1589 IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, N0.2, MARCH 1 9 2

Optimization of Permanent Magnet Motors Using Field Calculations of Increasing Precision

by P.A. Watterson, J.G. Zhu and V.S. Ramsden

School of Electrical Engineering, University of Technology Sydney,

PO Box 123, Broadway NSW 2007, Australia

Abstmct -- An algorithm is given for the fast optimization over a constrained domain of an objective function calculated iteratively. To apply the algorithm to optimizing the efficiency of permanent magnet synchronous motors, an iterative magnetic fleld calculation is employed, featuring a new procedure for achieving convergence of non-linear reluctivity. For each set of motor parameters constituting a point in the optimization search, the iteration of the field calculation is continued until the change in the objective function is below a preset sequence of tolerances.

I. INTRODUCITON

Design of permanent magnet motors is an example of an optimization problem in which accurate evaluation of the objective function is very expensive computationally. Specifically, each evaluation requires a magnetic field calculation in two (or three) dimensions, usually for materials of non-linear reluctivity. To reduce the total computation time, it is essential that an iterative field solution method be employed, as then the field solution for one motor in the optimization sequence can be used as a very good starting guess for the next field calculation.

This paper presents an algorithm designed to find the minimum in as few function evaluations as possible of an iterative function defined on a linearly constrained domain. The algorithm is based on Powell's method [1]-[2], which requires no derivatives, is quadratically convergent, and was found most efficient by Ramamoorty [3] for optimization when field calculation is by circuit analysis. Here the algorithm is applied to optimize the efficiency of a slotless motor with a solid Neodymium-Iron-Boron permanent magnet (PM) rotor. The magnetic field calculation is performed either by an analytical approximation, described in Section 111, or by a version of the 2-D finite-element code MOTOR-CAD [4], employing the iterative ICCG (Incomplete Cholesky Conjugate Gradient) solution method [5], in itself a minimization algorithm.

Manuscript received July 7, 1991. This work was supponed by the IRD Board Generic Technology Grant 15026, Australian Magnet Technology, Brook Cmpton Beus, Eledronic Drives and Email.

II. OPTIMIZATION ALGORITHM

The foundation of the optimization code is the Powell algorithm, which locates a local minimum via a sequence of line searches (the ordering of which is incorrect in [2]). Once a 'bracket' or interval containing a minimum is determined in a line search, the Brent [6] procedure is used, with parabolic interpolation between three evaluation points. To incorporate linear inequality constraints, bounds are placed on each line search, and if the search attempts to exceed them, a check is made on whether the function is still decreasing in the neighbourhood of the boundary. If a subsequent search direction from a boundary point would enter the prohibited region, the search direction is projected onto the boundary (or intersection of boundaries), as depicted in Fig. 1. The search over a boundary is effectively over one fewer dimension than the unconstrained search, making the algorithm more efficient than the use of penalty functions for such linear constraints. Non-linear constraints are incorporated via penalty functions, with a sequentially increasing penalty parameter.

inequality constraint

direction 1, linear

\, P;;;;;ted projected search ,

'. - - - - - -___ optimization onginal trqectory search direction

Fig. 1 Depicting projection of a search line along a constraint bqundaly.

true \ \

\ 1

/ /

/ \

'? \ \ 5 0

x' - -x_ x' 4 2

distance along search line ' Fig. 2 Apparent local maximum (at point 5) during a line search.

0018-!3464/92$03.00 0 1992 IEEE

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To cater for iterative functions, a tolerance on each loop of the Powell algorithm is assigned, and the function iteration at any point is stopped when the function changes by less then the tolerance. [Originally it was hoped not to assign a function tolerance, but to fix the number of iterations at some low value (even one). However, for functions dependent on additional hidden variables (such as reluctivity) the line search is then unable to resolve a quadratic minimum, due to the linear variation associated with change in the hidden variables.] The line search is also considered converged when the two lowest values are within the tolerance. The fluctuations make it possible for an apparent local maximum to occur during the line search, as depicted in Fig. 2, in which case the search is stopped and the function is reevaluated at the current minimum. To prevent an inaccurate low point enduring as a false minimum, the function is also reevaluated at the start of each Powell loop.

III. MAGNETIC FIELJI CALCULATION

Consider a motor with a solid linearly magnetized PM rotor and a slotless stator, as depicted in Fig. 3. If the reluctivities are taken as uniform within each material layer and the magnetic flux density B due to the winding is neglected, then in the 2-D model, B = V x (AZB has the very simple analytical form (following [7])

A, = (Vir + Wilr) coscp, (1)

where i goes from 1 to 4, through the PM, the air-gap and winding combined, the yoke and the surrounding air. Finiteness demands W1 = 0 and V4 = 0. The boundary conditions on Br and Hcp between layers gives a six-order linear system for the constants Vi and Wh The yoke iron reluctivity can be taken as a non-linear function of cB$ the average Bcp across the yoke at cp = 0 (after account is made for the yoke stacking factor, here 0.95). Fig. 4 shows the B-H characteristic assumed here, with linear segments being sufficient. The low-B relative reluctivity is 1.08 x

Slotless armature winding

Actual alr gap

core laminatlonr;

I I , 10000 12500 2 5 0 0 5000 7500 li

H 0" Fig. 4 Curve I - B-H for yoke iron; curve II - calculated average B and H c p c p

across yoke (at cp = 0, in iron) for radii I$,, = 20 mm. R2 = 30 mm. Lace numbered points - the new reluctivity algorithm. Small unnumtmed poims

(between large points 1 and 3) - permeabilj. relaxation, factor 0.1.

In the above analytic model, if the yoke reluctivity is assigned and varied, the average B and H across the yoke (at cp = 0 in the iron) vary exactly linearly, as illustrated in Fig. 4. The field solution lies at the intersection of this calculated B-H dependence and the material's B-H characteristic. A standard iteration to determine the intersection is to commence with the low B permeability, giving point 1 in Fig. 4, cross at fixed B to the B-H characteristic to identify a new permeability, but then to only proceed a relaxation factor, e.g. 1/10, towards it. The resulting iteration, shown in Fig. 4, has very slow convergence. A far superior method is to simply calculate two B-H points for arbitrary reluctivities and take the next reluctivity from the intersection of the interval joining them with the B-H characteristic. Here, point 2 is calculated for the reluctivity deduced by intersecting the B-H characteristic with a line of gradient -w from point 1. For this and other simple geometries (such as C-cores), the linearity of the calculated B-H dependence implies convergence in just three field calculations.

The above reluctivity iteration has been incorporated for each element in the 2-D finite element code, MOTORCAD [4]. The 'calculated' B-H dependence no longer now exists, except in an average sense over all elements. Sometimes variation in other elements causes a step of positive gradient (which may not even intercept the B-H characteristic) in which case the next reluctivity is obtained by dropping at constant H to the B-H characteristic. Use of constant H, rather than the standard constant B, is more stable (because

Fig. 3 Slotless motor cross-section. Here & = 40 mm, g = RI - It,,, = 1 mm. Assumed PM manence 1.2 T, relative diffenzntial reluctivity 0.95.

of the relative gradients of the B-H characteristic and the average calculated B-H dependence) and no relaxation factor

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is required. The near linearity of the average B-H dependence should imply even faster convergence than the Newton- Raphson method, which instead effectively follows tangents to the B-H characteristic, which is far from linear near the saturation point.

IV. APPLICATION

The optimization code is now applied, firstly on the analytic model, to determine the radii Rm and R2 and (finally) frequency f to minimize the total loss, i.e. the sum of copper, iron and friction losses. The 3-phase current (distributed in sectors) is iteratively determined to yield a specified mechanical power (500 W). End-windings are neglected and the motor length is fixed (80 mm). The specific iron loss (W/kg) is taken arbitrarily as

where B is in Tesla and f is in Hz. For the analytic model, B is taken as <B$, while in the finite element solution, 5 layers across the yoke are used with the maximum IBI in each. A friction torque of 5 x 10-7(2xf) ("/Hz) is assumed.

[io-lm1.6 + 10-3(m)21,

The relevant linear constraints are

(see Fig. 3). The first two constraints are made strict inequalities by adding a small distance ( m) to the right- hand sides. The yoke outer radius Ro is fixed (40 mm).

A. Analytic field calculation

Fig. 5 provides a contour plot of the loss for f = 50 Hz calculated by the analytic model. The separation in form of the loss contours can be attributed to the reduction in the variation of iron loss once the iron reaches saturation. Saturation may be inferred from the value of the peak Br at the yoke surface, also shown in Fig. 5. When the optimization search is commenced from point 1 on Fig. 5 (and also Table l), it terminates at the local minimum, point 2. When the search is commenced from point 3, it terminates at the global minimum, point 4, lying on the linear constraint at which the yoke thickness vanishes. At this point, the peak Br at the yoke surface is 0.34 T, exceeding the safety standard at 50 Hz of 0.5 mT.

The safety standard can be met at all frequencies by imposing the non-linear constraint,

@,.(surface) 10.025 THz. (3)

The resulting optimum at 50 Hz is at point 5. If f is also allowed to vary, the optimum is at point 6 in Table 1.

PM radius R,,, (mm)

/ /

IC.

/

/ /

I

e 30 I5

PM radius R,,, (mm)

Fig. 5 Evaluation by analytic model: contours of total loss (8,12, .A0 W); contours of peak Brat yoke surface (0.0005,0.001,.. 0.002,0.004,.. 0.01,

OM,.. 0.51 T); optimization trajectories to minima shown.

Typically, 30-40 points must be evaluated for optimization over two variables, but if penalty functions are present, the number of points increases to around 200.

TABLE 1 RESULTS OF OPTIMIZATION USING ANALYTIC MODEL

Pt. Description

1. Evaluation 2. Localminimum 3. Evaluation 4. Global minimum 5. Min. s.t. eq. (3) 6. As 5., f variable

20. 30. 15.96 31.30 25. 35. 30.44 40. 13.37 24.94 6.03 20.67

50. 30.27 50. 23.91 50. 25.21 50. 6.10 50. 32.91 304. 18.35

B. Finite Element Field Calculaiion

The field calculation by the finite element code should be more accurate given its inclusion of non-uniform reluctivity, though it loses accuracy in approximating A, = 0 at the yoke surface. The field solution for the radii of point 2 has peak <Bq> = 1.150 T, higher than the analytic model (because of lower net reluctance) but only by 0.010 T. Both the new reluctivity iteration and the standard permeability relaxation took about 9 function evaluations to compute the loss to

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within 0.001 W as 24.085 W, 0.17 W higher than by the analytic model. The number of reluctivity steps for each function evaluation was fixed at two, sufficient for the next reluctivity to be inferred by linear interpolation in the new method. The new method used a total of 126 conjugate gradient steps, compared to 211 by the permeability relaxation. Furthermore the permeability relaxation had not truly converged in H, though its variation in B was sufficiently slow that the function evaluation converged. This superiority of the new method is illustrated in Fig. 6, showing an expanded view of the convergence of the two algorithms. Though the new method appears more erratic, this makes it less likely for the function iteration to stop early, before proper convergence of the field solution. For relaxation factor 0.5, the permeability iteration failed to converge at all.

Starting an optimization search f" point 2 with tolerance of 0.001 W, the loss could only be reduced to 24.066 W at Rm = 16.01 mm, R2 = 30.95 mm, for which the field is as shown in Fig. 7.

The superiority of the new reluctivity iteration is borne out by the result of the optimization search starting from point 1. After three Powell loops with tolerances of 0.5,O.l and 0.01 W respectively, the new reluctivity iteration was at Rm = 16.05 mm and R2 = 30.80 mm, with loss = 24.08 W, very close to the accurate value. In contrast, the permeability relaxation had strayed to Rm = 15.46 mm and R2 = 31.38 mm, with loss = 24.35 W.

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Fig. 6 Finite element solution for R,,, = 20 mm, R2 = 30 mm. Calculated m n IBI and IHI in yoke middle at cp = 0: large points - the new reluctivity

iteration; small points - permeability relaxation, factor 0.1. Curve I - B-H for yoke iron.

Fig. 7 Finite element A, contours (interval lW3 Tm) for the radii of the 50 Hz local mini", = 16.01 m, R2 = 30.95 m.

V. CONCLUSIONS

The design of a slotless permanent magnet motor has illustrated the use of the optimization code, highlighting the need for optimization from several start points. The analytic field calculation available for this motor was of high accuracy, and will be used in extended versions of the model, including end-winding loss and eddy current losses in the windings and a metallic case. For most motors, especially slotted motors, a finite element solution is required. Preliminary results have shown a reduction in the computation time by using the new reluctivity iteration.

REFERENCES

[l] M.D. Powell, "An efficient method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives", Computer J . vo1.7,1964, pp.

[23 W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes. The Art of Scientific Computing, Cambridge: Cambridge University Press, 1986.

[3] M. Ramamoorty, Computer Aided Design of Electrical Equipment, Chichester: Ellis Harwood, 1988.

[4] A.Hameed and K.J.Binns, MOTOR-CAD. A Finite Element Package for Designing Electrical Machines with or without Magnets, Liverpool: Liverpool University, 1989.

[5] J. Meijerink and H. van der Vorst, "An Iterative Solution for Linear Systems of which the Coefficient Matrix is a Symmetric M-Matrix", Math. Comput., vol. 31,

[6] R.P.Brent, Algorithms for Minimization without Derivatives, Englewood Cliffs, N.J.: Prentice-Hall, 1973.

[7] D.E. Hesmondalgh and D. Tipping, "Slotless Construction for Small Synchronous Motors using Samarium Cobalt Magnets", IEE Proc., vol. 129 B,

155-162.

1977, pp. 148-162.

NO. 5,1982, pp. 251-261.