Permanent-magnet (PM) motors have been widely used in industrial applications and automobiles. However, owing to the design tolerances, manufacturing uncertainties, and material inconsistency, deviations from the ideal motor inevitably occur with parasitic effects, such as additional cogging torque and vibration. One of the most significant obstacles for evaluating these uncertainties is the large computational burden caused by the countless uncertain combinations, which must be computed via the finite-element method (FEM). Herein, a worst—uncertain—combination—analyze (WUCA) method is proposed to significantly reduce the computational cost. A widely used analytical method is modified to identify the origin of additional cogging-torque harmonics caused by uncertainties. Different types of uncertainties for the surface-mounted PM machine and interior PM machine can be analyzed simultaneously. With the WUCA method, the worst-case combinations can be confirmed theoretically; thus, FEM calculations for only a few combinations of uncertainties are required, rather than thousands. Compared with the widely adopted design of experiments based uncertain combining methods, the worst-case cogging torque obtained from the WUCA is higher. FEM verification of different pole/slot configurations revealed that the WUCA method was a general and effective method for quickly estimating the worst-case cogging torque under manufacturing uncertainties.

This paper discusses a method to consider the effects of the manufacturing uncertainties on the cogging torque harmonics, with which a sensitivity analysis can be incorporated into the design stage instead of just checking it afterwards. With the authors' previous work of the worst-uncertain-combination-analysis (WUCA) method, several design parameters and their effects on the additional cogging torque harmonics under uncertainties are discussed. It is demonstrated that a design with inherent robustness can be obtained through the properly selection of pole-arc coefficient and slot-opening width. A sensitivity index is proposed to estimate the effects of parameter uncertainties on torque harmonics. Some suggestions on improving the design robustness are consequently provided from the aspects of tolerances control and parameter design. The deductions are verified through finite element analysis (FEA) on different pole/slot configurations.

One of the main obstacles to the robust optimization for permanent magnet machines is the high computational burden, which is mainly caused by the robustness evaluation process considering the manufacturing uncertainties. In this paper, a sequential sampling Kriging model is adopted to estimate the design robustness, and the problem of high-dimensional variables for the surrogate model due to uncertainties is avoided through adopting the worst-case based approach. To further reduce the number of finite element analyses (FEA) required for constructing a metamodel, a multi-points sequential sampling process with a two-step optimization oriented surrogate model updating algorithm is proposed. Compared with the FEA directly based robust optimization in previous work, similar Pareto Fronts are achieved by adopting the meta-model proposed algorithm but the overall run time reduced by 75%. In the end, an optimization problem for a 1.2 kW motor is considered by applying the proposed algorithm, and two prototypes with deliberately designed tolerances, imitating the worst-case scenarios in the real world, are manufactured. The worst-case torque ripple is reduced from 7.6% to 4.8% after optimizing, and this verifies the efficacy of the proposed algorithm.

One of the main obstacles to applying the robust optimization for the permanent magnet motor might be the high computational burden, which is mainly caused by the robustness evaluation considering the manufacturing uncertainties. In this paper, efforts are made to speed up the process of robust optimization, from the aspects of reducing the computational cost and avoiding the computing resources being wasted. Firstly, several widely used methods, aiming to identify the worst-case performance under uncertainties, are examined and compared. And the efficient worst-uncertain-combination-analysis (WUCA) method is consequently adopted to significantly reduce the computational cost of robustness evaluation. From another aspect, the parameter rounding (PR) operation is inevitable due to the constraint of manufacturing accuracy, but its adverse effects on the optimization are often ignored. It is found that parts of the computing resources might be wasted during the optimization. A new PR during the calculation (PRDC) method is proposed and incorporated into the process of robust optimization. Compared with the conventional one, the similarly optimal results can be achieved with the computational time reduced by approximately 11% in the PRDC modified robust optimization.

The rotor eccentricity and manufacturing tolerances caused by unavoidable uncertainties in the process of manufacturing deteriorate the cogging torque of permanent magnet (PM) motors. To guide the design and optimization of PM motors, this paper proposes an analytical model to reveal the effects of mixed-eccentricity (ME) and manufacturing tolerances on the harmonic components of cogging torque. A coefficient is introduced to intuitively characterize the impact of ME on the distribution of the air-gap length. Then a modified analytical model is developed to theoretically identify the additional harmonic components (AHCs) of cogging torque associated with the eccentricity and tolerances. And the main AHCs of cogging torque are quantitatively determined through the finite element method. Finally, suggestions to improve the robustness considering uncertainties by selecting pole/slot combination and pole-arc ratio are proposed.

Conventionally, the small mean and variance of peak-to-peak cogging torque of permanent magnet (PM) machines considering manufacturing uncertainties can be achieved by a robust design or by reducing the uncertainties range. However, the consequent compromise of other design objectives or the increase in the manufacturing costs are frequently inevitable. In this paper, the combination sequence of the uncertainties is highlighted and implemented to achieve a stable performance even for a non-robust design without increasing the cost too much. A PM assembling approach is proposed to mitigate the influences of PM uncertainties on cogging torque by means of combining the uncertainties in a particular sequence, where the effects of uncertainties would counteract each other. Both the singular type of PM uncertainties and the combined ones are considered in the assembling approach. Furthermore, the proposed approach is verified by comparing the cogging torque performance with PM randomly assembled models, where several hundreds of models featuring different uncertainties are calculated through the finite element method. The proposed approach is discussed, particularly for the application of mass production and a small amount of prototypes. Two prototypes with different PM assembling sequences are fabricated and further verify the effectiveness of the proposed PM assembling approach.