole pairs. In axisymmetric geometry and with the given currents, we can write: A = ϕ A˜ ϕ (r, z)e j ωt, where tilde denotes phasors. Discarding e j ωt , after some manipulation, (4) is written as  ˜  ∂ 2 A˜ ϕ 1 ∂ A˜ ϕ A˜ ϕ ∂ 2 A˜ ϕ ˜ ϕ + v z ∂ Aϕ . + + − = μσ j ω A ∂r 2 r ∂r ∂z 2 r2 ∂z (3)

Fig. 6. Cross section of the machine characterized by the combined lamination. The stator windings and the mover are not shown. Light gray: gap between the teeth of the stators. White: disks that are stacked in the axial direction. Sliced gray: laminas stacked in the azimuth direction.

The back iron and the region of the teeth near to it are made by using comb laminas with short teeth stacked in the azimuth direction, whereas the region of the teeth near the air gap is made by disk laminas stacked in the axial direction. This is shown in Figs. 5 and 6. Besides the increased complexity in the practical realization, this combined lamination introduces a small air gap in correspondence of the interface of the regions characterized by different directions of lamination. Considering that the proposed machine is characterized by an increased distance between the stator teeth and iron layer inside the mover, the introduction of this new air gap is expected to marginally affect the reluctances of the magnetic circuits. For the motors that

Let Aϕ (r, ζ ) be the Fourier transform of A˜ ϕ (r, z), (bold italic arial font is used to denote the Fourier transform with respect z) ∞ 1 ˜ Aϕ (r, ζ ) e− j ζ z dζ (4) Aϕ (r, z) = 2π −∞

substituting in (3) and discarding e− j ζ z yields   (l) (l) ∂ 2 Aϕ 1 ∂Aϕ 1 2 − h l + 2 A(l) + ϕ =0 ∂r 2 r ∂r r

(5)

where the index l denotes the layer as in Fig. 2. In nonconductive material (l = 1, 2, 3, 7, 8, 9), we have h l2 = ζ 2 , while in conductors (l = 4, 5, 6) h l2 = j ζ 2(svs σ μ/ζ − j), where vs = ω/ζ is the synchronous speed and s = (vs − vz )/vs is the slip. The general solution of (5) is  A(l) ϕ (r, ζ ) = Cl I1 (h l r) + Cl K 1 (h l r )

(6)

where I1 and K 1 are the modified Bessel functions of first and second kind, respectively.

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 12, DECEMBER 2014

Because of (4), the normal component (radial) of the flux density is given by 1 B˜ r(l) (r, z) = 2π =

1 2π

∞ −∞ ∞





− jζz ∇ × A(l) dζ ϕ (r, ζ ) e

Br(l) (r, ζ ) e− j ζ z dζ

(7)

−∞

with  Br(l) (r, ζ ) = j ζ Cl I1 (h l r ) + Cl K 1 (h l r ) .

(8)

Similarly, the tangential (axial) component of the magnetic field strength can be written as hl  Hr(l) (r, ζ ) = Cl I0 (h l r ) − Cl K 0 (h l r ) . μ

(9)

Because of the asymptotic behavior of I1 and K 1 , we have C1 = C9 = 0, while the other unknown integration coefficients are evaluated by imposing the boundary conditions written as Br(i) (ri , ζ ) = Br(i+1) (ri , ζ )

i = 1, . . . , 8

H r(i) (ri , ζ ) = H r(i+1) (ri , ζ ) i = 2, i = 7 (2) (3) H r (r2 , ζ ) = H r (r2 , ζ ) + J int (ζ )

H r(7) (r7 , ζ ) = H r(8) (r7 , ζ ) + J ext (ζ )

(10)

where J int (ζ ) and Jext (ζ ) are the Fourier transform of (2)   Jint/ext L . (11) sin (k − ζ ) J int/ext (ζ ) = 2 k−ζ 2 Once the magnetic flux and the current density distributions on the mover are known, the average thrust force in the axial direction is 1 Fz = 2

2π r5 L/2

  Re J˜ϕ (r, z) B˜ r∗ (r, z) rdz dϕ dr.

(12)

0 r3 −L/2

The evaluation of the voltages at the terminals of the stator windings is discussed in [13], [24], and [25]. One simulation of the described model implemented in MATLAB, required ∼1 s on a desktop computer based on the i5 Intel processor. III. O PTIMIZATION P ROCEDURE To optimize the geometrical parameters of the machine, an EA has been used. A real parameter, stochastic, populationbased, global optimization algorithm has been proposed in [26]. This algorithm, denoted by self organizing centroids optimization (Soc-opt) is based on a modified version of the self-organizing map (SOM), which are neural networks for unsupervised learning [31]–[33]. The problem is the minimization of an objective function F (x), {F : ⊆ D → }, x ∈ , where is the search space.

A. Optimization Algorithm Description The Soc-opt creates a 2-D grid of P cells, where a distance metric d(c, i) is defined between two cells of index c, i = 1 . . . P. Each cell contains a centroid vector ci (t) ∈ D and a personal target vector t i (t) ∈ D , which is the local best solution of the cell, whereas the entire network tracks a global best solution t gbest (t) ∈ D . The time index t = 0, 1, 2 . . . Tmax represents the iteration index and Tmax is the maximum number of iterations. Random initialization is considered ci (0) = t i (0). Each cell knows the value of the fitness function in its target vector F(t i (t)), and the cell searches in the input space around the centroid ci (t) for a possible better target vector ti (t). This is accomplished by the following perturbation of the centroid: pi (t) = ci (t − 1) + δ i (t − 1)

(13)

in which pi (t) is a perturbed centroid, obtained from the centroid at the previous generation with the addition of a random perturbation δ i (t − 1) ∈ D , whose components are uniformly distributed in [−dneigh , dneigh ], where dneigh = ci − cneigh(i) , and neigh(i ) is a reference neighbor cell of the cell i . In each perturbed point that verifies the given constraints, the fitness value is calculated according to F(pi (t)). If the fitness in the perturbed points is better than the one of the targets, targets are updated  i   p (t) if F pi (t) ≤ F t i (t − 1) i t (t) = i (14) t (t − 1) otherwise this operation is equivalent to the selection in the evolutionary optimization algorithms. Then, the winning cell is selected based on the best value of the fitness function, gbest = arg min i (F(t i (t))). As a result of the selection (14), mini (F(ti (t))) always corresponds to the global best fitness value found in all generations. At this point, each centroid ci (t) moves in the direction of a convex combination xi (t) of its target vector ti (t) and the global best solution t gbest (t) ∈ D , given by

 xi (t) = λi (t) t gbest (t) + 1 − λi (t) t i (t) (15)  where λi(t) = exp −d (gbest, i )2 /2σλ2 is a neighborhood function as in the classical SOM algorithm [33]. The centroid output vector ci (t) tracks the input xi (t) according to the dynamic of a linear, time invariant, discrete time filter of order N, which has the following transfer function: G (z) =

b1 z N−1 + · · · + b N . z N − a1 z N−1 − a2 z N−2 · · · − a N

(16)

The transfer function (16) is defined using the Z-transform formalism, and the coefficients [b1, . . . b N ] and [a1 , . . . , a N ] are fixed and define the particular filtering action between the input sequence xi (t) and output sequence ci (t). To implement the filter in (16), a set of vectors that track the past values of the input and output sequences is introduced. We call them memory vectors: [xi1 , . . . , xiN ], [ci1 , . . . , ciN ], where cij ∈ D , xij ∈ D , i = 1 . . . P, j = 1 . . . N. At each iteration, after

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the creation of the input vector (15), the memory vectors are updated as follows, for j = 1 . . . N, i = 1 . . . P: ⎧ i i i i ⎪ ⎪ ⎨x0 (t − 1) = x (t) , c0 (t − 1) = c (t − 1) xij (t) = xij (t − 1) + λi (t) (xij −1 (t − 1) − xij (t − 1)) (17) ⎪ ⎪ ⎩ci (t) = ci (t − 1) + λi (t) (ci (t − 1) − ci (t − 1)). j

j −1

j

j

The output centroid vector is then defined as the following combination of the memory vectors, which implements the transfer function (16): ci (t) =

N 

b j xij (t) + a j cij (t).

Fig. 7.

Showing the geometry of the motor.

Fig. 8.

Thrust force as a function of mean radius.

(18)

j =1

Equation (17) performs a one-step time shift of the memory vectors taking into account the neighbourhood collaboration. This means that only the memories of cells near the winning one are substantially updated. Following the dynamics of the filter, the centroids track their best local solutions t i (t), and they are also affected by a global attraction toward the global best solution t gbest (t). After the centroid update in (18), a new perturbation step (13) is performed. At the end of the optimization, which is reached when a maximum number of function evaluations have been performed, we can say that the targets t i (Tmax ) represent the final candidate solutions and t gbest (Tmax ) is the global best solution found by the algorithm. The proposed algorithm exhibits a good behavior, when compared with well known evolutionary optimization algorithms [26]. B. Constrains Handling A set of constraints for the search space is considered by defining maximum and minimum values for each optimization variable. These are direct constraints on the solution vector, which are imposed in the initialization step and in the perturbation step (13). If an element of the randomly perturbed vector pi (t) falls outside of the allowed interval, it is forced to a new random value that lies within the interval. A number of indirect constraints are also defined on some figures of merit, which have a complex dependence on the optimization variables. As an example, in order to prevent iron saturation and material overheating, the maximum magnetic flux density and the maximum-induced current density are, respectively, constrained. These quantities are calculated at the same time of the objective function calculation on perturbed vectors F( pi (t)), i = 1 . . . P, after (13) and after forcing the direct constraints. Vectors pi (t) are then separated in two groups. One group contains the vectors that do not respect the indirect constraints, while the other contains the vectors that respect all these constraints. If the second group is empty, the optimization process is stopped. Otherwise, each vector of the first group is discarded and substituted with a vector randomly selected from the second group. Constraints handling is carried out before the selection step (14); in this way, target vectors ti (t), which are the candidate solutions, always verify both direct and indirect constraints.

IV. I NFLUENCE OF L EADING D ESIGN PARAMETERS FOR A P RELIMINARY D ESIGN In order to provide criteria for a preliminary design of the machine, the dependence of the thrust force (the objective function) and of the current and magnetic flux densities (indirect constraints) are discussed for a sample motor, characterized by a stator length L S = 1 m, with two pole pairs. A double layer winding is arranged in 24 slots. Fig. 7 shows a portion of the diametral cross section of the motor. Computations by the proposed semi-analytical model have been performed for different values of mean radius of the mover in the interval 7 < Rav, mov < 21 (dimensions are in centimeters). The thickness of the central layer of the mover is δfe, mov = 8 mm (massive iron with μr,fe = 1000 and σfe = 1 · 107 S/m), and the thickness of both the conductive layers (aluminum with σ Al = 3.5 · 107 S/m) is δAl,int = δAl,ext = 4.5 mm. The speed is vz = 14 m/s. The length of both the air gaps is δair = 5 mm. In the first group of simulations, the size of the slots is kept constants (z slot,int = z slot,ext = 27 mm and rslot,int = rslot,ext = 28 mm) and the current density in the conductor is set at 4 A/mm2 . Fig. 8 shows the thrust force as a function of the mean radius, whereas Fig. 9 shows the behavior of the maximum and average-induced current densities in the two conductive layers of the mover. As expected, the thrust force shows a linear dependence with the radius, whereas the current densities assume constant values when the radius is >15 cm. Another indirect constraint considered in the optimization is the peak value of the magnetic flux density. We preliminary

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Fig. 9.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 12, DECEMBER 2014

Current density on the mover as a function of radius.

Fig. 10. Peak value of the maximum magnetic flux density in the teeth of the inner stator.

look for the more stressed portions of the iron that are easily located in correspondence of the conjunction of stator teeth with the back iron. Using the semi-analytical model, the mean flux density is evaluated in the teeth, as the superposition of the flux entering the tooth trough the cylindrical surface facing the mover and of the flux from the slots adjacent to the tooth. Figs. 10 and 11 report the behavior of the maximum peak value of the mean magnetic flux density in all the teeth of the two stators (inner and outer, respectively) as a function of the mean radius of the motor. As expected, the last teeth (those on the exit end) are characterized by the highest values of the magnetic flux density for every radius [36]. Figs. 12 and 13 report the feeding voltages for the three phases on the inner and outer stators, respectively, together with the voltage drops on the leakage impedances. These voltages have been obtained considering 20 conductor per layer (40 conductors per slot). The feeding voltages can be introduced as a further indirect constraints if they are not allowed to exceed an assigned value. The figures put into evidence the importance of the voltage drops on the leakage impedances, whose reactance is related to the leakage fluxes, which may assume high values due to the large distance between the stators teeth and the iron part of the mover.

Fig. 11. Peak value of the maximum magnetic flux density in the teeth of the outer stator.

Fig. 12. Feeding and leakage voltages (rms) in the three phases of the inner stator.

Fig. 13. Feeding and leakage voltages (rms) in the three phases of the outer stator.

In order to investigate the influence of the variation of the slot size on the feeding voltage and on the fluxes in the teeth, simulations have been performed for Rav, mov = 15 cm, varying the thickness of the slots and maintaining the total current in the slots as well as and the number of conductors per layer. The depth of the slots is rslot,int = rslot,ext = 40 mm, whereas the thickness varies in the range 16.5 < z slot,int =

MUSOLINO et al.: FORCE OPTIMIZATION OF A DSTLIM

Fig. 14. Feeding and leakage voltages (rms) in the three phases of the inner stator as a function of the slot thickness.

Fig. 15. Feeding and leakage voltages (rms) in the three phases of the outer stator as a function of the slot thickness.

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Fig. 16. Peak value of the maximum magnetic flux density in the teeth of the inner stator as a function of the slot thickness.

Fig. 17. Peak value of the maximum magnetic flux density in the teeth of the outer stator as a function of the slot thickness.

V. A PPLICATION E XAMPLES z slot,ext < 33 (dimensions are in millimeters). All the other parameters are the same as the one above described. The feeding voltages are changed in order to maintain the thrust force unchanged. However, the correction to the feeding voltage is within ±4% with respect to the value above. Figs. 14 and 15 show the feeding voltages for the three phases on the inner and outer stators, respectively, together with the voltage drops on the leakage impedances. As expected, the feeding voltages decrease with the slot thickness because of the reduction of the leakage inductance, meanly as a consequence of the increased air gap in the path of the leakage flux, since the two curve families are substantially parallel. This reduction of the feeding voltages is characterized by an increase of the flux density in the teeth. In fact, the flux from the mover, which remains roughly unchanged for a given thrust force, is concentrated in a smaller section. Moreover, despite the reduction of the leakage inductance, and consequently of the leakage flux, the related flux density in the teeth increases because of the reduced cross section. This behavior is evidenced by Figs. 16 and 17, which, respectively, show the flux density in the teeth of both the stators.

In this section, we consider two examples of optimization of the proposed DSTLIM characterized by different operation requirements. We look for the maximum thrust force on the mover with respect to some geometrical and feeding parameters. Some geometrical parameters as well as the electric conductivity and the magnetic permeability are assigned. The main geometrical parameters of the motors are shown in Fig. 7. Since we are interested in the thrust force maximization, we imposed the rated current density in the stator coils ( Jcoil ), corresponding to the planned service for the particular application (continuous, impulsive, and so forth). As indirect constraints, we introduced the maximum allowable current density in the conductive parts of the mover (Jmax ) and the maximum value of the average flux density in the teeth near the exit end (Bmax ), where the flux density is expected to assume its maximum. A. Small Size Motor at Standstill As a first example, we considered a small size motor designed for discontinuous operation whose prescribed geometric parameters are reported in Table I.

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TABLE I P RESCRIBED G EOMETRIC PARAMETERS OF THE M OTOR

TABLE II O PTIMAL G EOMETRIC PARAMETERS OF THE M OTOR Fig. 19. Evaluated points in a subspace of the search space, related to the parameters of the inner part z slot,int , rslot,int , δAl, int .

Fig. 18. Behavior of the best value as a function of the number of evaluations.

Fig. 20. Comparison between the distribution of the current density in the conductive layers of the mover.

1) Optimization: We look for the maximum thrust force at standstill with respect to the following seven parameters: the width and the depth of the slots in both the inner and the outer stators, the thickness of the conductive sheets on the mover and the feeding frequency. According to Fig. 7, the optimization vector is: [z slot,int, z slot,ext , rslot,int , rslot,ext , δAl, int, δAl, ext , freq]. The current density in the conductors of both the inner and outer stators is Jcoil = 6 A/mm2 (rms value). The following direct constraints are imposed (all the bounds are expressed in millimeters): 4.05 < z slot,int < 9.45 , 8.5 < rslot,int < 30, 1 < δAl, int < 10, 4.05 < z slot,ext < 9.45, 8.5 < rslot,ext < 30, and 1 < δAl, ext < 10. As indirect constraints, we assumed Jmax = 19 A/mm2 (rms value) and Bmax = 2 T (peak value). The optimization has been carried out using the Soc-opt algorithm, with a population size of P = 32 and maximum number of evaluations 16 000. The results are reported in Table II, and the optimal feeding frequency is 14.16 Hz. Fig. 18 shows the behavior of the best global value as a function of the number of evaluations. The optimal thrust force on the mover is 1115 N. The windings on both the stators are distributed in 24 slots, two coils per slot. The number of turns of each coil is 42. Fig. 19 shows the evaluated points in terms of the components of the optimization vector related to the inner stator

(z slot,int, rslot,int , δAl, int ). As can be observed, the proposed algorithm explores a large portion of the search space. The best solution is found in the region more intensively searched. The force is limited by the action of the direct constraints on the external stator and by indirect ones in the inner stator. 2) Comparison With Numerical Model: As an example of validation of the semi-analytical model, we performed a comparison of the optimized configuration with the results obtained by a FEM analysis [34]. The numerical axisymmetric model contains ∼2 · 105 elements and takes 2 min of CPU time on a Linux machine based on a i5 Intel processor with 24 GB of RAM. Fig. 20 shows the comparison between the current density in the middle of the two aluminum sheets of the mover (rint = 67.9 mm, rext = 76.05 mm). The agreement is fully satisfactory, also in the regions corresponding to the exit end, where the difference about the numerical and the analytical model is ∼8%. Fig. 21 reports the comparisons between the radial components of the magnetic flux density at the interface between the conductive parts of the mover and the air (rint = 66.7 mm, rext = 77 mm). The waveforms produced by the numerical analysis are affected by a pronounced ripple because of the proximity with

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TABLE III P RESCRIBED G EOMETRIC PARAMETERS OF THE M OTOR

Fig. 21. Comparison between the distribution of the radial component of the magnetic flux density on the conductive layers of the mover.

The applied voltages cannot be symmetric because of the end effects. The analytical model yields Vext = (176, 192, 179) V and Vint = (109, 119, 111) V, while the numerical model gives Vext = (175, 197, 184) V and Vint = (113, 124, 115) V. B. Medium Size Motor at Speed

Fig. 22. Comparison between the average flux density inside the last four teeth of the external stator. Solid lines: analytical model. Dotted-dashed lines: numerical one. Tooth #25 is on the exit end of the mover.

the teeth of the stators. The mean values of the numerical solutions, taken on a window of two slot pitches, are substantially coincident with the analytical solutions. The force on the mover as predicted by the numerical model is 1135 N, with an error near 2% with respect to the analytical model. This small difference is a consequence of the agreement between the numerical and analytical solutions in terms of the current density and of the magnetic flux density distributions on the conductive parts of the mover as shown in Figs. 20 and 21. The average flux density in a stator tooth can be evaluated as the sum of two components: 1) one is due to the flux lines, which enter the tooth by the surface facing the air gap and 2) the other is due to the flux lines, which enter the tooth by the adjacent slots; details are reported in [13]. Fig. 22 shows the comparison of the average flux density inside four external stator teeth as a function of the distance from the axis of the motor. Solid lines refer to the analytical model, dotted-dashed lines to the numerical one. Agreement is very good, also in consideration of the simplifying hypotheses; errors are less than 10% also in correspondence of the teeth on the exit edge. Similar results have been obtained for the inner stator.

As a second example, we considered an intermediate section of a multistage device intended for the acceleration of heavy masses at relatively high speed, e.g., those used in crash-test equipments. The design of a high-performance PMTLM is discussed in [35], where the considered current density in the stator windings (at room temperature) is comparable with the one assumed in the following. The configuration here investigated is characterized by a long stator made by several sections. The length of the mover is about twice or three times the length of a single stator section. Moreover, because of the typical operating condition (a stator winding is fed when the mover has entirely entered the corresponding section), the analytical model is expected to accurately describe the device. Assuming a track length of 20 m and a final speed of 20 m/s, an average acceleration of 10 m/s2 is required. Considering a vehicle mass of 2000 kg and a mover mass of 300 kg, the required average thrust force is 23 kN. We considered here the optimization of a section characterized by an average speed of the mover of 14 m/s. The prescribed geometric parameters (Fig. 7) are reported in Table III. The geometry of the mover is assigned, as it would be useless optimizing the mover in a context where only one stator section is considered. The search space is constituted by the slots dimensions (width and depth) of both stators and by the feeding frequency. The current density in the both the stators coils is Jcoil = 8 A/mm2. The following direct constraints have been assumed (bounds are in millimeters): 12.5 < z slot,int < 29.2, 10 < rslot,int < 40, 12.5 < z slot,ext < 29.2 10 < rslot,ext < 40. The indirect constraints are: 1) Jmax = 22 A/mm2 and 2) Bmax = 1.85 T. The results of the optimization procedure are reported in Table IV and the optimal feeding frequency is 35.12 Hz. The windings on both the stators are distributed in 24 slots, two coils per slot. The number of turns of each coil is 20. The average of the three phase voltages are 1145 Vrms and 950 Vrms for the outer and inner stator, respectively. The thrust force on the mover is 23.2 kN.

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TABLE IV O PTIMAL G EOMETRIC PARAMETERS OF THE M OTOR

Fig. 25. Average flux density inside the five teeth of the internal stator near the exit end. Tooth #12 is on the exit end (peak values). Solid lines: analytical model. Dotted-dashed lines: numerical one.

Fig. 23. Evaluated points with respect to the slot depth and width of the inner stator.

the center of the winding. The tooth corresponding to the exit edge is the #12. Comparison with the results by the numerical model, indicated by the dotted-dashed lines, has been fully satisfactory in this case too. VI. C ONCLUSION An optimization procedure based on an EA in conjunction with an analytical model is proposed to maximize the thrust force of the DSTLIM. The effects of the finite length of the stator on the current density and on the magnetic flux distributions are taken into account. Moreover, an approximate evaluation of the average flux density in the teeth is considered with the aim of correctly imposing the indirect constraints. Comparison with FEM analysis is used to validate the analytical model for the evaluation of the output (the thrust force) and of the indirect constraints. The FEM results are in very good agreement with the analytical results; this confirms the validity of the analytical model and of the optimal design.

Fig. 24. Current density distribution in the middle of the conductive parts of the mover.

Fig. 23 shows the evaluated points with respect to the values of the depth and width of the inner stator slots. It can be observed that the best found solution point is on the boundary of the search space. The slot depth is limited by a direct constraint, whereas the slot width is limited by indirect constraints. The axisymmetric FE model with 2.6 · 106 elements required ∼5 h of CPU time, and the thrust force was 22.4 kN. Fig. 24 shows the current density distribution in the conductive parts of the mover (rint = 158.7 mm, rext = 171.3 mm). The figure also shows the results obtained by the numerical model. Because of the relatively high speed of the mover, the current density extends outside of the region correspondent to the fed section (−0.5 < z < 0.5) [36]. Finally, Fig. 25 shows the average magnetic flux density inside five teeth near to the exit end of the internal stator. Numbering of the teeth starts with 0 assigned to the tooth in

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