Optimisation of a switched reluctance motor using experimental design method and diffuse elements response surface M.C. Costa, S.I. Nabeta, A.B. Dietrich, J.R. Cardoso, Y. Mar ! echal and J.-L. Coulomb Abstract: The experimental design method is applied to the construction of a response surface based on diffuse elements as preliminary steps in the optimisation process of the magnetic torque of a switched reluctance motor. The experimental design method is used to find and choose the most significant parameters related to the magnetic torque of the motor, reducing the number of parameters to be optimised. The diffuse element method is used to build a response surface that represents an approximation of the real objective function. The optimised values of the parameters are obtained after the application of genetic algorithms in the response surface, reducing drastically the final computation time. 1 Introduction The optimisation of electromagnetic devices can become complex and require a high computation time when the problem is described by a large number of parameters to be optimised. A solution to this problem is to optimise only the parameters that have a significant contribution in the value of the objective function. The application of the experimental design method (EDM) allows the identification of these parameters, as well as its more significant interactions, using a reduced number of objective function evaluations to this end, as shown by Gillon and Brochet [1] and Sado and Sado [2] . Another solution for the high computation time problem is replacing the objective function by an approximation with fast evaluation. This approximation is called response surface and it can be obtained using the diffuse element method, as shown in Costa et al [3]. To show the efficiency of these methods, we applied them to the optimisation of the magnetic torque of a switched reluctance motor. 2 Description of switched reluctance motor The goal of this application is to optimise the geometry of a 4/6 switched reluctance motor to maximise the magnetic torque developed by the machine. Figure 1 shows the geometry of the machine, the parameters to be optimised and the rotor position where the magnetic torque is computed. Table 1 presents the initial values of the parameters and their variation ranges. Note that r 3 depends on the value of r 2 , so the air-gap value remains constant in all configurations. To evaluate the developed torque we used a magnetostatic finite-element model. 3 Experimental design method analysis The experimental design method is an analysis tool that allows one to quantify the contribution of each parameter in the objective function value. The method comprises the performance of some evaluations of the objective function, modifying the value of each parameter in a reasonable way [2] . This set of evaluations represents a factorial design whose number of experiments to be performed is given by N exp ¼ n k ð1Þ where k ¼ number of parameters and n ¼ number of levels used for each parameter. One can see that the number of experiments becomes prohibitive when the number of parameters increases. To avoid this problem we use a ‘fraction’ of the complete design. This type of design is known as fractional or incomplete factorial design [2] . The r 4 r 5 r 0 r 3 r 2 r 1 β 1 β s Fig. 1 Geometry of switched reluctance motor M.C. Costa, S.I. Nabeta, A.B. Dietrich and J.R. Cardoso are with the LMAG – PEA, Escola Polit! ecnica da Universidade de S* ao Paulo, Av. Prof. Luciano Gualberto, trav. 3 - n. 158, S* ao Paulo- SP, 05508-900, Brazil Y. Mar! echal and J.-L. Coulomb are with the LEG - Laboratoire d’Electrotechnique de Grenoble, France, ENSIEG - INPG - CNRS UMR 5529, St. Martin d’H" eres BP46 - 38402, France r IEE, 2004 IEE Proceedings online no. 20040951 doi:10.1049/ip-smt:20040951 Paper first received 10th March 2004 and in revised form 5th July 2004. Originally published online: 8th December 2004 IEE Proc.-Sci. Meas. Technol., Vol. 151, No. 6, November 2004 411
Transcript
Optimisation of a switched reluctance motor usingexperimental design method and diffuse elementsresponse surface
M.C. Costa, S.I. Nabeta, A.B. Dietrich, J.R. Cardoso, Y. Mar!echal and J.-L. Coulomb
Abstract: The experimental design method is applied to the construction of a response surfacebased on diffuse elements as preliminary steps in the optimisation process of the magnetic torque ofa switched reluctance motor. The experimental design method is used to find and choose the mostsignificant parameters related to the magnetic torque of the motor, reducing the number ofparameters to be optimised. The diffuse element method is used to build a response surface thatrepresents an approximation of the real objective function. The optimised values of the parametersare obtained after the application of genetic algorithms in the response surface, reducing drasticallythe final computation time.
1 Introduction
The optimisation of electromagnetic devices can becomecomplex and require a high computation time when theproblem is described by a large number of parameters to beoptimised. A solution to this problem is to optimise only theparameters that have a significant contribution in the value ofthe objective function. The application of the experimentaldesign method (EDM) allows the identification of theseparameters, as well as its more significant interactions, using areduced number of objective function evaluations to this end,as shown by Gillon and Brochet [1] and Sado and Sado [2].
Another solution for the high computation time problemis replacing the objective function by an approximation withfast evaluation. This approximation is called responsesurface and it can be obtained using the diffuse elementmethod, as shown in Costa et al [3].
To show the efficiency of these methods, we applied themto the optimisation of the magnetic torque of a switchedreluctance motor.
2 Description of switched reluctance motor
The goal of this application is to optimise the geometry of a4/6 switched reluctance motor to maximise the magnetictorque developed by the machine. Figure 1 shows thegeometry of the machine, the parameters to be optimisedand the rotor position where the magnetic torque iscomputed. Table 1 presents the initial values of theparameters and their variation ranges. Note that r3 dependson the value of r2, so the air-gap value remains constant in
all configurations. To evaluate the developed torque weused a magnetostatic finite-element model.
3 Experimental design method analysis
The experimental design method is an analysis tool thatallows one to quantify the contribution of each parameterin the objective function value. The method comprises theperformance of some evaluations of the objective function,modifying the value of each parameter in a reasonable way[2]. This set of evaluations represents a factorial designwhose number of experiments to be performed is given by
Nexp ¼ nk ð1Þwhere k¼ number of parameters and n¼ number of levelsused for each parameter. One can see that the number ofexperiments becomes prohibitive when the number ofparameters increases. To avoid this problem we use a‘fraction’ of the complete design. This type of design isknown as fractional or incomplete factorial design [2]. The
r4
r5
r0
r3r2 r1
β1
βs
Fig. 1 Geometry of switched reluctance motor
M.C. Costa, S.I. Nabeta, A.B. Dietrich and J.R. Cardoso are with the LMAG –PEA, Escola Polit!ecnica da Universidade de S*ao Paulo, Av. Prof. LucianoGualberto, trav. 3 - n. 158, S*ao Paulo- SP, 05508-900, Brazil
Y. Mar!echal and J.-L. Coulomb are with the LEG - Laboratoired’Electrotechnique de Grenoble, France, ENSIEG - INPG - CNRS UMR5529, St. Martin d’H"eres BP46 - 38402, France
r IEE, 2004
IEE Proceedings online no. 20040951
doi:10.1049/ip-smt:20040951
Paper first received 10th March 2004 and in revised form 5th July 2004.Originally published online: 8th December 2004
most-used fractional designs are the ones given byTaguchi’s tables [1, 2], which represent a predefined setof fractional designs. The use of fractional factorialdesigns reduces the number of experiments, but at thesame time it gives a great problem: the presence ofconfusions. Confusions represent a merge between thecontribution of main effects of the parameters and of theirinteractions. For example, if we consider a four-parameterproblem and use only eight experiments to perform theEDM analysis, we obtain the confusions presented inTable 2.
In this situation one cannot identify if the contributionCA of the contrast (a set of confusions) A is due to theparameter p1, the interaction p2 p3 p4 or both. To avoid theinfluence of these confusions in the analysis of EDM weconsider the following hypotheses, as described in [2]:
� The high-order interactions (more than two) can beconsidered negligible.
� Once a contrast is negligible, all the effects that composethis contrast can be considered negligible too.
� Two significant factors (parameters) can also result in asignificant interaction. However, two nonsignificant factorsdo not give significant interactions.
Based on these hypotheses we analyse the results from theapplication of Taguchi’s L16 design in the analysis of ourswitched reluctance motor.
3.1 Application of L16 designTable 3 shows the application of Taguchi’s L16 design whereone can see the parameters values at every performedexperiment, as well as the respective magnetic torquecomputed by finite elements.
Table 4 presents some contrasts obtained by the analysisof L16 design where we verify that only the contrasts B, Eand F have more than 5% of contribution and can be
considered significant. Another significant contrast obtainedby this design is
r1r5 þ brbs þ r1r2r4bs þ r1r2r4r5br ¼ 29:62%
From our analysis of the Taguchi L16 design, we identifythe factors r2, br, bs and the interaction brbs are significantcontributors to the torque, and thus have significance forthe objective function.
3.2 Application of complete factorial designAiming to validate the previous analysis we have made ananalysis of our machine using the complete factorial design[1, 2], which allows us to identify the effect of eachparameter and each interaction, without confusionsbetween them. This step is only for the verification of theresults and it is not usually used in the procedure ofparameters screening.
Table 5 shows the contributions obtained after theevaluation of 64 (26) experiments for the completefactorial design analysis. Based on these values we canidentify the following significant parameters: br, bs andr2, as well as the following interactions: brbs and r2brbs.All the other interactions presented a 5% lowercontribution so they can be considered negligible, whichvalidates the results obtained by the application of L16
design.
Table 1: Optimisation parameters
Parameter Min. value Max. value Initial value
r1 8mm 18mm 12mm
r2 20mm 35mm 25mm
r3 r2+0.25mm r2+0.25mm r2+0.25mm
r4 46mm 53mm 50mm
r5 58mm 65mm 58mm
br 0.40rad 0.90 rad 0.6283rad
bs 0.40rad 0.72 rad 0.5585rad
Table 2: List of confusions of fractional factorial design
Contrast Confusions Contribution (%)
A p1+p2p3p4 CA
B p2+p1p3p4 CB
C p3+p1p2p4 CC
D p4+p1p2p3 CD
E p1p2+p3p4 CE
F p1p3+p2p4 CF
G p2p3+p1p4 CG
H p1p2p3p4 CH
Table 3: Experiments performed for Taguchi’s L16 design
Once we identified the significant parameters (r2, br and bs)of our optimisation problem we used the diffuse elementmethod to build a response surface that represents anapproximation of the objective function. We used astandard diffuse element response surface [3] with a regulardiscretisation of nine points by direction of the domain. Thecreation of this response surface required 729 (¼ 93) finite-element computations.
Figure 2 shows a 3D representation of the responsesurface considering r2¼ 35mm, where we verify a stronginteraction between parameters br and bs.
After creation of this response surface we applied anoptimisation method based on genetic algorithms [4] with aconfiguration of 30 individuals and 300 generations to findthe optimal values of the parameters, which are presented inTable 6. Figure 3 shows the optimised geometry of theSRM. The optimised value of r2 represents its upper limit,mainly due to its direct relation with magnetic torque. This
limit, as well as the lower one, is related to geometryconstraints.
5 Conclusions
We have verified that both the experimental design methodand diffuse elements response surface are useful tools in theoptimisation process of electromagnetic devices. Weproposed and validated some useful considerations in thetreatment of confusions when using fractional factorialdesigns. After EDM application, the initial number ofoptimisation parameters of the SRM decreased more than50 %. After optimisation, the magnetic torque increasedmore than 40 %.
6 References
1 Gillon, F., and Brochet, P.: ‘Screening and response surface methodapplied to the numerical optimisation of electromagnetic devices’, IEEETrans. Magn., 2000, 36, pp. 1163–1166
2 Sado, G., and Sado, M.C.: ‘Les plans d’exp!eriences: de l’exp-!erimentation "a l’assurance qualit!e’. Afnor Technique, France, 1991
3 Costa, M.C., Coulomb, J.-L., Mar!echal, Y., and Nabeta, S.I.: ‘Anadaptive method applied to the diffuse-element approximation in theoptimisation process’, IEEE Trans. Magn., 2001, 37, pp. 3418–3422
4 Alotto, P., Eranda, C., Brandst.atter, B., F.urntratt, G., Magele, C.,Molinari, G., Nervi, M., Preis, K., Repetto, M., and Richter, K.R.:‘Stochastic algorithms in electromagnetic optimisation’, IEEE Trans.Magn., 1998, 34, (5), pp. 3674–3684
Table 5: Contributions obtained by complete factorialdesign analysis
Parameter or interaction Contribution (%)
r1 0.01
r2 21.52
r4 0.94
r5 0.80
br 9.29
bs 12.78
brbs 35.79
r2brbs 7.62
y o5.0
4
3
2
1
0
0.7
0.6
0.5
0.4 0.40.5
0.60.7
0.80.9
Fig. 2 Representation of response surface considering r2¼ 35 mm
