Thermal design optimization of electric machines using a global sensitivity analysis

Fast iron losses model of stator taking into account the flux weakening mode for the optimal sizing of high speed permanent internal magnet synchronous machine

This chapter presents a brief summary of the design fundamentals including the analysis models and methods for electrical machines and drive systems, based on our design experiences, particularly for permanent magnet electrical machine with soft magnetic composite cores. Because of the multi-disciplinary nature, these design models and methods will be investigated at the disciplinary level, including electromagnetic, thermal, mechanical, power electronics, and control algorithm designs. Several design examples will be presented to illustrate the corresponding design models and methods based on our research findings, such as the finite element model for design analysis of motors, and the model predictive control algorithm and its improvement form for the drive systems. These models and algorithms will be employed in the design optimization of electrical machines and drive systems in the following chapters.

Axial flux magnetic-geared double-rotor machines (AMGDRMs) are characterized as a potential core component of the power splitting system in hybrid electric vehicles (HEVs). Conventional AMGDRMs suffer from low mechanical strength due to high axial magnetic force. In this paper, an interior-modulating-rotor axial flux magnetic-geared double-rotor machine (IMR-AMGDRM) is proposed to solve this problem. An analytical model of the IMR-AMGDRM is proposed to predict the magnetic field distribution and electromagnetic performances. Since analytical model is based on 2-D geometry, a 2-D approximation of IMR-AMGDRM which has similar output performance to 3-D finite element analysis (FEA) is presented based on the permanent magnet (PM) magnetic energy equivalent principle. Utilizing subdomain method, the magnetic field distribution and electromagnetic performances are calculated. Finally, the accuracy of the proposed analytical model is validated by 3-D FEA.Index Terms—Analytical model, axial flux, magnetic-geared machine, double-rotor machine.I. IntroductionAxial flux magnetic-geared double-rotor machines (AMGDRMs), which enable high torque density, have been considered a promising candidate for power splitting in hybrid electric vehicles (HEVs) [1]. However, conventional AMGDRMs suffer from low mechanical strength due to high axial magnetic force. To enhance mechanical strength, an interior-modulating-rotor axial flux magnetic-geared double-rotor machine (IMR-AMGDRM) which can reduce axial magnetic force dramatically is proposed in this paper. Fig. 1 (a) and (b) illustrate the configuration of the proposed IMR-AMGDRM.As axial flux type machine, IMR-AMGDRMs have complex 3-D magnetic circuits. 3-D finite element analysis (FEA) can offer accurate results by taking into account the nonlinear factors such as magnetic saturation and flux leakage. But the major drawback of 3-D FEA is serious time cost. Analytical model which is much lower time-consuming provides an alternative for electromagnetic performance prediction in the initial design procedure.In this paper, an analytical model of the IMR-AMGDRM is proposed. The geometry of analytical model is derived based on PM magnetic energy equivalent principle to guarantee similar output performance to 3-D FEA. The solution to analytical model is calculated by subdomain method. To evaluate the accuracy of the proposed analytical model, the results computed by analytical model and that of 3-D FEA are compared.II. Configuration of IMR-AMGDRMIn conventional configuration, the inner and outer rotors are the permanent magnet (PM) rotor and the modulating rotor, respectively. Since the modulating rotor is closer to the PM rotor than stator, the modulating rotor suffers from high axial magnetic force. Furthermore, the modulating rotor is hollow and thus the low mechanical strength of modulating rotor becomes the major drawback of conventional configuration [1]. To solve this problem, the IMR-AMGDRM which switches positions of the PM rotor and the modulating rotor is proposed. In interior-modulating-rotor configuration, as shown in Fig. 1 (a) and (b), the PM rotor is between the modulating rotor and stator, i.e. the inner and outer rotors are the modulating rotor and the PM rotor, respectively. Owing to that the PM rotor suffers from reverse axial magnetic forces from both sides simultaneously, the total axial magnetic force is dramatically reduced. Hence, although the outer rotor is hollow in both configurations, the axial magnetic force of interior-modulating-rotor configuration is significantly decreased, leading to higher mechanical strength and reliability.III.Geometry of 2-D ApproximationGenerally, the unrolled section of axial flux machine at mean radius is used as 2-D approximation. However, few references are found to evaluate the electromagnetic performance equivalence of such approximation. To make the electromagnetic performance of 3-D model and 2-D approximation equal, the amount of magnetic energy of the PM should remain unchanged after approximation, i.e. PM magnetic energy equivalent principle [2]. A 2-D approximation of IMR-AMGDRM which has identical output performance is presented by using PM magnetic energy equivalent principle.IV. Analytical Calculation of IMR-AMGDRMThe proposed analytical model is illustrated in Fig. 1 (c). It can be seen that the model consists of four subdomains. Region i, I, II and III represent the ith slot of the modulating rotor, the inner air gap, the PM rotor and the outer air gap, respectively. Utilizing subdomain method, the magnetic field distributions in the inner and outer air gaps are obtained by the solution of Laplace’s and Poisson’s equations. Then, the electromagnetic performances such as no-load back EMF, torque and axial force can be derived from the calculated magnetic field distributions.Comparisons of computation accuracy between analytical model and 3-D FEA is shown in Fig. 2. As can be observed, the results of analytical model are well consistent with that of 3-D FEA, validating the proposed method. **

Purpose – The purpose of this paper is to investigate the impact of different mathematical formulations of the problem of optimal design of electrical machines on the results obtained using a local optimization solver. The aim is to investigate the efficiency and reliability of standard local solvers when handling different mathematical formulations. This could provide guidelines for designers in practical engineering applications.

Electrical machine design optimization is an expensive procedure as it contains numerous variables and multiple objectives. Therefore, it might require hundreds of time‐consuming finite element analyses (FEA). Surrogate models are one of the superior alternatives that can overcome the computational burden of FEA. However, in optimization problems with sensitive input‐output relationships, surrogate models can lack accuracy or suffer from unreasonable initial FE‐computational cost. To tackle this problem, this work presents a novel strategy called Waveform‐Targeted Surrogate Modeling (WTSM) that improves the computational efficiency of surrogate‐based design optimization of electrical machines by modifying the model construction process. In this paper, an Interior Permanent Magnet Machine (IPMSM) is studied to evaluate the proposed strategy's performance compared to the conventional Black Box Modeling (BBM) method. A two‐step assessment procedure consisting of multiple scenarios has been developed to analyse the reliability of the WTSM concerning different training‐validation datasets. Meanwhile, the Kriging model and Latin‐Hypercube‐Sampling (LHS) method are employed for surrogate model construction purposes. From the discussion and the results, it can be found that the proposed WTSM strategy can significantly increase the accuracy of the surrogate modelling procedure while the required computational cost can be reduced.

In electric machine design, efficient methods for the optimization of the geometry and associated parameters are essential. Nowadays, it is necessary to address the uncertainty caused by manufacturing or material tolerances. This work presents a robust optimization strategy to address uncertainty in the design of a three-phase, six-pole permanent magnet synchronous motor (PMSM). The geometry is constructed in a two-dimensional framework within MATLAB®, employing isogeometric analysis (IGA) to enable flexible shape optimization. The main contributions of this research are twofold. First, we integrate shape optimization with parameter optimization to enhance the performance of PMSM designs. Second, we use robust optimization, which creates a min–max problem, to ensure that the motor maintains its performance when facing uncertainties. To solve this bilevel problem, we work with the maximal value functions of the lower-level maximization problems and apply a version of Danskin’s theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower-level problems with gradient-based optimization. The paper concludes by presenting numerical results showcasing the efficacy of the proposed robust optimization framework. The results indicate that the optimized PMSM designs not only perform competitively compared to their non-robust counterparts but also show resilience to operational and manufacturing uncertainties, making them attractive for industrial applications.

The suppression of the rotation speed degradation of the Superconducting Magnetic Bearing (SMB) with the Permanent Magnet (PM) rotor including the layered insulator thin films is studied. The SMB system consists of the superconductor stator and the PM rotor, and is a no-contact bearing system using the levitation force between them. The SMB was expected to be applied to the electric power storage flywheel system because friction causes the rotation speed degradation and decreases the energy storage capability. However, it was found that the electromagnetic friction of the SMB was not zero, due to electromagnetic forces; 1) the magnetic force interaction between the inhomogeneous magnetic field of the PM rotor and the eddy current it induces in the cryostat, 2) the magnetic force between the inhomogeneous magnetic field of the PM rotor and the shielding current it induces in the superconductor, 3) the magnetic force between the magnetic field of superconductors yielding the levitation force and the eddy current induced in the PM rotor. This magnetic field of superconductor is inhomogeneous too, because the superconductors consist of several HTSC bulks. In this research, it was found that the 3rd phenomenon is most significant. So, we use the advanced PM rotor including the layered insulator thin films to suppress its eddy current and the degradation of the rotation speed. Using our SMB equipment, the rotation speed degradation was measured using the PM rotors with/without the insulator thin films and the electromagnetic frictions were compared to each other.

The main focus of this thesis is the uncertainty propagation (UP) and global sensitivity analysis (GSA) in complex hydrogeological numerical models. Various methods are presented with applications on numerical models for the groundwater flow and mean lifetime expectancy (MLE) in the scope of Andra's (French National Radioactive Waste Management Agency) project for the geological disposal of high-level and intermediate-level long-lived radioactive wastes in a highly impermeable layer from Callovo-Oxfordian age (COX) in France. <br> A state of the art is provided for the theory of uncertainty propagation and for the methodologies of sensitivity analyses in a broad sense. Methods for UP are provided from 2-levels Factorial Designs to quasi-random samplings. GSA techniques encompass screening methods such as the Morris Measures and the Derivative-based Global Sensitivity Measures (DGSM), and also the so-called Sobol' indices based upon the variance decomposition of the response of interest. Meta-modelling techniques such as polynomial regression and Polynomial Chaos Expansions (PCE) meta-models are employed as surrogate models for UP and GSA purpose at negligible computational-costs. A comparison of GSA techniques upon various complex analytical test-functions was undertaken with the purpose of determining a relevant method to be employed in the context of “screening” out unimportant variables in computer-intensive, high-dimensional models. <br> A numerical model of groundwater flow and lifetime expectancy is employed for assessing the effect of uncertain advection-dispersion parameters and their spatial distributions upon the MLE from a target zone inside the domain. The model is a 2-dimensions synthetic cross-section of the eastern region of the Paris Basin (Meuse/Haute-Marne sector). This model was used as an exploratory tool for sensitivity analysis methods applied upon numerous sets of uncertain hydrodynamic and dispersion parameters in 15 layers. The uncertainty characterizing the permeability-porosity values in aquifer formations encompassing the COX have proved to add much of variability to the MLE calculated from the target zone. The model also served at exploring the effect of the spatial variability of permeability-porosity parameters in two main aquifer sequences on the groundwater flow rates and MLE in the model. The variabilities of the output responses are mainly due to the uncertainty upon the means and variances of the permeability-porosity distributions, as well as the longitudinal correlation lengths, in each sequence. <br> Then, a 3-dimensions high-definition hydrogeological model representing the Meuse/Haute-Marne sector in the eastern region of the Paris Basin is a more comprehensive numerical model incorporating realistic geometries, fractures, heterogeneities and discontinuities encountered on field. A sensitivity analysis of the MLE from a given zone located in the middle of the COX layer was performed by perturbing the hydraulic conductivities and porosities values in fourteen hydrogeological formations. The uncertain permeability-porosity parameters in the aquifer formations from Bathonian in the Dogger sequence, and Rauracian-Sequanian in the Oxfordian sequence, have strong and rather non-linear effects on the variability of the output response of interest. <br> The methodologies for UP and GSA employed in the present thesis have proved to be very efficient when applied to large hydrogeological models of groundwater flow and MLE. In particular, quasi-random sampling methods offer a flexible frame for providing the uncertainty distribution of the output response of interest at low computational costs. Screening techniques provide a fast estimation for the overall contribution of each input variable to the variability of the output. Meta-modelling techniques such as PCE proved highly accurate in individualising the low- and high-order effects of each input variable upon the output response of interest.

The optimal design of electric machines requires a significant amount of computation time, since the design is usually performed based on finite-element analysis (FEA) for better accuracy. For this reason, the development of an optimization algorithm is regarded as a prerequisite to reduce the number of objective function call, which is directly related to computation time. The local search algorithm does not guarantee that the global optimum solution can be searched effectively. Therefore, in this paper, we introduce the novel optimization method, search region management (SRM), to improve the efficiency of the local search algorithm. SRM uses a guided random bit generation based on Gray code for the region selection when the local search iterates the optimum search process for a multi-start. In addition, the SRM saves and manages the information of previous searched points to prevent overlapping calculation and to increase the effectiveness of the local search during every iteration process. To clarify the effectiveness of the proposed method for electric machine design, it is applied to the optimal design of brushless dc motor coupled with FEA minimizing torque ripple.

In this paper, optimization algorithm that improves simplex method is proposed. Simplex method searches optimal point in the opposite direction of the worst point, which eliminates the worst point in next generation search. However this method has its weakness on fast searching. To supplement limitation of simplex method, proposed method is adopting algorithm to search optimal point by crossover toward the best point, not eliminating the worst one. Lastly, optimization algorithm is applied to designing electric machine with Finite Element Method (FEM) to validate algorithm.

The efficient design optimization of electric machines for electric power steering (EPS) applications poses challenges in meeting demanding performance criteria, including high power density, efficiency, and low vibration. Traditional optimization approaches often fail to find a global solution or suffer from excessive computation time. In response to the limitations of traditional approaches, this paper introduces a novel methodology by incorporating a Gaussian process-based adaptive sampling technique into a surrogate-assisted optimization process using a metaheuristic algorithm. Validation on a 72-slot/8-pole interior permanent magnet (IPM) machine demonstrates the superiority of the proposed approach, showcasing improved exploitation–exploration balance, faster convergence, and enhanced repeatability compared to conventional optimization methods. The proposed design process is then applied to two surface PM (SPM) machine configurations with 9-slot/6-pole and 12-slot/10-pole combinations for EPS applications. The results indicate that the 12-slot/10-pole SPM design surpasses the alternative design in torque density, efficiency, cogging torque, torque ripple, and manufacturability.

This paper presents the use of Global Sensitivity Analysis (GSA) methods to analyze the sensitivity level of the performance of the impedance measurement algorithm of SEL-421 distance relay protection when used to perform transmission line protection with compensation circuit. This GSA technique was developed to see the dominance effect of system uncertainty (or factors). Two techniques that might be used for sensitivity analysis using GSA are Quasi-Monte Carlo (QMC) and data sampling with Sobol technique. Experimental is done with DIgSILENT software to do modeling of the power system, simulation, and calculation of impedance and algorithm error. The testing is carried out automatically with the help of the developed algorithm using DPL (DIgSILENT Programming Language). SIMLAB software, in this case, (with Sobol technique) is applied to generate some non-linear factor data and analyze the output of variations in impedance measurement errors.

In pharmacometrics, understanding a covariate effect on an interested outcome is essential for assessing the importance of the covariate. Variance-based global sensitivity analysis (GSA) can simultaneously quantify contribution of each covariate effect to the variability for the interested outcome considering with random effects. The aim of this study was to apply GSA to pharmacometric models to assess covariate effects. Simulations were conducted with pharmacokinetic models to characterize the GSA for assessment of covariate effects and with an example of quantitative systems pharmacology (QSP) models to apply the GSA to a complex model. In the simulations, covariate and random variables were generated to simulate the outcomes using the models. Ratios of variance explained by each factor (each covariate and random effect) over the overall variance of the outcome were used as sensitivity indices. The sensitivity indices were consistent with the effect size of covariate. The sensitivity indices identified the importance of creatinine clearance on the pharmacokinetic exposure for a renally-excreted drug. These sensitivity indices could be applied to plasma concentrations over time (repeated measurable outcomes over time) as interested outcomes. Using the GSA, each contribution of all of the covariate effects could be efficiently identified even in the complex QSP model. Variance-based GSA can provide insight when considering the importance of covariate effects by simultaneously and quantitatively assessing all covariate and random effects on interested outcomes in pharmacometrics.

We consider parameter selection and verification techniques for models having one or more parameters that are noninfluential in the sense that they minimally impact model outputs. We illustrate these techniques for a dynamic HIV model but note that the parameter selection and verification framework is applicable to a wide range of biological and physical models. To accommodate the nonlinear input to output relations, which are typical for such models, we focus on global sensitivity analysis techniques, including those based on the Pearson correlation coefficient, Sobol indices based on second-order model representations, and Morris indices, as well as a parameter selection technique based on standard errors. A significant objective is to provide verification strategies to assess the accuracy of those techniques, which we illustrate in the context of the HIV model.

The magnetic-field-modulated brushless double-rotor machine (MFM-BDRM), composed of the stator, the modulating ring rotor, and the permanent-magnet (PM) rotor, is a new power-split device for hybrid electric vehicles (HEVs). Compared with traditional double-rotor machines (DRMs), the MFM-BDRM shows more complicated electromechanical energy conversion relations, due to its special operating principle—the magnetic field modulation principle. To analyze the speed relation in the MFM-BDRM, a diagrammatized method is proposed. It shows that the speeds of stator magnetic field, modulating ring rotor, and PM rotor present a collinear speed characteristic. On this basis, the torque relations of stator, modulating ring rotor, and PM rotor are investigated from the view of a conservative lossless system. Then, a lever-balanced torque map is proposed to analyze their torque characteristic. It shows that the torques of stator, modulating ring rotor, and PM rotor can be calculated by the lever balance principle. The power flow map is further proposed to analyze the power flow characteristic among three ports. In addition, comparison of the MFM-BDRM and the planetary gear shows that the MFM-BDRM can be totally equivalent to an electrical machine and a planetary gear, making it gain a great advantage particularly when the MFM-BDRM is used in HEVs. The electromagnetic performance of MFM-BDRM is investigated by a finite-element method, which shows that the MFM-BDRM has advantages of fine sinusoidal back electromotive force and low torque fluctuation. Finally, the speed and torque analysis and FE results are verified by experiment.