Numerical optimization (Electrical Machine) Part 2

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Optimisation of an actuator using a magnetic equivalent circuit model

The MEC method and the mentioned combined numerical optimisation algorithm will be applied for the example. The task is to optimise the shape of a brushless DC motor. Fig. 10.50 shows the initial shape of the electromagnetic device. The used complete equivalent magnetic circuit can be taken out of the introductory section about numerical techniques.

The armature winding is fixed to closed stator slots and the rotor is axially assembled out of permanent magnet rings. The magnet material used is a plastic bonded NdFeB grade (MQ1). The objective is to minimise the material costs of the construction under the assumption of the same torque production as the initial construction. Objective variables can be taken from Fig. 10.49.

Material costs are estimated and set to 0.7 US$/kg for the lamination, 4.1 US$/kg for the used copper volume of the armature winding and 110 US$M/kg for the used magnet material. The resulting shape of the motor can be seen in Fig. 10.51.

Using a quality function with K as the sum of all material costs, the start value of quality is approximately 42 US$. After the optimisation, the overall material costs decreases to 11 US$. Fig. 10.51 shows the reduction of the cost intensive permanent magnet volume. Due to the expensive armature winding compared to the lamination cost, wide stator teeth can be noticed. Maximum flux density of 1,04 T is enumerated in the teeth.

Objective variables.

Fig. 10.49. Objective variables.

Initial geometry. 1 permanent magnet; 2 stator yoke; 3 area of armature winding; 4 shaft.

Fig. 10.50. Initial geometry. 1 permanent magnet; 2 stator yoke; 3 area of armature winding; 4 shaft.

Optimised shape of the DC actuator.

Fig. 10.51. Optimised shape of the DC actuator.

Design of a lifting magnet

FEM method and the numerical optimisation algorithms are applied to the shape optimisation of a lifting magnet The objective is the reduction of the weight of the device at constant lifting force. There are no additional geometrical constraints. The problem is formulated with 10 free parameters to be optimised as indicated in Fig. 10.52.

Obviously, the problem has to be defined as a non-linear magneto-static field problem. During optimisation an accuracy of at least 1% was required for the FEM field calculation. The size of the mesh was restricted to 4000 elements. The optimisation method used was the evolution strategy (4/3, 12).

Lifting magnet with initial proportions.

Fig. 10.52. Lifting magnet with initial proportions.

Arrows indicate the admissible variation of design parameters.

This method offers a compromise between reliability and performance. A plus-strategy with a smaller number of children and parents would result in faster convergence. The initial and adaptive generated final mesh for the FEM calculation can be seen in Fig. 10.53.

a) Initial and b) adaptive generated final mesh for the optimised geometry.

Fig. 10.53. a) Initial and b) adaptive generated final mesh for the optimised geometry.

The resulting field plot of the optimised lifting magnet is shown in Fig. 10.54. The optimisation resulted in a reduction of weight of approximately 6% in comparison to the initial geometry.

The dependence of step length on the iteration steps is illustrated in Fig. 10.55. It serves as convergence and stopping criterion. About 40 iterations, each involving 12 objective function evaluations, seem to be sufficient for an geometrical accuracy of 1 mm. Fig. 10.55 shows that the most significant reduction in weight is achieved during the first 30 iterations. The improvement in the following iterations is less than 1%.

Field distribution of the optimised lifting magnet.

Fig. 10.54. Field distribution of the optimised lifting magnet.

Step length and weight of the lifting magnet versus iteration count during optimisation.

Fig. 10.55. Step length and weight of the lifting magnet versus iteration count during optimisation.