Multicriteria Optimal Latin Hypercube Design-Based Surrogate-Assisted Design Optimization for a Permanent-Magnet Vernier Machine

Abstract

This paper proposes an efficient surrogate-assisted design optimization method based on multicriteria optimal Latin hypercube design (LHD) for multi-objective optimization of a permanent-magnet vernier machine (PMVM). By combining simulated annealing algorithm, the uniformity and orthogonality of spatial distribution of the LHD are optimized to efficiently capture data features over the optimization parameter ranges. Consequently, an accurate surrogate model with high generalization capability can be built with a small training set by using the optimal LHD. The optimal design of the PMVM is obtained by multi-objective design optimization based on the surrogate model. A prototype is made and tested to validate the proposed method.I. IntroductionDue to high torque capability at low speeds, permanent-magnet vernier machines (PMVMs) are suitable for direct-drive applications such as wind power generation and electric drilling [1]. To deal with the issue of massive finite-element analysis (FEA) for multi-objective design optimization, this paper proposes a novel surrogate-assisted multi-objective design optimization method for PMVMs. A multicriteria optimal Latin hypercube design (LHD) employing simulated annealing (SA) algorithm is proposed for efficient date feature capturing that enables to use a small training dataset for obtaining an accurate surrogate model. The sparrow search algorithm (SSA)-optimized multi-output least squares support vector machine (MLS-SVM) [2] and the non-dominant sorted genetic algorithm-III (NSGA-III) [3] are combined to build the multi-output surrogate model and to perform the multi-objective design optimization. A prototype is made based on the obtained optimal design and tested for validation.II. PMVM and Sensitivity AnalysisA surface-mounted PMVM with 18 slots and 15 pole pairs is studied as an example, whose topology is illustrated in Fig. 1 (a). The four optimization objectives are average torque, power factor, THD of back-EMF, and cogging torque. To improve the optimization efficiency, a global sensitivity analysis (GSA) of four optimization targets is performed using the Sobol’s method [4], the stack bars of total sensitivity coefficient (TSC) for optimization parameters are given in Fig. 1 (b), which shows the geometrical parameters that have the greatest influence on design objectives. Thus, five geometrical parameters including PM thickness (tPM) and width (wPM), yoke thickness (ty), slot width (ws), and stator split ratio (k=Rsi/Rso) are selected for optimization while stator outside diameter, axial length, air-gap length, and current density are fixed.III. Surrogate-Assisted Optimization with LHDThe optimal LHD employs SA to minimize a composite indicator ψp that consists of the Morris-Mitchell index Φp and the correlation coefficient of the test factors ρ2 to guarantee the spatial distribution uniformity and orthogonality as given in (1), where ω1 and ω2 are the weight factors. Subsequently, a 270*5 LHD is built to capture the data feature over the optimization parameter ranges more efficiently as shown in Fig.2 (a), in which 20 sets of data are used for testing the regression results.minψp = ω1Φp + ω2ρ2(1)Finite element analysis (FEA) simulations are only performed within the optimal LHD to acquire the training data. The surrogate model based on MLS-SVM are trained using the data of the optimal LHD. The radial basis function (RBF) kernel is adopted for MLS-SVM and SSA is combined with MLS-SVM to find an optimal hyper-parameter set that produces a surrogate model with the highest average determination coefficient R2. The four-objectives optimization is performed by NSGA-III and an optimal design is selected from the Pareto front shown in Fig. 2 (b). A prototype is made based on the optimal design and tested, Figs. 2 (c) and (d), to validate the proposed method and the optimization.The proposed method needs only 270 FEA simulation cases during the whole design optimization procedure. For the conventional FEA-based parametric optimization method, even when only 5 levels are selected for each of the five geometrical parameters, 3125 FEA cases are required. Meanwhile, since the parameter space is discretized, the results of the conventional optimization method are likely to be locally optimal. For the method employing randomly generated LHDs [5], it requires nearly 500 FEA cases to achieve consistent R2s, which consumes 2 times the time of the proposed method. Thus, the proposed method can secure both efficient and accurate global design optimization of PMVMs.IV. ConclusionsThis paper proposes a novel optimal LHD-based surrogate assisted design optimization method for multi-objective optimization of PMVMs, which can perform high efficiency and high validity simultaneously. The proposed method and optimal design are validated by experiments on a prototype. Details of the design optimization and test results will be given in the full paper. **