Multiobjective Optimization of IPM synchronousmotors using Response Surface Methodology and

filtered Monte Carlo approach

Robert SeifertElektrotechnisches Institut

Dresden University of TechnologyEmail: Robert.Seifert@mailbox.tu-dresden.de

Ramon Bargallo PerpinaDepartment for Electrical Engineering

Polytechnic University of Catalonia, BarcelonaEmail: ramon.bargallo@upc.edu

Abstract—Permanent Magnet Synchronous Motors offer highefficiency and power density besides low assembly effort andhence have been established in a wide market over the past years.Especially buried magnets enable a superior field weakeningability but require an exceptional design effort. Costly FiniteElements computations are inevitable for consideration of theoccurring non-linearities and non-trivial magnet shapes. TheResponse Surface Methodology can reduce the number of FEruns significantly by introducing a acceptably exact second orderregression model based on a few carefully chosen design samples.Instead of commonly used, but time-consuming EvolutionaryStrategy methods, the Monte Carlo approach is applied foroptimisation. Using simple filter algorithms, distinctive Paretofrontiers can be determined quickly and related to their causativemotor designs.

I. INTRODUCTION

Interior permanent magnet synchronous motors (IPMSM)became established in a wide field of applications, especially incase compact designs with high power capability are required.Due to the absence of rotor and commutator losses theyare characterised by an excellent efficiency. The additionalreluctance torque provides a high torque over a wide speedrange. Most likely inverter driven they provide a superiorcontrollability and a high power density, when operated atthe optimum torque-to-ampere trajectory. However despiteall advantages high magnet costs hamper the entering intothe mass market of electrical excited machines. In spite ofthe exploding world-market prices for rare earth materialsand magnets1, their benefits over ferrite magnets in terms ofsize, weight and performance are so conclusive, that they arestill dominating the market. In contrary to surface mountedmagnets, which are often bent and skewed and adapted forcertain rotor designs, interior magnets get along with simpleshapes, which even can be used in different layouts of differentsizes. This way the wastage and therefore production costs canbe minimized.

The excitation through buried magnets enable a ruggedrotor construction which allow high speeds and requires lowmaintenance, as the magnets are physically protected and fixedwithout the need of bandages at high speeds. This in turnallows small air gaps and a further increased efficiency. Techni-cally the introduced steel poles alter the magnetic circuit, weak

1A rise in prices about a thousand percent from 2010 to 2012 is reportedin [1]

the d-axis path and hence capacitate the complete cancellationof the excitation field. On the other hand they strengthenthe q-axis due to an dominating quadrature axis inductance,are leading to a high saliency of the magnetic circuit andintroducing an additional reluctance torque component, whichmakes IPMSM superior for field weakening demanding ap-plications, like pump/fan drives or machine tool servo drives.Because low flux linkages at high speeds not automaticallycause decreased torques at low speeds, as the reluctance torqueis independent of the permanent magnet flux. Cogging torquecan be minimised by a deliberate rotor design, even withoutthe otherwise commonly used skewing, which is only hardlyfeasible for IPMSM. Air barriers to prevent magnetic short cir-cuits and the occasionally wide spread steel poles cause a non-trivial flux distribution regarding the rotor, which complicatesthe design procedure considerably. Analytic models relatingrotor geometry and air gap field fail to describe satisfactorybasic correlations. The use of the finite element analysis isinevitable to determine the fundamental waves in the air gap,whose evaluation then can be applied on analytic models. Theobjective of this paper is to introduce the combined use ofFinite Element Analysis, Response Surface Methodology andMonte Carlo Method for an effective reduction of computationtime.

II. ANALYTIC MODEL OF THE IPMSM

The first attempts of analyzing the field weakening abilitiesusing the linear lossless model of IPMSM date back to the80’s. Inverter voltage and stator currents are related to torqueand power output considering d- and q-axis inductance asconstant motor parameters. The influence of the saturation ofthe magnetic circuit is neglected as well as stator and ironlosses. However the model is sufficient to describe the drivecharacteristics and power capability of the motor over theentire operating range and allows a comparison of differentmotor designs, but is not capable of determining realistic motorcharacteristics and is therefore disqualified for optimizationpurposes.

A. Non-linear model

The development of the analytic model is highly shortenedand just relevant correlations are shown. For an extensivederivation I refer to [2] for further reading, a comparison ofanalytic model and FE-analysis is shown in [3].

Based on the magnetic flux λm and the d/q- inductancesas well as currents the torque can be calculated by

T =3

2p(λmIq + (Ld − Lq)IqId) (1)

In this equation becomes apparent why IPMSM are of-ten denoted as a hybrid combination of the conventionalsynchronous-reluctance SM and surface mounted PMSM. [4]The first term of the equation describes the interaction of theq-axis current with the magnetic flux due to the permanentmagnets, orientated to the d-axis, and is called field-alignmenttorque Tal equally to machines with surface mounted magnets.Additionally the second term represents the reluctance torquewhich is generated by interaction of the stator current compo-nents and its orthogonal magnetic flux components. It is highlydependent on the saliency ratio ξ.

Tal =3

2pλmIq (2)

Trel =3

2p(λsdIq − λsqId) =

3

2p(1− ξ)IqIdLd (3)

Under saturation the flux linkages, especially the q-axis flux,are not proportional to the stator currents. The inductancesdecrease significantly in the most saturated driving regions formaximum q-axis current (β = 90). Rather its use as parameterfor torque optimization should be avoided completely.2 Insteadall calculations are carried out using the flux linkages directly,computed by the FE-analysis. To contain the number ofnumeric computations the stator current is fixed at the ratedcurrent circle and therefore the variable parameters are reducedto a single one, the electric angle β.3

Id = Ir cos(β) and Iq = Ir sin(β) (4)

Consequently the flux linkages for rated current Ir are justfunctions of β, while it is sufficient to observe them in therange of β = [90, 180]

λd = λm + λsd(Id) = λd(β) (5)λq = λsq(Iq) = λq(β) (6)

For the torque follows

T =3

2pIr

(λd(β) sin(β) + λq(β) cos(β)

)(7)

and the voltages are determined by

Vd = −ωλq(β) and Vq = ωλd(β) (8)

B. Drive operation limits

Besides mechanical restrictions, the main limitations of thedrive regions are the maximum output voltage of the inverterand the maximum acceptable stator current under thermalaspects.

I2d + I2

q ≤ I2r and V 2

d + V 2q ≤ V 2

max (9)

2In the experiments the q-axis inductance was reduced about 24% formaximum q-axis current, equally the saliency ratio as the direct axis was justslightly influenced. Further analysis regarding saturation effects was made bySoong and Miller [5].

3Assuming d-axis and phase A axis are aligned.

defining an ellipse

(LdId + λm)2 + (LqIq)2 ≤ (

Vmaxω

)2 (10)

which has its center at the point [CVL, 0], with

CVL = −λmLd

=λmλsd

Id(β = 180) = −λmλsd

Ir (11)

Later will be shown that the ratio λmλsd

, respectively the dif-ference λm − λsd, which defines this point, is a very suitablecriterion to optimize the constant power speed range (CPSR) asit directly expresses the field weakening ability. It is describedby the ratio Ω2/Ω1 of maximum speed Ω2 to minimum speedΩ1 = Ωr at which rated power can be achieved. Furthermorehyperbolas of constant torque can be defined using eq. 1:

Iq =2T

3pLd

(λmLd

+ (1− ξ)Id)−1

(12)

Practically every specific value of torque (below Tmax) canbe attained by a indefinite number of current vectors. Tomaximise efficiency Jahns [4] introduced a current trajectoryfor an optimal ’maximum torque per stator current ampere’characteristic, in which the current vector follows the normalvectors of the torque hyperbolas, in other words it is to choosethe point on every torque hyperbola as close as possible to theorigin of the d/q-plane. (Mode 0 in Fig. 1) Equation 10 showsthat the voltage ellipse shrinks for increasing speed and on theother hand the voltage demand increases (eq: ??) for constanttorque and therefore constant flux linkages and currents. Byderivation of eq. 1 and transposition, the current angle β formaximum torque can be determined [6] to:

βTmax = sin−1(λm +

√λ2m + 8(ξ − 1)2L2

dI2r

4(ξ − 1)LdIr

)(13)

For speeds below base speed Ωb the voltage demand, whichis necessary to maintain the maximum torque, reaches the limitof the inverter (Point A in Fig. 1). So torque and power arejust limited by the current (Mode I), above Ωb the currentvector is following the current limit circle (Mode II), wherevoltage limit ellipses and constant torque hyperbolas haveintersections. Which leads us to the fact, that a classificationin two machine types is necessary.

Class I — Finite speed IPMSM: Machines of class I areusually excited by rare earth magnets with a high remanentflux density and have a high saliency. Even the maximumnegative d-axis current is not able to completely cancel out themagnetic flux, which entails a finite maximum speed accordingto equation ?? at point C. The current vector remains at therated current circle and the machine is kept in Mode II foroptimal field weakening performance.

Class II — Infinite speed IPMSM These machines areexcited by weaker magnets, e.g. ferrite, and/or have a relativelylow saliency. Moving on the current limit circle the total d-axis flux would become negative at a certain angle β (point B)and decrease the torque rapidly. For an optimum torque/currenttrajectory (Mode III) the d-axis current needs to be reduced insuch a away that constant torque hyperbolas and voltage limitellipses osculate tangential. As the latter is centred inside thecurrent circle, the speed can theoretically rise to infinite.

Mode II

Mode I A

maximum speed

C

voltage limit elipse

Mode 0

CVL > Ir

Iq current limit

circle

Id

Mode II

Mode I A voltage limit

elipse Mode 0

CVL > Ir

Iq

current limit circle

Id

C

infinite speed

Mode III

B

a) Finite Type b) Innite Type

Fig. 1: Drive operation limits of IPMSM, for a) finite type andb) infinite type [6] [7]

This differentiation was first made by Morimoto [6] and ex-tended by Soong and Miller [7] by classifying and comparingthese two types with other classes of brushless synchronousAC motor drives. Together with Jolly [3] they all refer to theoptimal field weakening design at the changeover of these twoclasses. [7] The basic concept is to reduce q-axis as well asd-axis flux to zero to achieve the maximum field weakeningability. For the q-axis this is generally the case for β = 180

as Iq is zero. So the point of neutralization of d-axis stator fluxλsd and magnetic flux λm has to coincide with this electricangle. In this case the speed reaches its maximum or infinitefor total neutralization.

Ωmax =Vmax

p√λd(β = 180)2 + λq(β = 180)2

(14)

As mentioned above for infinite speed IPMSM, this can beobtained by regulating the stator current. But in this mannerthe utilization of the machine is below capacity as the thermallimits allow higher currents. Hence the neutralization is desiredfor rated current. Thus the voltage limit ellipse is centered atthe rated current circle. The essential design objectives foroptimal field weakening performance, first stated by Schiferl[8], are:

λmλs

∣∣∣∣Id = −Ir, Iq = 0

= 1 (15)

III. FINITE ELEMENT ANALYSIS

To determine all parameters of the non-linear model theFE-analyis is the first choice, as it takes into account allsaturation effects by regarding the whole second quadrant ofthe rated current circle. Since all computations are executed atzero frequency, iron losses due to hysteresis effects and eddycurrents are still neglected. The real frequency is not knownin advance, given that it is controlled by the velocity which

is in turn dependent on the geometry. However the evaluationof the momentary current peak values ensure maximum com-parability. Consequently all achieved values of flux linkagesare peak values as well. Disregarding the harmonics of air gapflux and e.m.f back voltage does not cause problems, as theanalytic model is based on fundamentals anyway.

IV. RESPONSE SURFACE METHODOLOGY

The Response Surface Methodology (RSM) was first intro-duced by Box and Wilson in the Journal of the Royal StatisticalSociety [9] in 1951 for the purpose of optimizing conditionsin chemical investigations and became well established in thisfield. It originated from the difficulty that a process can notalways be described by physical laws and analytic modelsor simply involves indeterminate factors. Rather RSM createsan empirical model that relates the process response to wellknown input parameters. Its application to electromagneticproblems started in the 90’s for instance by Rong and Lowther[10], those who applied RSM to a non-linear optimizationproblem of an electromagnetic actuator. The subject of thispaper was already analyzed by Jabbar [11] and Jolly [3] in thelast decade.

The objective is to achieve a set of design variables γ inwhich the response η is maximum or minimum within theexperimental region R, that is defined by practical limitationslike geometry, mechanical restrictions or saturation. The k in-dependent design variables are normalized to equally boundedfactors x1, x2, ..., xk and need to be capable of exact mea-surement. In order to determine the response y the regressionmethod least squares is applied to multiple order polynomialsto fit the sure values y of n design samples computed byFE-analysis. To minimize the computation time it is desiredto keep n as small as possible, which is dependent on thepolynomial order and the number of factors k. For satisfyingresults the Design of experiments (DoE) to determine y hasto be chosen advisedly. RSM can not substitute measurementsand additional FEM computations in the experimental regionaround the optimized response, but reduce them to a feasiblenumber.

A. Procedure

Every process is characterized by a true response such asthe actual torque of a real synchronous motor, which can bemeasured and hence underlies a measurement error εm. It isself-explanatory that there are no prototypes available for everydesign sample to verify the FEM computations. For this reasona single computation result itself is assumed to be an adequateimage of a real machine and defined as the true response η .However the evaluation of the FE-analysis is not always trivialand underlies an error εeva. Considering this, the so-called surevalue or mean response y is described as:

y = η + εeva (16)

The evaluation error is random and assumed to have zeromean value and thereby y is conditioned to be an adequaterepresentation of the true response η. To relate response andinput variables the approximation function f needs to beattained.

y = f(x1, x2, ..., xk) + εeva (17)

Which is realized by means of low-degree polynomials rep-resented by the matrix of arguments X and the coefficientmatrix β:

y = X ′β + εeva (18)

Depending on the degree d of the polynomials, the matricesfor three independent variables are composed as follows forthe second-degree model (d = 2):

X ′ =[

1 x1 x2 x3 x1x2 x1x3 x2x3 x21 x2

2 x23

](19)

β′′ = [β0 β1 β2 β3 β12 β13 β23 β11 β22 β33 ] (20)

In general the number of terms and coefficients of k-variated-degree polynomials can be determined to

c =

(k + d

d

)=

(k + d)!

k!d!(21)

The design samples are represented by a matrix D containingthe various normalized geometries:

D =

x11 x12 ... x1k

x21 x22 ... x2k

. . ... .

. . ... .

. . ... .xn1 xn2 ... xnk

=

D1

D2

.

.

.Dn

(22)

For every design sample a mean response y is known, com-puted by the previous FE-Analysis. Furthermore Xn can becreated for every geometry by inserting Dn in X , which leadsus to the following overdetermined linear system of equations:

y1

y2

.

.

.yn

=

X ′1X ′2...X ′n

· β +

ε1ε2...εn

(23)

and accordingly the vector ε of evaluation errors εeva,n:

Y = Z · β + ε (24)

The coefficient vector β is determined under the approach oflinear least squares [12], that means the sum of the squares ofthe errors in ε is minimised. It can not get compensated to zerosince the equation system is overdetermined and doesn’t havejust one exact solution. The so-called ordinary least-squaresestimator of β is obtained by:

β = (ZZ′)−1Z′Y (25)

Henceforth the predicted response y can be determined forevery single point in the experimental region R, that meansX can reflect any possible combination of design variables[x1, x2, ..., xk].

y = X′ · β (26)

Its variance is given as:

Var (y) = σ2X′(ZZ′)−1X (27)

TABLE I: Statistical attributes

Name Equation

Mean value of all mean responses y =y1+y3+...+yn

n

Average relative error |∆y/y| = 1n

∑ni=1

∣∣∣ yi−yiyi

∣∣∣Maximum relative error max |∆y/y| = max

∣∣∣ yi−yiyi

∣∣∣Total sum of squares (variance of samples) SST =

∑ni=1(yi − y)2

Regression sum of squares (var. of responses) SSR =∑ni=1(yi − y)2

Residual sum of squares (var. of errors) SSE =∑ni=1(yi − yi)2

Coefficient of determination R2 = 1− SSESST

Adjusted coefficient of determination R2 = 1− (1− R2) n−1n−c−1

B. Error estimation

There are various statistical attributes, which can evaluatethe accuracy of a regression model (Table I). In case the regres-sion model just involves interpolations and no extrapolations,the summation of SSE and SSR equals the total sum ofsquares SST . For instance in case the whole experimentalregion R lies inside of the space framed by the Full FactorialDesign. The explained coefficient of determination R2 iscommonly used for the rating of regression models (1 =ideal regression), even though it is not utterly correct for allcases. Preferable is the unexplained coefficient R2, which alsoinvolves the number of sampling points and the degree of thepolynomial represented by its number of terms c (eq. 21).

C. Introduction in different RSM designs

Generally it is desired to minimize the number of samples,which need to exceed the number of coefficients c of theunderlying multi-order trivariate polynomials. Consequentlyfor second-order polynomials 10 coefficients are required,which can be provided by all of the following introduced RSMdesigns, but just the Full Factorial Design is applicable to third-order polynomials involving 20 terms.

1) Full Factorial Design (FFD): also called 3k FactorialDesign, is commonly used due to its high accuracy, though it isthe most time-consuming design considering 3k = 27 designsamples. Every design variable can take a value [−1 0 1].It has the characteristic property of orthogonality. A designis orthogonal in case the correlation matrix ZZ′ results in adiagonal matrix. For this reason the significance of unknownparameters can be evaluated easier, as both β and ε (assumingnormal distribution) are statistically independent.

2) Box Behnken Design (BBD): Box and Behnken devel-oped an efficient alternative in 1960 [13]. It is a simplificationof the FFD and omits the vertex and face center points andtherefore requires only 13 computations. On closer inspectionit becomes apparent, that every level of every design variableis represented by an equal number of four points, the originexcluded. In some scientific problems it is desired to analyzethese layers, called blocks, independently under comparableconditions. The response variance within a single block isequal and uncorrelated to nearby blocks, which is calledorthogonal blocking.

−1 0 1

−1

0

1

−1 0 1

a) b)

−1 0 1 −1 0 1

c)

Fig. 2: Full Factorial Design (FFD), Central Composite Design(CCD) and Box Behnken Design

3) Central Composite Design (CCD): is referred as one ofthe most popular ones [12] and was introduced by Box andWilson in 1951 [9]. It emanates from the usual approach ofstarting with a first-order polynomial model, considering justthe vertexes of a simple cube, to identify initial tendencies.Subsequently further points are added to improve the model.Besides the origin these are usually face centred points (FaceCentered Design) or star points, which then should enable therotatability of the design. That means the prediction varianceVar (y) (eq. 27) is constant for all responses equidistant fromthe origin, a property which becomes useful for sphericalproblems. The required level of the star points for rotatabilityis F k/4.

D. Design of experiments

Within the design of experiments (DoE) is declared howthe experimental region R is embedded in the chosen RSMdesign. Certain levels [x1, x2, ..., xk] are defined by physicalboundaries, which should be identified correctly to excludeinadequate responses and keep R as small as possible.

The parameter levels are generally coded as:

xγ = αSP2γ − (γmax + γmin)

γmax − γmin(28)

in which γ is the physical design variable and αSP the starpoint factor, in case the Central Composite Design is used.IndeedR is considerably larger than the space actually coveredby Full Factorial and Box Behnken Design, so outer responsesneed to be extrapolated which can cause a loss of accuracy.This difficulty appears since this paper intends to comparedifferent RSM designs while introducing the optimization pro-cess. If a motor design is supposed to be optimized exclusivelyby FFD or BBD, their design matrix D is adapted fittinglyto R (compare TABLE IIb). For the optimization problemdiscussed in this paper, the design variables γ are the magnetposition δR (distance from shaft) and the outer arc length ofthe air barriers, defined by the angles α1 and α2 (Fig. 5).

E. Comparison of RSM designs

1) General comparison: For the most optimization prob-lems the maximum torque T is the main objective. It is decisivewhether a motor can be used for certain applications or not.Furthermore it highly influences the efficiency and poweroutput. Within R it varies only slightly, because the probedgeometry is not as crucial as electro-magnetic factors like theremanent field strength of the permanent magnets or the actualapplied current. Hence good regression results by the use ofRSM can be expected.

Except the first-order FFD1 all RSM designs yield tosatisfactory regression results and even the former would

TABLE II: Errors and coefficients of determination

FFD1 FFD2 FFD3 CCD BBD

a) for TmaxR2 0.59 0.96 0.98 0.96 0.96R2 0.52 0.93 0.90 0.87 0.74

|∆y/y| 2.54 0.65 0.36 0.66 0.67max |∆y/y| 6.14 3.82 3.14 2.07 3.58|∆y|/|y| 2.51 0.65 0.35 0.66 0.66

b) for Tmax and reduced RR2 0.56 0.98 1.00 0.96 0.98R2 0.48 0.97 0.99 0.87 0.87

|∆y/y| 2.56 0.50 0.16 0.62 0.51max |∆y/y| 6.14 0.90 0.33 2.07 1.24|∆y|/|y| 2.53 0.50 0.16 0.63 0.51

c) for λπR2 0.90 0.98 0.98 0.98 0.98R2 0.88 0.97 0.92 0.94 0.87

|∆y/y| 53.87 14.24 4.80 5.42 18.38max |∆y/y| 1430.74 286.52 49.76 24.69 424.25|∆y|/|y| 9.02 3.98 2.41 3.92 4.18

d) for CPSRR2 0.17 0.47 0.73 0.35 0.08R2 0.02 0.13 -0.16 -1.26 -4.50

|∆y/y| 185.14 180.51 186.74 254.24 16.41max |∆y/y| 545.61 689.60 695.92 884.81 91.19

derived from λπ :|∆y/y| 12.20 8.09 7.57 7.89 8.45

max |∆y/y| 90.6 43.43 69.08 33.84 65.86

e) for NmaxR2 0.17 0.46 0.73 0.35 0.07R2 0.01 0.12 -0.17 -1.28 -4.56

|∆y/y| 265.10 255.60 277.35 363.57 20.65max |∆y/y| 863.27 1096.36 1140.47 1409.30 92.57

derived from λπ :|∆y/y| 11.61 6.29 5.30 5.52 6.95

max |∆y/y| 92.45 46.12 78.70 32.8 69.00

be suitable for the prediction of tendencies. Although CCDand BBD stand back behind the comparable FFD2 design,their relative errors are virtually identical. The use of higherpolynomials, if possible, can further improve accuracy. For aproper DoE or reduced R, FFD2 and BBD outperform or atleast match CCD in terms of accuracy and FFD3 minimizesits errors to almost zero (TABLE IIb).

Another often desired motor design objective is the fieldweakening ability, characterized by the maximum speed Nmaxand the constant power speed range CPSR. In TABLE IId,ebecomes apparent, that these variables are not suitable asoptimization objectives at all. So the d-axis flux λd for β = π(henceforth denoted as λπ) is used as it directly quantifiesthe field weakening. The accuracy for the regression of λπ islower than for Tmax (TABLE IIc), but much more importantas a certain value λπ = 0 needs to be achieved for optimalperformance. Manual optimization by trial-and-error in thepredicted optimum region is inevitable. Due to the smallabsolute values of λd around zero, the relative errors increasesignificantly. For this reason the ratio of average absolute errorto the average response value |∆y|/|y| is determined, to showthat the absolute error still remain very small and allow asatisfactory regression. The vertex point D22 = [−1, 1,−1]is excluded in the BBD, which is the most influential designsample for the optimization of the field weakening region. Forthis purpose BBD is discarded.

0 0.05 0.10

50

100

λπ in Wb

CP

SR

0 0.05 0.10

2

4

6

λπ in Wb

Nm

ax in

105 r

pm

Fig. 3: Regression of CPSR and Nmax as a function of√λ2π

2) Application on the non-linear model: The field weak-ening ability can be determined by a single FEM computationfor Iq = 0 and Id = Ir, but for maximum torque the electricangle β is not constant within R. Hence it is necessary toevaluate the entire second quadrant of the rated current circle,considering all saturation effects.

3) Selection of suitable optimization objectives: It wasclaimed before, that Nmax and CPSR are no suitable op-timization objectives, as they are able to reach an infinitevalue. This behavior can not be handled by the regressionmodels. The occurring errors are unacceptable. Rather the fluxlinkage λ is used as optimization objective as it is the actualinitiator of the hyperbolic behavior The maximum speed cantherefore be estimated indirectly using eq. 14, reducing theerrors to satisfactory levels and enable rough predictions, assmall estimation errors of λ are still increased substantial bythe hyperbolic function.

Nmaxrpm

=Vmax · 60

2πp√λ2π

Wb

V= 1.0982× 10−4 ·

(√λ2π

Wb

)−1

(29)

For the regression of CPSR no trivial equation is availableand a xa polynomial is used, fitting Yλ with an accuracy ofR2 = 0.9999.

CPSR = 0.285 ·(√λ2

π

Wb

)−0.913

(30)

F. Conclusion

All introduced designs of the Response Surface Methodol-ogy can adequately predict motor characteristics and param-eters based on few advisedly chosen motor design samples.It became apparent that the choice of the regression model isonly secondary as long as the significance of its shortcomingsis considered and the input variables are selected wiselyavoiding substantial non-linearities. Sensitive parameters canbe estimated using robust substitutes deduced from the analyticmodel.

The preferential decision criterion should rather be thecomplexity of the subordinated experiments or computations.Since magnet and circuit properties can be easily changedusing scripts, the determination of the response vector isautomatable and suggests the use of high-order polynomialsand the FFD. On the contrary optimization problems con-sidering geometrical motor parameters usual require manualmodifications and are often initiated exerting first-order modelswhich will be rearranged or improved in a second step. In thiscase flexible and efficient designs like the CCD are suited. Itis the first choice for the optimization process in this paperas it provides the most precise regression results in the fringeareas of the exemplary experimental region. The BBD would

be appropriate for an adjusted design of experiments as wellas the FFD, if one can condone the high experimental costs.

V. OPTIMIZATION

In motor design naturally appear multi-objective optimi-sation problems which lead to compromises, as the optimumdesigns for every objective are usually mutually exclusive. Forinstance high efficiency contradicts to low production costsor, like in this case, a high torque excludes a high maximumspeed. There are various approaches to solve this problems,for instance evolution strategy methods (ES) [14] and inparticular genetic algorithms (GA) [15]. They are alreadyhigh developed and use genetic operators, like recombinationand mutation, to manipulate prior solutions. Enhancementsare then considered in further variations. They all tend tofind Pareto optimal solution sets, means vectors of designvariables Dγ whose variation does not decrease any responsewithout increasing at least one another, while one is optimalregarding the others. The entirety of all vectors forms oneor multiple so-called Pareto frontiers. Basically ES methodsare controlled trial-and-error methods. The genetic operatorsminimize the number of trials by evaluation of every iterativestep. Furthermore partial solutions and finally Pareto optimalsolutions are connected to its causative machine design. Buttechnically the optimal solution can be found without anyknowledge about the underlying design parameters. For thispurpose the most rudimentary of all statistical approaches isused, the Monte Carlo Method. Within a few seconds a largenumber of random vectors of design variables is generated andthe responses are calculated. The visualization of the resultsreveals the Pareto frontier precisely.

A. Pareto optimization with Monte Carlo Method

Unavoidable improving one criterion adversely affects an-other. However a so-called Pareto optimal solution set ofcompromises can be found in between. For one preferablevalue of criterion A, contrary criterion B is optimized. Theentirety of all optimized combinations of A and B leads tothe Pareto frontier, in case of two-dimensional problems astrictly monotonic function in the AB plane. For more thanthree objectives the visualization is hardly possible and theuse of complex evaluation algorithms is inevitable.

In general a multi-objective optimization problem can beformulated as:

F (x) = min(f1(x), f2(x), . . . , fk(x)

)(31)

wherein f are functions representing the k design objectivesregarding the decision makers x ∈ R, which are in this casethe design variables. If an objective function is to be maxi-mized, it needs to be transformed into its inverse monotonicfunction. The global optimum does not exist generally and istherefore called utopian solution.

xutopian ∈ R if∀x ∈ R, fi(xutopian) ≤ fi(x) for i ∈ [1, 2, . . . , k] (32)

A solution is Pareto dominant, if

xdominant ∈ R iffi(xdominant) ≤ fi(x) for ∀i ∈ [1, 2, . . . , k]

fj(xdominant) < fj(x) for ∃j ∈ [1, 2, . . . , k] (33)

0 50 100 150 200 0 5 10

0 2 4 6 8 103.2

3.4

3.6

3.8

4T m

ax in

Nm

CPSR0 0.2 0.4 0.6 0.8 1

3.2

3.4

3.6

3.8

4

Nmax

in 10 6rpm

Fig. 4: Pareto optimal solutions of CPSR and Nmax inrelation to Tmax

and Pareto optimal, if it is the most dominant. [16]

xP = xpareto ∈ R if and only if there exist nofi(x) ≤ fi(xpareto) for ∀i ∈ [1, 2, . . . , k]

fj(x) < fj(xpareto) for ∃j ∈ [1, 2, . . . , k] (34)

The Pareto optimal design vectors DPγ are to be determined,

which minimize the magnetic flux λd at β = π to zero:

min(fλ(Dγ)

)= 0 (35)

and maximize the maximum torque Tmax, respectively mini-mize its negative −Tmax:

max(

(fT (Dγ))⇐⇒ min

(f−T (Dγ)

)(36)

Hence the two-objective optimization problem is

F (Dγ) = min(fλ(Dγ), f−T (Dγ)

)(37)

and has the Pareto optimal solutions DPγ , that means there

exists no vector Dγ with:

f−T (Dγ) ≤ f−T (DPγ ) and fλ(Dγ) < fλ(DP

γ ) (38)

The functions f−T (Dγ) and fλ(Dγ) are unknown. Indeed thesecond-order polynomials, provided by the RSM, can emulatethem, but still an analytic optimization is not constructive.Rather the Pareto solutions are determined visually from thesolutions attained by the Monte Carlo Method3. The readingerror is less than the regression error arising from the RSM.The particular design objective in this case is:

Tmax = f(λπ

)= f

(λm, Ld, Lq, βTmax

)!= max (39)

The Monte Carlo experiments are regressions results of theResponse Surface Methodology and expected to be underneath:

yTmax ≤ f(yλπ

)(40)

For a precise mapping of potential Pareto optimal solutions,up to 200000 random design vectors within R are generated.Using the polynomials provided by the CCD regression model,the predicted responses y for Tmax and λπ are determinedin order to achieve a distinctive boundary. Fig 5 shows thepredicted maximum for Tmax, it can be seen that for theoptimum design the air barriers prevent the flux lines to bendexcessively within the rotor path, force its bending into the airgap and therefore maximize the Maxwell forces. In contrary to

3The Monte Carlo Method was developed by S. Ulam in the late 40s and isbased on the Law of large numbers. It postulates that the result of an randomexperiment converges to its expected value if its number of trails is sufficientlyhigh.

Fig. 5: Bending of flux lines in rotor path for different anglesα2

Fig. 6: Motor design for (almost) total cancellation of air gapflux

the torque, the optimum field weakening characteristic λπ = 0is clearly defined. It theoretically allows infinite speed andCPSR (Fig. 4), with a clear understanding that these speedsare mechanically impossible. Consequently the optimum fieldweakening design will seldom be applied practically as itdecreases the torque for the whole speed range. Unless themaximum constant power output is desired up to the pointof maximum speed. However motor designs which allowthe complete cancellation (Fig. 6) of the magnetic excitationfield are theoretically interesting in terms of this paper asit illustrates the ability of the RSM to determine concreteobjectives. So it will be shown that maximization problemsare uncritical, since an exact value is irrelevant, as long as themaximum is distinctive. But a minimization to a certain value’zero’ requires several iterations and subsequently verificationsof the predicted responses.

VI. CONCLUSION

Traditional design procedures, based on analytic modelsand physical prototyping yield to satisfactory results for a widerange of motor types. Induction machines, synchronous motorswith distributed windings and even salient pole SM and surfacemounted PMSM can be modeled analytically accurate enough.The finite element analysis is then primarily used for the finaloptimization.

However the negligence of saturation for IPMSM is notacceptable anymore and occasionally extraordinary shapedmagnet circuits prohibit the use of analytical models andnecessitate FE-analysis already during the design process. Thisis basically a time-consuming process as CAD drawings needto be created and modified and the computations are costly.With the Response Surface Methodology a procedure wasintroduced minimizing the effort to a few design samples.Which, defined by a sophisticated design of experiments, areinterpolated and therefore establish a consistent non-linear

0 0.02 0.04 0.06 0.08 0.1 0.123.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

λπ= 0.0957 Wb

Tmax

= 3.8 NmT

cog= 0.195 Nm

x1= −1.65

x2= −1.44

x3= 0.858

α1= 34.1 °

α2= 19.7 °

δR

= 11.6 mm

λβ=π = f ( 1/ Ωmax ) in Wb

λπ= 0.0157 Wb

Tmax

= 3.41 NmT

cog= 0.113 Nm

x1= 0.26

x2= 1.61

x3= −0.631

α1= 39.8 °

α2= 36.6 °

δR

= 5.37 mm

T max

in N

m

T cog in

Nm

0.1

0.15

0.2

0.25

response

normalised parameters(input of RSM pattern)

input parameters(e.g. actual geometry)

pareto optimal solution set

example B

example A

Fig. 7: Scatter plot of 200,000 responses, filtered for a two-objective Pareto optimal solution set of maximum torque and field weakeningability (cancellation for λπ = 0) and adjusted for invalid parameter combinations; cogging torque as informative additional objective (colorcoded); immediate output of responses and their corresponding (normalized) design parameters; computation time less than 5 seconds on anaverage workstation assuming existing training data (by previous FE computations)

model of the machine taking into account all saturation effects.Motor parameters and performance characteristics can be

estimated adequately for the entire space of design parameters.The determination of torque and power capability shows onlyminor regression errors of less than 2%, while the field weak-ening ability is merely vaguely predictable as the influencesof frequency dependent effects like iron losses and the skineffect of the windings are regarded inadequate. In case this isthe main object of investigation, the model can be enhancedby including stator resistance, end-winding inductance and aniron loss factor. But even so the resisting torque due to frictionconstrains the maximum speed to less than a hundred thousandrpm, though the development of magnetic bearings obliteratethis limitation.

REFERENCES

[1] K. Lewotsky. (2012, May) Dn insight: What rare earth shortages meanfor engineers. [Online]. Available: http://www.designnews.com/author.asp?section id=1386&doc id=240050

[2] T. J. E. Miller, Brushless permanent-magnet and reluctance motordrives. Oxford : Clarendon Press ; New York : Oxford UniversityPress, 1989.

[3] L. Jolly, “Design optimization of permanent magnet motors usingresponse surface methodology and genetic algorithms,” Master’s thesis,National University of Singapore, 2005.

[4] T. Jahns, G. Kliman, and T. W. Neumann, “Interior permanent-magnetsynchronous motors for adjustable-speed drives,” Industry Applications,IEEE Transactions on, vol. IA-22, no. 4, pp. 738–747, 1986.

[5] W. Soong and T. Miller, “Field-weakening performance of brushlesssynchronous ac motor drives,” Electric Power Applications, IEE Pro-ceedings, vol. 141, no. 6, pp. 331–340, 1994.

[6] S. Morimoto, Y. Takeda, T. Hirasa, and K. Taniguchi, “Expansion ofoperating limits for permanent magnet motor by current vector controlconsidering inverter capacity,” Industry Applications, IEEE Transactionson, vol. 26, no. 5, pp. 866–871, 1990.

[7] W. Soong and T. Miller, “Theoretical limitations to the field-weakeningperformance of the five classes of brushless synchronous ac motor

drive,” in Electrical Machines and Drives, 1993. Sixth InternationalConference on (Conf. Publ. No. 376), 1993, pp. 127–132.

[8] R. Schiferl and T. Lipo, “Power capability of salient pole permanentmagnet synchronous motors in variable speed drive applications,”Industry Applications, IEEE Transactions on, vol. 26, no. 1, pp. 115–123, 1990.

[9] G. E. P. Box and K. B. Wilson, “On the experimental attainment ofoptimum conditions,” Journal of the Royal Statistical Society. Series B(Methodological), vol. 13, no. 1, pp. 1–45, 1951.

[10] R. Rong, D. Lowther, Z. Malik, H. S., J. Nelder, and R. Spence,“Applying response surface methodology in the design and optimizationof electromagnetic devices,” Magnetics, IEEE Transactions on, vol. 33,no. 2, pp. 1916–1919, 1997.

[11] M. Jabbar, L. Q., and L. Jolly, “Application of response surface method-ology (rsm) in design optimization of permanent magnet synchronousmotors,” in TENCON 2004 IEEE Region 10 Conference, vol. C, 2004,pp. 500–503 Vol. 3.

[12] A. I. Khuri and S. Mukhopadhyay, “Advanced review: Response surfacemethodology,” WIREs Computational Statistics, vol. 2, pp. 128–149,2010.

[13] G. E. P. Box and D. W. Behnken, “Some new three level designs forthe study of quantitative variables,” Technometrics, vol. 2, no. 4, pp.455–475, 1960.

[14] D. A. Van Veldhuizen and G. B. Lamont, “Multiobjective evolutionaryalgorithms: Analyzing the state-of-the-art,” Evol. Comput., vol. 8, no. 2,pp. 125–147, 2000.

[15] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitistmultiobjective genetic algorithm: Nsga-ii,” Evolutionary Computation,IEEE Transactions on, vol. 6, no. 2, pp. 182–197, 2002.

[16] P. Ngatchou, A. Z., and M. El-Sharkawi, “Pareto multi objectiveoptimization,” in Intelligent Systems Application to Power Systems,2005. Proceedings of the 13th International Conference on, 2005, pp.84–91.