3D Analytical Model for a Tubular Linear Induction Generator in a Stirling Cogeneration System

3D Analytical Model for a Tubular Linear Induction Generator in a Stirling Cogeneration System Pierre Francois, Isabelle Garcia Burel, Hamid Ben Ahmed, Laurent Prevond, Bernard Multon

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Pierre Francois, Isabelle Garcia Burel, Hamid Ben Ahmed, Laurent Prevond, Bernard Multon. 3D Analytical Model for a Tubular Linear Induction Generator in a Stirling Cogeneration System. IEEE IEMDC 2007, May 2007, ANTALYA, Turkey. pp.392-397. hal-00676232

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3D Analytical Model for a Tubular Linear Induction Generator in a Stirling cogeneration system P. François, I. Garcia Burrel, H. Ben Ahmed, L. Prévond, B. Multon SPEELabs/SATIE (UMR CNRS 8029), Ecole Normale Supérieure de Cachan [email protected]

Abstract— This article sets forth a 3D analytical model of a economical, quiet, ‘clean’, compact and robust. The associated tubular linear induction generator. In the intended application, electrical system for performing the electromechanical the slot and edge effects as well as induced current penetration conversion of energy has been designed as a tubular induction phenomena within the solid mover cannot be overlooked. generator with a solid mover. The global optimization of this Moreover, generator optimization within the present context of engine–generator–electronics–load–control cogeneration cogeneration has necessitated a systemic strategy. Reliance upon system requires, for each subsystem, a set of flexible models an analytical modeling approach that incorporates the array of typically-neglected phenomena has proven essential to offering adapted to a systemic approach. Against this backdrop, an greater computational and analytical flexibility. This article will analytical modeling strategy thus seems best suited for our describe the electromagnetic model of the generator and draw generator. However, given the highly-dynamic operating comparisons with a finite element model, in addition to mode and maximum allowable dimensions, the model must be identifying the elements of equivalent electrical diagram and built to incorporate skin effects along with edge effects. displaying results from the multi-objective optimization study performed using a genetic algorithm. II. DESCRIPTION OF THE COGENERATOR Index Terms— Linear Induction Generator, edge effects, Stirling engine, multi-objective optimization As shown in Figure 2, the drive assembly is composed of two Stirling engines working in opposition [2]. I. INTRODUCTION The thermal and electric parts are thus very tightly integrated, which serves to constitute a compact electric generating set. he production of electrical energy must be integrated into The Stirling external-combustion engine is particularly well- an optimal strategy with respect to sustainable T adapted to the cogeneration mode, as its pumping chamber development goals. As such, it would appear that a sizable may be assimilated with a boiler [3]. Besides this aptitude for and growing share of all electricity consumption stems, to an cogeneration operations, such a configuration also proves increasing extent, from a decentralized cogeneration-based beneficial through: simplifying the kinematics by introducing (electricity and heat) or trigeneration-based (electricity, heat linear motion; taking advantage of external combustion in the and air conditioning) production system [1]. steady state; and facilitating the guidance of mobile parts Figure 1 : cogénérator components thanks to the induction system. Such characteristics would

vector electrical suggest much longer life cycles with much less maintenance. control load

equivalent dynamic electrical displacer diagram rectifier inverter Mover flux cold part Is psi coils generator Fg : generator opposing strengtn opposing strength PWM Fth : stirling motor strength Yp : axial piston position stator

mover / piston linear induction Fg stirling piston position generator engine catcher heat part Yp Fg* = Fth Yp ---> Fth, thermic model

Figure 2: Layout of the cogenerator The purpose of the present study lies within this specific scope The generator studied is of the tubular linear induction type and proposes exploring new solutions for the autonomous (see Fig. 3). The mover comprises the drive engine pistons production of electricity in cogeneration mode. While low- and is made of a solid, conductive non-magnetic material like power cogenerators do exist, their flexibility in making aluminum. thermal and electric power adjustments is inadequate The operating mode (very fast alternating motions), along and their life cycle generally proves too short with respect to with the unique structure and its dimensions, make it the type of applications involved. necessary to develop specific models that incorporate the edge The cogenerator studied herein, composed of a Stirling engine effect as well as dependence of the generator's coupled directly to a linear generator, is intended to be

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z E electromagnetic parameters on frequency variations (i.e. a mover dynamic model). To proceed with this approach, the following hypotheses have been adopted: τ A=0 All materials used are isotropic; periodic conditions Magnetic saturation never occurs; λs The spatial distribution of primary windings (stator) is presumed to be symmetrical three-phase. A=0 Moreover, the frequency-based approach defined herein has à t=0 been based on a power supply composed of three-phase s r2 λ sinusoidal currents balanced at a frequency of ωs. This step r3 will subsequently lead to easily identifying components of the Figure 4: Slotless, infinite-length model equivalent electrical diagram and thereby extend the design to other types of non-sinusoidal power supplies. According to our initial hypotheses, the vector potential can be written as: r ur jkz−ω t L z ()s r external yoke Az,r,t()=ϕ() re uθ (7)

5 mover E A comprehensive understanding of the vector potential at any 3 ε r5 fixed primary E point within the computation field thus requires determining r1 1 its radial distribution ϕ()r .

Based on Maxwell's equations and the relation in (5) above, Figure 3: Generator design geometry we can derive a quite remarkable differential equation whose The search for generator performance (in terms of axial force, general solution consists of a sum of modified Bessel electromagnetic power, losses, etc.) necessitates solving functions of both the 1st and 2nd kind. The vector potential Maxwell's equations, i.e.: distribution function in each region (index n) of the uuuuuuurur r rot H= J (1) computation space can then be expressed in the following () general form, equation (8) : ur ur ∂B E =− (2), with: []nnnnn⎡⎤[] [] [] [] jkz( −ωs t) ∂t AX.I.rY.K.r()z,r,t=γ+γ() () .e ur uur ⎣⎦⎢⎥11 B.H=µ (permeability µ is assumed to be constant) (3) where for each region [n], we have defined a coefficient rurur JEB=σ( +ν∧ ) (4) []n []nnn [] [] γ=k² +ωµσ js g . ur uuuuuuuruur BrotA= () (5) The coefficients X[]n and Y[]n are constants determined from In order to describe the analytical modeling steps carried out the region boundary conditions. g[]n , µ []n and σ [n] are the more simply, we adopted a gradual approach in which both slip, magnetic permeability and electrical conductivity of the edge and slot effects are first neglected [4,5,6] and then region [n], respectively. get taken into account via a sequence of appropriate modulation functions. Based on this solution, it becomes straightforward to deduce, for each region [n], the expressions for both the axial III. GENERATOR ALYTICAL MODEL component of the magnetic field and the azimuthal component Let's begin by considering an infinite-length geometry along of the electric field, equations (9a) et (9b) : the z-axis. []nnn[] [] []nnnnn1A⎛⎞∂γ A [] [] [] [] jk.z()−ω t H z, r, t=+=γ−γ⎡⎤ X .I .r Y .K .r .e s z00()[]nn⎜⎟[]⎣⎦() () µ∂µ⎝⎠rr A. Infinite length analytical model []nnnnn[] [] [] [] jk.z()−ω t Ez,r,tjAjX.I.rY.K.r.e()=ω =ω⎡⎤() γ +() γ s 1) Slotless stator θ ss1⎣⎦ 1 The power exchanged between the mover and the fixed parts The primary current is first modeled by means of a surface (expressed in complex values) corresponds to the difference in current sheet with a perfectly-sinusoidal distribution given by input and output power through the mover's exterior and the following expression (see Fig. 4): interior surfaces S [n] ,

PPrPr=− j()kz−ω t ( 32) ( ) λλ()zt,.= ˆ e s (6) tr ss These power values correspond to the flux of the Poynting 2 with: j =−1 , ω is the primary (stator) current angular n s vector through surface S[ ] : frequency and k a propagation coefficient: k=π . τ []n 1 []n*nnn[] [] [] Pr()= E(γγ[] n.r) .Hz ( .r ) .S (10) 2 θ

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z Kc.E Inserting Equations (9a) and (9b) into (10) yields a power mover function independent of the time of value: (-a) []n []nnnn1 γ [] [] [] Pr()=− j ω .XIK*YIK*.S⎡⎤() + −() − (+b) 2 s0011µ[]n ⎣⎦ τ (-c) A=0 periodic conditions The real part of Ptr corresponds to the transmitted (+a) electromagnetic power. The axial electromagnetic force F λs z (-b) therefore simply equals: (+b) A=0 ⎡⎤ Pr− Pr ℜ ⎢⎥( ) ( ) (-a) à t=0 F = ⎣⎦32 (11) z v λs s Figure 6: Slotted, infinite-length model Figure 5 presents, for a given generator geometry, the evolution in axial force vs. rotor current frequency output by The expressions for the axial component of the magnetic field the analytical model and the finite element model. These and the azimuthal component of the electric field, under these results reveal the very strong correlation between these two conditions, are, equations (14a) and (14b) : ∞ []n []nmnnnnγ [] [] [] [] jk() .z+χω t models in the case where both edge and slot effects have been H() z, r, t=γ−γ⎡⎤ X .I .r Y .K .r .e ms zm0mm0m∑ []n ⎣⎦() () neglected m1,5,..= µ ∞ []nnnnn[] [] [] [] jk() .z+χω t Ez,r,t=χωγ+ j⎡⎤ X.I.rY.K.r.e γ ms θ ()∑ sm1m⎣⎦() m 1m() m1,5,..= A comparison between the analytical model and finite element model is displayed in Figure 7 below, with results obtained for the same geometry as before. Here once again, a very strong level of correlation between the two models has been demonstrated.

Figure 5: Flux density calculated using finite elements for v=vs (a), and the force developed for a slotless, infinite-length machine vs. the rotor current frequency (b)

2) Slotted stator

Slot effects will now be taken into account by focusing on: 1- increase in the actual airgap by means of the Carter Figure 7: Flux density calculated using finite elements for coefficient K (see Fig. 6) [7,8]; c v=v (a), and the force developed for a slotted, 2- modulation in the current density due to s infinite-length machine vs. the rotor current frequency (b) discretization of the primary winding: m=∝ 6 ⎛⎞mπ jmkz+χω t ˆ ()s B. Finite length analytical model λλss()zt,.sin.= ∑ ⎜⎟ e (12) m=1,5,.. mπ ⎝⎠12 In order to take account of the edge effects produced by the ⎧ −=11,7,...for m finite length of the stator, we have undertaken the same where: χ = ⎨ approach as before, yet this time in introducing a new ⎩+=15,11,...for m amplitude ( ) modulation function of the current The solution, for each harmonic m , of Maxwell's equations ±1 ∆s density λ (see Fig. 8). The modulation length L , chosen now enables writing, equation (13): s

∞ [] []n [] [] [] jk( z+χω t) arbitrarily, is such that: L >> L. AX.I.rY.K.rnnnn()z,r,t=γ+γ⎡⎤ .e ms ∑ ⎢⎥mm() m() m ∞ m1,5,..= ⎣⎦11 42⎛⎞⎛ςπL ςπ ⎞ ∆=()zzsin .cos (15) []nnnn [] [] [] s ∑ ⎜⎟⎜ ⎟ with: γ=k² +χωµσ j g , km= π , ς =1,3,.. ςπ. ⎝⎠⎝LL ⎠ mmsm m τ The expression of the modulated primary current then gmg=+11χ ( −) m becomes (equation (16) : ∞∞ The synchronous speed for each harmonic of space m equals: jkIII z++χω t jk z χω t 12 1 ⎛⎞⎛⎞mLπςπ⎡⎤()mmςςss( ) λλ()zt,sinsin=+ˆ e e τ ssπς2 ∑∑m ⎜⎟⎜⎟12 L ⎢⎥ νχωss= . ς ==1,3,..m 1,5,.. ⎝⎠⎝⎠⎣⎦ m mπ with: kmI =+πτς 2 , kmII =−πτς 2 mς τ ( L ) mς τ ()L

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the ratio τ ) has been done. We show figure 10 the end effect -1 L mover reduction factor.

z Kc.E

(-a)

(+b) 1,00

(-c) A=0 +1 0,95

(+a) L/2 L ∆s 0,90 (-b) 0,85 (+b) A=0

(-a) 0,80

0,75 Fz / Fz infinite length Fz / infinite 0,70 -1 0,65 0 2 4 6 8 10121416182022242628303234 Figure 8: Slotted, finite-length model Length / pole pitch Fz / Fz_infinite_length

Using the same equations as above, the expression derived for Figure 10 : Strength réduction versus (generator length / pole vector potential is, equation (17) : pitche) ∞ ∞ jkI z+χω t []nII∑ ⎡⎤[]nn[]nn[] [] ( mς s ) A ()z,r,t= ∑ ⎢⎥ Xmmςς .I11γ+ .r) Y m ς .K γ m ς .r) e ς=1, 3,. . ⎣⎦( ( m1,5,..= C. Schéma électrique equivalent ∞ ∞ jkII z+χω t +γ+γ∑ ⎡⎤X[]nn .III[]nn .r Y[] .K II [] .r e ( mς s ) ∑ ⎢⎥mmςς11( ) m ς( m ς) ς=1, 3,. . m1,5,..= ⎣⎦ Afin d’établir les équations électriques représentatives du In[] I In [] [] n [] n générateur asynchrone et son schéma équivalent, il est where: γ=k² +χωµσ js g , mmςς m ς nécessaire de calculer les différentes inductances et resistances γ=II[] nk² II +χωµσ j g II [] n [] n [] n primaire et secondaire. mmςςs m ς I1 I2 ⎛⎞2τ 2τ I , II ⎛⎞ R1 L1 L2 ggm=+11χς() −⎜⎟ + ggm=+11χς() −⎜⎟ − mς ⎝⎠L mς ⎝⎠L I0 Remark: L1 : primary leakage flux V L2 : secondary leakage flux For τ 1, we return to the case of an infinite-length m R2 L  X R2 = Rr / g machine.

The finite stator length also necessitates modulating the airgap permeance, which translates the increase in airgap at the L’inductance et la résistance primaires d’une phase sont generator edges. By applying the same modulation function as aisément définies : the one described above over the magnetic field, we obtain: 2 σ.π.D () []nn[] R1= p. . (19) []ns∆ z ⎛⎞AA∂ n Hz,r,t=+ (18) S s z () []n ⎜⎟ µ∂⎝⎠rr 2 L = n D (20) 1 µ .π. .h+Dm.e where A[]n is given in equation (17). 0 b (3 )

With, p, length pitche pole number, n, spire number, b, coil thickness, D, mean coil diameter, Dm, mean airgap diameter, h, coil height, e, airgap thickness

Se, airgap surface, (airgap cut by (θ,r) plane)

Figure 9: Force developed for a slotted,finite-length machine Au secondaire, nous utilisons le modèle analytique. Il nous permet de connaître les puissances actives et réactives aux

Furthermore, the expression for electric field Eθ may be frontieres des regions, fer, entrefer, mover en integrant sur ces deduced directly from the vector potential expression: frontieres le vecteur de Poynting (voir paragraphe III.A.1). De EjA= ω . ces puissances, nous pouvons déduire les impédances du θ s secondaire, ainsi que l’impedance magnétisante en fonction de As shown in Figure 9 results, obtained using the same la fréquence des courants induits. generator dimensions as before, the discrepancy between the 1 * = (γ . ) . ( γ . ) . Sn,, two models remains relatively small. P(sn) 2Eθ n rn H z n rn An exhaustive study on the validity of the analytical model for various geometric parameters (with special emphasis on

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1 ⎛ 1 ⎞ et, Q(sn) = Eθ(γ .rn ) . H z ( γ .rn ) . Sn, ⎛ &y&p ⎞ ⎛Kpp Kpd Kpd ⎞⎛ yp ⎞ ⎛Dpp 0 0 ⎞⎛ y& p ⎞ ⎜ m ⎟ 2 n n ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ p y = ⎜K K 0 ⎟ y +⎜D D 0 ⎟ y +⎜ 0 ⎟F La Figure 11 présente les puissances active et réactive qui sont ⎜ &&d1 ⎟ dp dd ⎜ d1 ⎟ dp dd ⎜ &d1 ⎟ ⎜ ⎟ géné ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ à l’origine des valeurs de R2 et L2. ⎝&y&d2 ⎠ Kdp 0 Kdd ⎝ yd2 ⎠ Ddp 0 Ddd ⎝ y&d2 ⎠ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2000 Ce modèle d’état s’apparente à un ressort couplé à un élément 1600 visqueux. Nous pouvons en déduire que le moteur Stirling se

1200 comportera comme un oscillateur si la force appliquée par le générateur, compense le terme visqueux. Nous obtenons alors 800 l’équation de base d’un oscillateur : yp =kyp , équation 400 &&

Ptr (W) and Qtr (VAR) représentative d’un fonctionnement ‘à piston libre’, en 0 particulier. 0 1020304050

eddy current frequency -Ptr_VI_var_fr Qtr_VI_var_fr B. Contrôle-commande du cogénérateur

Le contrôle-commande et l’optimisation globale du Figure 11:Ptr Qtr versus mover versus eddy currents frequency cogénérateur sont repris des travaux d’Isabel Garcia Burrel [11]. IV. ETUDE GLOBALE DU COGÉNÉRATEUR Le fort couplage entre le moteur Stirling et le générateur électrique nécessite de les ‘contrôler’ simultanément. La Comme il est rappelé dans l’introduction, l’étude du grandeur nous permettant d’assurer le pilotage de l’ensemble cogénérateur a été partagé selon trois disciplines, en boucle fermée est la position du piston, pièce commune au électrotechnique, thermique et automatique. La modélisation moteur et au générateur du générateur présentée dans cette article est complétée d’une modélisation du moteur Stirling en vue d’optimiser globalement le cogénérateur et de le controller. A. Modélisation du moteur stirling

La modélisation du moteur thermique est reprise des travaux de Julien Boucher [12]. Le cycle thermodynamique du moteur Stirling comporte deux isochores et deux isothermes. Par hypothèse, les échanges se réalisent en mode adiabatique. ( P.Vγ= constante). Le moteur est à double effet, à piston libre. Le point de fonctionnement du moteur se caractérise par la couple de valeurs (ypmax, fosc), où ypmax est l’amplitude de la course du piston et fosc sa fréquence d’oscillation. Ce point de Figure 12: cogenerator control-command fonctionnement (ypmax, fosc) dépend de la différence de Comme l’exprime l’équation d’état du modèle thermique, ce température entre les chambres chaude et froide. Ce point de contrôle a pour but d’assurer, à tout instant, l’équilibre entre fonctionnement impose, de facto, la puissance thermique l’effort moteur (moteur Stirling) et l’effort résistant fournie et la puissance électrique générée. (générateur asynchrone), équilibre indispensable au

Par hypothèse, la variation de vitesse du piston et des fonctionnement ‘à piston libre’. déplaceurs est supposée sinusoïdale. Comme nous l’avons vu au paragraphe IV.A, c’est cette position du piston qui est la consigne entrée pour contrôler le ypref =ypmaxsin()2πfosc.t générateur. Ce contrôle est de type vectoriel. yd1=yd max.sin(2π f osc.t+Φ) yd1=yd max.sin(2π f osc.t+Φ) C. Optimisation globale du cogénérateur

Il s’en suit que, pour un point de fonctionnement donné, la Cette etude se devait d’être complétée par une demarche position du piston est connue à chaque instant. Cette position d’optimisation globale, il est evident que l’optimisation de sera la position-reference prise comme consigne de la chacun des composants pris isolément ne conduit pas commande du générateur (yref). nécessairement à l’optimisation du cogénérateur pris dans son ensemble. L’équation d’état du moteur est de la forme (equation (21): Cette optimisation avait pour but de répondre à deux objectifs minimiser les pertes du générateur. minimiser la taille des composants de l’onduleur.

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L’optimisation a été réalisée avec un logiciel utilisant un our new model makes it possible to identify elements of algorithme génétique NSGA II. L’avantage d’un tel equivalent dynamic electrical diagram [9]. algorithme est la possibilité de prendre en compte un grand Also, this article highlights the multi-objective optimization nombre de paramètres, associés à des objectifs et des studies conducted within the specific scope of co-generation contraintes multiples, ces objectifs et ces contraintes étant operations [11] through applying a genetic algorithm [13]. calculés avec les modèles électriques et REFERENCES thermiquesprécedemment décrits. [1] H.I. Onovwiona, V.I. Ugursal, “Residential cogeneration systems: Review of current technology”, Renewable and sustainable energy review, 2004 (Online) available: www.sciencedirect.com. [2] P. François, L. Prévond, G. Descombes, International Patent N° PCT- Objectives Constraints Parameters studied FR02-00173, 2002. output electric power, induction ( < 1,6 T), mover radius, [3] S. Backhaus, G.W. Swift, ”A thermoacoustic Stirling heat engine, yield ... eddy current mover thickness, detailed study”, J. Acoust. Soc. Am. Proc. 107 (6), 2000, CD-ROM. ( < 10 A/mm2) yoke thicknesse, ... airgap thickness, [4] B. Alvarenga, I. Chabu, J.R. Cardoso, “Modeling and testing of a Ring eddy current, cage tubular linear induction motor for an oil pumping system”, 1993. pitch pole length, [5] A.L. Cullen, TH. Barton, “A simplified electromagnetic theory of the coil spire number induction motor, using the concept of wave impedance”, IEE Proceedings, Vol. 105, Part C, No. 8, pp. 331-336, September 1958. generator dynamic model, [6] M.V. Zagirnyak, R.M. Pai, S.A. Nasar, “Analysis of tubular linear electric equations computation induction motors, using the concept of surface impedance”, IEEE Transactions on Magnetics, Vol. MAG-21, N°04, pp.1310-1313, July

R1, L1, tau, sigma 1985. [7] F.W. Carter, “Note on air gap and interpolar induction”, J. IEE, 1929. Optimization [8] A. Balakrishnan, W.T. Joines and T.G. Wilson, “Air gap reluctance and inductance calculation for magnetic circuits using a Schwarz- Algorithm, vectorial command, NSGA II correctors adjustment Christopheffel transformation”, IEEE trans. On Power Electronics, Vol. 12, N° 4, 1997. Uo, Imu Ti, Ki [9] R.M. Pai, I. Boldea, S.A. Nasar, “A complete equivalent circuit of a linear induction motor with sheet secondary”, IEEE Trans. On cogenerator dynamic simulation Magnetics, Vol. 24, N°1, January 1988, pp. 639-654. thermic model involvement [10] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. On Objectives results Evolutionary Computation, vol. 6, Issue 2, April 2002, pp. 182-197. [11] I. Garcia Burel, L. Prévond, S. Le Ballois, E. Monmasson, “A Stirling micro-cogenerator emulator”, IECON Proc., Paris, November 2006. Figure 13: Optimisation process [12] P. Nika, J.Boucher, F. Lanzetta, P. François, L. Prévond, B. Multon, La figure [14] donne un exemple de résultat d’optimisation. Il 30mai-2juin 2005. “La cogénération Stirling à pistons libres”, Congrés concerne la minimization des pertes dans le générateur et la Français de Thermique, SFT 2005, Reims [13] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, “A fast and elitist minimization de la taille des semi-conducteurs de l’onduleur. multiobjective genetic algorithm: NSGA-II, IEEE Trans. On Evolutionary Computation, Vol. 6, Issue 2, April 2002, pp. 182-197.

Figure 14 : looses:generator reduction and electronic components inverter optimization

V. CONCLUSION

This article has set forth the steps behind development of a 3D analytical model for the tubular linear induction generator with inclusion of both slot and edge effects. Utilizing a frequency-based approach, this model proves to be relatively accurate with respect to the finite element model. Besides calculating the electromagnetic performance of the generator,