IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 715

Automatic Generation of Sizing Static Models Basedon Reluctance Networks for the Optimization

of Electromagnetic DevicesB. du Peloux1;2, L. Gerbaud1, F. Wurtz1, V. Leconte2, and F. Dorschner2

Laboratoire d’Electrotechnique de Grenoble, 38042 St Martin d’Hères, FranceSchneider Electric Industries SAS, 92500 Rueil Malmaison, France

During the presizing of electromagnetic devices, the designer needs fast calculation tools allowing him to make optimizations. Reluc-tance networks are thus very often used as approximate models: they are easy to interpret and provide very short computing times.However, to set them to equations is a tiresome task and often leads to errors. The paper proposes to automate this task. Starting from asimple and intuitive graphic interface, the model and its partial derivatives are automatically generated. Thus, optimization sizing andsensibility computation can be carried out.

Index Terms—Electromagnetic device, model generation, numerical method, optimization, reluctant network, sizing.

I. INTRODUCTION

THE way to design new products in industry has markedthese last years a significant turning point. In order to re-

duce design times and to optimize manufacturing costs, the useof design software tools is needed. Thus, many simulation soft-ware tools have gradually reduced the very expensive steps ofprototyping.

In the electromagnetic field, those are mainly based on thefinite element formalism which, for precise results, requires onthe one hand a particular competence of the user, and on theother hand important computing times that are not compatiblewith the optimization of numerous parameters.

An alternative solution is to use approximate models suchas reluctance networks [1], [2], because they provide very fastcalculation time. Unfortunately, even if the topology is knownby the designer, it is time consuming to set such models intoequations and translate them into a programming language,especially when saturable elements leads to implicit equationsystems.

To avoid this and to make that the designer saves time, thepaper provides a tool that uses graphically user defined reluc-tance networks to automatically generate computable models.The originality is to provide the model partial derivatives in thesame time. That is particularly efficient to couple model withgradient based optimization algorithm.

First, the paper justifies the symbolic formulation of reluctantmodels from the optimization viewpoint. Second, it presents theway that the models can be generated by the tool ReluctTool andhow an analytical model is buit from a user reluctant networkdescription. Third, it compares the method used in ReluctTool tosolve the implicit equation system due to saturable reluctanceswith a method developed before [3]. Fourth, the paper discussesthe shape of the generated model: its implementation and its

Digital Object Identifier 10.1109/TMAG.2006.872010

architecture; it also shows how it can be used with solvers oroptimizers. Finally, it details an application.

II. OPTIMIZATION

The mathematical formulation of a constrained problem isformulated by

where is a vector containing the model parameters (it includesmodel inputs Pi and model outputs Cj), is the objective functionto be minimized, are inequalities depending on model parame-ters, are equalities depending on the model parameters.

Several deterministic algorithms allowing to solve such prob-lems use the gradient of the outputs according to the inputs ofthe model used for the sizing in order to find the direction of alocal minimum.

Because the models based on reluctance networks have manyparameters, and since symbolical model outputs derivative canbe expressed according model inputs, the models generated byReluctTool can be constrained and optimized by gradient-basedalgorithms.

III. MODEL BUILDING

A. How the Designer Describes His Model

ReluctTool has been developed with the aim to make reluc-tance network description easy and intuitive for designers. Oncethe designer knows his model topology, he has just to drag anddrop suitable elements (reluctances and flux sources) on a layerand to connect them together (see Fig. 1).

Usually used elements are available with a predefined asso-ciated model: air reluctance, linear reluctance, saturable reluc-tance with H(B) models, or amp-turns source. Each one can beeasily configured throughout a specific dialog box.

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716 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Fig. 1. GUI screenshot of an automotive claw-pole alternator model [4].

During the element configuration, the designer can set numer-ical or literal values. The literal expressions will be parsed andthe parameters will then appear as model inputs or outputs.

In a complementary window, the designer can add analyticalequations in order to complete his model. Commonly, it is pos-sible to express a force in function of a flux and a section.

B. Upgradeable Basic Element Library

The basic elements used to build reluctance networks are pro-vided into a library that can easily be upgraded. Indeed, everyelement (reluctance or flux source) is wholly described into anXML file where their associated models and the correspondingparameters are specified.

C. Sizing Model Generation

From the designer graphical description of a reluctance net-work, the paper now presents how a complete sizing model canbe automatically generated.

Three main steps are identified.1) To Translate the Reluctance Scheme Into Equa-

tions: First, the graphical representation of the networktopology has to be analyzed in order to establish the meshequations of the circuit. This task is possible thanks to anobject oriented graph representation of the network topologybuilt while the designer lay out its network. In fact, only a setof independent meshes are extracted from the graph, what isrequired for the model solving.

In parallel, the basic element submodels are converted intoequations or functions depending on the parameters the designerset into the configuration boxes. So the mesh equations refer tothe reluctance and the amp-turns source expressions.

At the end of this task, all the mesh equations have thus beengenerated. They constitute an implicit system (due to the re-luctance expressions that depend on the fluxes). The results ofthis system correspond to composed fluxes from the indepen-dent meshes.

Then, and always thanks to the graph analysis, a set of explicitequations is generated in order to calculate the fluxes valuesfrom the results of the implicit system solving.

Fig. 2. Method A: the implicit system is solved by the optimizer.

Finally, all these equations are gathered with the other modelequations specified by the designer (force calculation ). Atthis stage, every equation that composes the model is known.A first XML file corresponding to a nonordered model is gener-ated. These XML data are the starting point of the next genera-tion step.

2) To Arrange the Equations Into a Solving Sequence: Inthis second step, starting from XML data, every model equationis parsed and all the relationships between the parameters are putin an occurrence matrix. In this way, the dependencies betweenthe parameters are thus known. So the model inputs and outputsof the model can be separated and equations can be organizedin a solving sequence.

This step is also ended with a new XML representation ofthe model that specifies all the parameter dependencies and thesequence in which equations have to be solved.

3) To Derivate and to Translate the Model Into a Pro-gram: Finally, thanks to a rule-set-based tool [8], all thesymbolic derivatives of the model outputs according to itsinputs are made. Indeed, although a numerical algorithm isused to solve the implicit system, it is possible to symbolicallydetermine all its output derivatives according to its inputs [6].

At the end of this process, the model (since representedthrough XML data) is converted to a portable programminglanguage (Java) and compiled. The resulting files are thenlinked with numerical libraries (dedicated to solve the implicitsystem) and packed in a Jar archive file.

IV. WAYS TO SOLVE IMPLICIT SYSTEM

A. Method A

In previous works [3], the implicit equation system dueto flux computation was solved thanks to an optimizationprocess. Indeed, in order to cancel the implicit equations

, a criteria wasdefined. Then, the optimization process has to find the meshfluxes that allow to cancel those criteria. So the modelhad to be optimized before to be solved (see Fig. 2).

B. Method B

With the other approach [6], the solving and the optimizationare separated. It is the way that is chosen in ReluctTool (seeSection IV-D for more details about method B efficiency). Thegenerated model is now linked to an appropriate algorithm that

DU PELOUX et al.: AUTOMATIC GENERATION OF SIZING STATIC MODELS 717

Fig. 3. Method B: the implicit system is solved inside the model.

TABLE ICOMPARISON OF METHODS A AND B

allows to solve the implicit system at the appropriate momentduring its computation. So the algorithm is now inside the model(see Fig. 3).

C. Algorithm [5]

Because the used algorithm has been written and configuredspecially in order to solve implicit systems generated by reluc-tance networks, it is really more efficient than a global optimiza-tion process.

The algorithm used into models generated by our tool(method B) is a Newton–Raphson coupled with minimizationof the implicit system norm. In this way, there is the benefit ofthe quadratic convergence of Newton–Raphson method whenthe algorithm is near the solution, and the minimization methodalso ensures the convergence from any starting point.

D. Efficiency

Tests have been made on a caw-pole alternator reluctantmodel (see Fig. 1). This model leads to an implicit system often equations. Results are shown on Table I.

As expected, method B is faster (around 60 times). Note thatwith method A, every iteration of the optimization process haveto compute the full model; whereas, in method B, the implicitsystem is only valued.

Furthermore, with method A, if constraints are not satisfiedfor the implicit system (i.e., ), the results mean nothing;contrary to method B where every result is correct.

With small systems, the two methods need similar computingtime, but the more the size of the system increases, the moreefficient approach B is.

E. Changes for Model Optimization

1) With Approach A: The initial values of the fluxes appearas input parameters of the model, so the designer has to set theirinitial values. Besides, this method requires virtual output pa-rameters (corresponding to implicit function values) that are tobe cancelled by the optimizer. So, the optimization specifica-tions have to take into account those parameters. Two ways are

Fig. 4. Simplified view of a calculation component: E are input parameters,and S output ones (that will be used as sizing criteria).

possible: to integrate the sum of their square into the objectivefunction that will be minimized (not always possible), or to con-strain those parameters to zero.

So, there are two consequences for method A: 1) the opti-mization specifications are not representative of the physicalissue and 2) the optimization routine has to play with more pa-rameters than in method B, so convergence will be more difficultand slower.

2) With Approach B: The solving of the implicit system istransparent: the model includes it. So, the optimization specifi-cations will be more understandable and fully dedicated to de-signer’s issue.

V. GENERATED SOFTWARE COMPONENT

A. Standalone Software Component

As shown in Fig. 4, the generated software componentthat represents the reluctant model appears as a single andautonomous entity. In short, it just implements a set of inter-faces that allows external environment to set its inputs, to runits computation, and to get its outputs and their derivativesaccording to its inputs (its jacobian).

As a matter of fact, this component respects a standard de-veloped by our research team Interfaces for Component Archi-tecture (ICAR). This allows various software, such solvers oroptimizers, to use components generated from different modelgenerators (our reluctant model generator is one among others).

B. ICAR Standard [7]

ICAR standard specifies an interface for all components thatallows the access to their input and output names, and the cor-responding object oriented representations. Then this interfaceprovides the access to the different component services. A ser-vice is defined by a task that the component is able to do, any-thing is allowed so the notion of ICAR component is reallygeneral.

With this notion of service, components may have severalfacets and are able to do many things. For instance, in the paper,the component computes itself or computes its jacobian: thesetwo tasks are two different services.

From the use viewpoint, the component inputs have to be setwith appropriate data. Then component has to be droved to run aparticular service (or more), that acts on the component outputs.Finally, the component outputs can be received (Fig. 5 shows anexample).

718 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Fig. 5. Instance of ICAR component used by an optimizer.

Fig. 6. ICAR component into its environment.

C. ICAR Components Into Their Environment

The main advantage of this standard is to place the compo-nent into a central position of the sizing process (see Fig. 6). Onone hand, there are specific generators that build ICAR compo-nents from different dedicated environments (the paper presentsone of them). On the other hand, there are software that use suchcomponents thanks to their services to provide models uses: cal-culator, optimizer, etc.

VI. APPLICATION

Working with electromechanical relays, a typical issue is tomaximize the static electromagnetic force due to the permanentmagnet in neutral position while minimizing the electromag-netic force due to amp-turns used to trip the relay (Fig. 7).

After having set all the element parameters and added a fewequations to compute forces, a simple button-click allows togenerate automatically the corresponding software component.With composition property of such components, it is then easyto generate a second one that includes two basic components inorder to compute each working point.

Thus, the generated component is used in an optimizing envi-ronment to find geometrical variables and magnet residual mag-netism values. Fig. 8 shows some parameters evolution duringthe optimization process. Finally, the magnetomotive force isreduced by 50%.

VII. CONCLUSION

This tool provides an efficient way to generate models ofstatic reluctance networks as software components. The de-signer has only to focus on his fundamental work ie the design.He is free from setting his model in equations and encodingthem in a programming language, what is time consuming andoften leads to errors. And thanks to the symbolic valuing ofthe output derivatives, an optimization using gradient-basedalgorithms can be used for sizing.

Fig. 7. Modeling of a circuit breaker: right part is drawn by the designer andis used to automatically generate a programming language that implements themodel (see part C).

Fig. 8. Sizing parameter evolutions during the sizing process. Bottom-rightone is the objective function.

Furthermore, the solving of the implicit system is includedinto the solving of the model itself: the model is then au-tonomous and can be solved outside of an optimization process.Optimization specifications of the model are also more under-standable. Convergence is faster and more efficient.

In future developments, dynamic aspects will be taken intoaccount, taking into account energetic aspects.

REFERENCES

[1] J. Turowski, “Reluctance networks,” in Computational Magnetics, J. K.Sykulski, Ed. New York: Chapman & Hall, ch. 4.

[2] H. C. Roters, Electromagnetic Devices. New York: Wiley, 1941.[3] A. Delale, L. Albert, F. Wurtz, and L. Gerbaud, “Automatic generation

of sizing models for the optimization of electromagnetic devices usingreluctance networks,” IEEE Trans. Magn., vol. 40, no. 2, pp. 830–833,Mar. 2004.

[4] L. Albert, C. Chillet, A. Jarosz, J. Rousseau, and F. Wurtz, “Design ofClaw-Pole alternator using magnetic equivalent circuit,” in Proc. IEEE11th Biennial Conf. Electromagn. Field Comput. (CEFC 2004), Seoul,Korea, Jun. 2004, p. 190.

[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-merical Recipes in C: The Art of Scientific Computing. Cambridge,U.K.: Cambridge Univ. Press.

[6] C. Coutel, F. Wurtz, and J. Bigeon, “A comparative study of two methodsfor constrained optimization with analytical models dealing with im-plicit parameters,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1738–1, May1999.

[7] V. Fischer and L. Gerbaud, “CoreLab: A component-based integratedsizing environment,” Int. J. COMPEL, vol. 24, no. 3, pp. 753–766, 2005.

[8] V. Fischer, L. Allain, and L. Gerbaud, “RAMA: A lightweight rule-basedtool for expressions analysis and code generation,” in Proc. 15TH Eur.Simul. Symp. Exhibition, Delft, The Netherlands, Oct. 2003, p. 4.

Manuscript received June 25, 2005 (e-mail: Bertrand.Du-Peloux@leg.en-sieg.inpg.fr).