On the Use of the Cumulative Distribution Function for Large-Scale Tolerance Analyses Applied to Electric Machine Design

In the field of electrical machine design, excellent performance for multiple objectives, like efficiency or torque density, can be reached by using contemporary optimization techniques. Unfortunately, highly optimized designs are prone to be rather sensitive regarding uncertainties in the design parameters. This paper introduces an approach to rate the sensitivity of designs with a large number of tolerance-affected parameters using cumulative distribution functions (CDFs) based on finite element analysis results. The accuracy of the CDFs is estimated using the Dvoretzky–Kiefer–Wolfowitz inequality, as well as the bootstrapping method. The advantage of the presented technique is that computational time can be kept low, even for complex problems. As a demanding test case, the effect of imperfect permanent magnets on the cogging torque of a Vernier machine with 192 tolerance-affected parameters is investigated. Results reveal that for this problem, a reliable statement about the robustness can already be made with 1000 finite element calculations.