PARAMETRIC MODEL ORDERREDUCTION OF INDUCTION

HEATING SYSTEMAnanya Roy, M. Nabi

Department of Electrical EngineeringIndian Institute of Technology, DelhiHauz Khas, New Delhi - 110016, India

Email: roy.ananya2009@gmail.com, mnabi@ee.iitd.ac.in

KEYWORDS

Finite Element Modeling; Parametric ModelOrder Reduction; Induction Heating System

ABSTRACT

Induction Heating system have been widelyused nowadays. Modeling of such system remainsa challenge. In this paper, modeling of a axi-symmetrical structured induction heating systemis carried out. Few geometrical parameters affectthe modeling of the system. Variation of theseparameter values leads to multiple simulations,making the design procedure computationally ex-pensive. Original system is produced from finiteelement method and is reduced to lower order sys-tem to make the simulation cost effective. As thesize of system matrices are different with differentstate space models, matrix interpolation is usedwith two stage model reduction. Simulation re-sults of the models at different parameter valuesare shown.

Introduction

Induction Heating process has wide applica-tion areas from industrial area to household ap-pliances. Induction heating process is a multi-physical system. Uniform current density is ap-plied to the copper coils. It produces a magneticfield in the system. Due to ohmic loss of eddy cur-rent induced in the iron body, heat is generated.Experimental design of induction-heated systemsis not only tedious but also expensive. Thus, itnecessitates the need for a precise mathematicalmodel with help of simulation software[1]-[3].

Most mathematical models of the inductionheating system are expressed as distributed pa-rameter models with partial differential equations(PDEs). These PDEs are converted to ordinarydifferential equations (ODEs) with Finite element(FE) analysis. The generated ODEs are large

in number and make the system computation-ally costly. The original system can be projectedto a lower dimensional subspace to produce itslower order approximation using Model OrderReduction (MOR) techniques. There are vari-ous methods for model order reduction depend-ing on the need such as Singular value decom-position method, Krylov subspace based method,structure-preserving model order reduction etc[4]-[8].

Sometimes few analytically inexpressible pa-rameters like geometrical parameters in FE modelbased large systems, play important role in thedesign process. Changing such parameter valueswhile using standard MOR techniques are compu-tationally costly because of repetitive simulations.In such problems Parametric MOR (pMOR) tech-niques are used. It tries to preserve the param-eter dependency of the original systems in thereduced models so that multiple simulations fordifferent values of the parameters can be carriedout in the reduced space. While considering ge-ometric parameters regular pMOR cannot be ap-plied, although the geometric parameters does ef-fect the FE model and solution implicitly. Ma-trix Interpolation (MI) based pMOR framework[9] has been proposed for such scenarios. In [9],system matrices of reduced order models usingMOR techniques, were transformed as the re-duced states do not have same physical interpre-tation. Then weighted interpolation is performedto get a parametric reduced model. In the presentinduction heating model, it is not directly applica-ble as the higher order FE models are of differentsizes for different parameter values. Hence somemodifications are proposed with some results.

Modeling of Induction Heating System

Here we are considering an induction heatingsystem as shown in Fig. 1. The model consists ofa iron cylinder and copper wires around it. The

Proceedings 32nd European Conference on Modelling and Simulation ©ECMS Lars Nolle, Alexandra Burger, Christoph Tholen, Jens Werner, Jens Wellhausen (Editors) ISBN: 978-0-9932440-6-3/ ISBN: 978-0-9932440-7-0 (CD)

Fig. 1: Three dimensional model of InductionHeating System

copper coils carry alternating current producingelectromagnetic effect. Due to ohmic losses ofeddy current in the iron cylinder, heat is gener-ated. The distance p, between copper coil andiron core, is considered as a parameter.

The governing equation describing the mag-netic vector potential A is given as

1

µO2A− jνA+ Js = 0 (1)

where Js is the current density and it varies sinu-soidally in time with a single frequency, ω. As it istime harmonic in nature, it is sinusoidal functionof time. So it can be represented in terms of vec-tor phasors that depend on the space coordinatesbut not on time. The permeability is representedwith µ and ν is a scalar quantity with ν = ωσwith σ being electrical conductivity.

The induced heat (Q) can be written as

E.J = σA2 = Q (2)

where E and J are electric field intensity and cur-rent density respectively. As only magnitude ofMagnetic vector potential is considered, Q is ascalar quantity. Due to the ohmic losses of theeddy currents in iron, heat is produced. The timedependent temperature profile T (r, z, t) can begiven as

cρ∂T (r, z, t)

∂t+ O(κOT (r, z, t)) = Q (3)

where the model is considered to be thermallyinsulated and heat is not convected or radiatedfrom the iron body. Here, specific heat and massdensity of the materials are represented with c and

ρ, whereas the thermal conductivity is denoted byκ.

After discretizing the model, the Finite Ele-ment representation of (1) can be written as

[K + jS]A = F (4)

where K and S are the real, sparse, symmetricmatrices and K has full rank.

The heat transfer equation can also be writtenas

CdT (t)

dt+ KT (t) = Q (5)

Again, C and K are sparse, real and symmet-ric matrices with K having full rank. The inputvector is denoted by Q. Specific heat constantand mass density are consumed in C and κ inK. Hence, (4) & (5) is the coupled model of theinduction heating system.

Parametric Model Order Reduction

There are some geometrical parameters whichaffect the performance of the system. Such a pa-rameter, the distance between the coils and thecylindrical core, is shown in Fig. 1. As the dis-tance increases, the magnetic field or the magni-tude of magnetic vector potential in the iron bodydecreases and heating time increases.

The FE electromagnetic equation where the pa-rameter varies implicitly, can be written as

[K(p) + jS(p)]A = F (p) (6)

Similarly, the FE heat transfer equation can bewritten as

C(p)dT (t)

dt+ K(p)T (t) = Q(p) (7)

where, p ∈ R.FE generated ODEs are quite large in number

making it complex to handle. These systems canbe replaced with reduced order model projectingthem to lower dimensional subspaces. The pa-rameters which influences the system modeling,can not be preserved using conventional model or-der reduction methods. Parametric model orderreduction (pMOR) techniques are very useful forpreserving the parameter in the reduced models.

All the reduced models from pMOR have equalnumber of states. But they do not interpretethe same physical quantities. This necessitatesthe need of state transformation. The requiredstate transformation can be achieved employingthe matrix interpolation method as described in[9]. Here, all the system matrices are consideredto be equal and this leads to equal size of project-ing matrices.

While modeling it has been seen that withchange in the parameter value, p the size of sys-tem matrices (K,S,C, K ∈ Rni×ni) changes.Therefore, the projecting matrices are of differ-ent sizes. Usual matrix interpolation method asdescribed in [9] is inapplicable. So, a two stagemodel order reduction method is suggested. Inthe first stage reduction, the original systems areprojected to a moderately lower dimensional sub-space making all the system matrices of same size.In the second stage, these moderately large sys-tems are reduced to get lower order approxima-tions.

In the first stage of order reduction the originalsystem is projected to a lower order (Rr1) planeusing one sided Arnoldi algorithm. The reducedsystem can be written as

[Kr1(p) + jSr1(p)]Ar1 = Fr1(p) (8)

Cr1(p)dTr1(t)

dt+ Kr1(p)Tr1(t) = Qr1(p) (9)

Here, system matrices are represented as Kr1 =VT

r1KVr1 , Sr1 = VTr1SVr1 , Fr1 = VT

r1 F , Cr1 =

VTr1CVr1 , Kr1 = VT

r1KVr1 and Qr1 = VTr1Q

with Vr1 as the projection matrix.As described earlier, (8) & (9) are moderately

large systems. So, it can be reduced further byprojecting it to a lower order subspace Rr2 withVr2 . The new reduced order model becomes

[Kr2(p) + jSr2(p)]Ar2 = Fr2(p) (10)

Cr2(p)dTr2(t)

dt+ Kr2(p)Tr2(t) = Qr2(p) (11)

For each discrete value of parameter, large scalemodel and reduced models are obtained employ-ing projection matrices (Vr2,i) with one-sidedArnoldi algorithm.

The transformed reduced systems can be ex-pressed as

[MiKr2,iT−1i + jMiSr2.iT

−1i ]Ar2 = MiFr2,i

(12)

MiCr2,iT−1i

dTr2(t)

dt+MiKr2,iT

−1i Tr2(t) = MiQr2,i

(13)Now, following the standard MI procedure, thetransformation matrices can be found from

Ti = RTVr2,i and Mi = (VTr2,i

R)−1

For obtaining R, whose q columns span the uni-versal subspace of the MOR strategy, the follow-ing has been suggested. The projection matricesVr2,is form a pool of independent directions andthus are accumulated in

Vall = [Vr2,1 Vr2,2 · · ·Vr2,k]

followed by its SVD, given by Vall = UΣNT .The first q columns of U span q different direc-tions and serve as R.

Results and Simulations

The actual model of Induction Heating sys-tem is three dimensional. But due to its axi-symmetrical structure, two dimensional model isconsidered as shown in Fig. 2. The dimensions

Fig. 2: Parameter affecting the Induction HeatingSystem

of the model considered is given in Table 1. The

TABLE I: Geometric characteristics of the prob-lem domain

p1, p2, p3 5mm, 10mm, 15mmd1 70mmd2 60mmd3 70mmd4 71mmd5 6mmd6 10mmd7 71mmt1 25mmt2 5mmt3 55mm− 65mm

geometric parameter considered here is the dis-tance between coil and the iron core. The pa-rameter value varies from 5 mm to 15 mm. Thematerial properties considered here are given inTable 2. Uniform current density of 8A/m2 isapplied to the copper coils. Here zero boundaryconditions are assumed for (4). As the bound-aries are far away from the model, they do nothave any effect of the magnetic field. Produced

magnetic vector potential in the system is shownin Fig 3. The temperature profile in iron core isshown in Fig 4. The thermal field is simulated for40 sec. It is clear from Fig. 4 that the heatingtime varies with the change in the distance be-tween coil and core. As the FE generated modelsare of different sizes, in first stage all the modelsare projected to lower dimensional subspace of or-der 1000. In the second stage all the models arereduced to an order of 16. With the help of matrixinterpolation, interpolated reduced system is gen-erated. Fig. 5 shows the bode plots of the originalsystem, first stage reduced system, second stagereduced system and matrix interpolated reducedsystem. From the plots it can be suggested thatthe reduced models and the matrix interpolatedreduced model can replace the original system forlower frequencies.

Fig. 3: Magnetic field of Induction Heating Sys-tem

TABLE II: Material Properties

Parameters Iron Copper AirRelative Permeability(N/A2) 1500 1 1Electrical Conductivity(S/m) 107 5.96 × 106 3 × 10−15

Mass Density(kg/m3) 7874 8940 1.225Specific Heat(J/kg −K) 450 385 1000Thermal Conductivity(W/m−K) 83.5 401.0 0.001

CONCLUSIONS

In this article, finite element modeling of induc-tion heating system is described. In the designprocess, variation of the parameter value leads

Fig. 4: Temperature plot in Iron core

Fig. 5: Bode Plot of Induction Heating System(Thermal Model)

to repetitive simulations which is computation-ally costly. So, the original system is reduced em-ploying parametric model order reduction. Thedistance between copper coil and iron core is theparameter. As the size of system matrices changeswith the variation of parameter values, standardmatrix interpolation method cannot be applieddirectly. Hence, the original finite element modelis reduced to a lower order approximation in twostages. Simulation results show the efficacy of theproposed modification.

References

[1] C. Chaboudez, S. Clain, R. Glardon, D. Mari,J.Rappaz, M. Swierkosz, ‘Numerical modeling in in-duction heating for axisymmetric geometries’, IEEETransactions on Magnetics 33.1 (1997): 739-745

[2] M. H. Tavakoli, H. Karbaschi, F. Sama-vat,‘Computational modeling of induction heatingprocess’, Progress in Electromagnetics researchletters, 11, 93-102

[3] O. Klein, P. Philip,‘Correct voltage distribution foraxisymmetric sinusoidal modeling of induction heat-ing with prescribed current, voltage, or power’,IEEE transactions on Magnetics, 38(3), 1519-1523.

[4] J. N. Reddy, ‘An introduction to the finite elementmethod’, Vol. 2. No. 2.2. New York: McGraw-Hill,1993.

[5] Klaus-Jurgen Bathe, E. L. Wilson, ‘Numerical meth-ods in finite element analysis’ (1976): 6-12.

[6] B. Salimbahrami, B. Lohmann, ‘Order reduction oflarge scale second-order systems using Krylov sub-space methods’, Linear Algebra and its Applications415 (2006) 385 - 405.

[7] Zu-Qing Qu, ‘Model Order Reduction Techniqueswith Applications in Finite Element Analysis’,Springer, 2013.

[8] A. C. Antoulas, D. C. Sorensen, ‘Approxima-tion Of Large-Scale Dynamical Systems: AnOverview’, Int. J. Appl. Math. Comput. Sci.,2001,Vol.11,No.5,1093-1121.

[9] H. Panzer, J. Mohring, R. Eid, B. Lohmann,‘Parametric Model Order Reduction by MatrixInterpolation’, Automatisierungstechnik Methodenund Anwendungen der Steuerungs- Regelungs-und Informationstechnik, Volume 58, Issue 8,

Pages 475 - 484, ISSN (Print) 0178-2312, DOI:10.1524/auto.2010.0863, August 2010.

Ananya Royis a Research Scholarin Control And Automation Re-search Group, in the Department ofElectrical Engineering in Indian In-stitute of Technology, Delhi, India.

M. Nabi is an Associate Profes-sor in Control And Automation Re-search Group, in the Department ofElectrical Engineering in Indian In-stitute of Technology, Delhi, India.