An Accurate Multiobjective Optimization Strategy for Surface-Mounted Permanent-Magnet Machines Based on Nonlinear Finite-Permeability Subdomain Model

This article presents an accurate multiobjective optimization strategy for surface-mounted permanent-magnet machines (SMPMMs) by combining a nonlinear finite-permeability subdomain model (FPSM) with the nondominated sorting genetic algorithm II (NSGA-II). The nonlinear FPSM is developed by introducing a nonlinear iterative algorithm (NIA) to consider the magnetic saturation of soft magnetic materials and applied to SMPMMs. For the NIA, two iterative solving methods (ISMs), namely bisection and relaxation methods (RMs), are compared for the convergence speeds through statistical analysis of 500 designs. This analysis shows that the RM is the best in terms of computation time. When the relaxation coefficient equals to 1, the number of iterations comes down to 2 for this specific machine with the given allowable error. The electromagnetic performances of one optimal case are validated by finite-element models (FEMs) to demonstrate the effectiveness of the presented optimization strategy. The strategy proposed in this article can help designers to find the optimal designs for SMPMMs.


I. INTRODUCTION
P ERMANENT-MAGNET (PM) machines are widely used in industrial manufacture and household appliances due to their high efficiency and torque density.A fast and accurate modeling approach is essential for motor design and optimization.
Finite-element models (FEMs) have proven to be a highly effective technique for designing PM machines because they can consider magnetic saturation and complex geometries.However, the design process using FEM is too timeconsuming.Subdomain (SD) models, as a favorable alternative to FEM, have faster calculation speed [1], [2].However, the traditional SD models assume that the permeability of iron parts is infinite, and the magnetic saturation effect of ferromagnetic materials is hence ignored.Two different finitepermeability subdomain models (FPSMs) have been proposed to consider the magnetic saturation effect.The first one gives the general solutions of Maxwell's equations for every SD by considering nonhomogeneous Neumann boundary conditions and both θ -edge and r -edge interface conditions [3].The other type of FPSM divides the machine into an arbitrary number of homogeneous or nonhomogeneous layers where the permeability in the stator or rotor slotting is represented as a Fourier series along the direction of permeability variation [4].
Many studies on the optimal design of electric machines are primarily based on FEM [5], [6], [7], which is very time-consuming because of the many iterations needed during the optimization process.Few studies combine the nonlinear FPSM with an optimization algorithm to accelerate the optimization process.Zhao et al. [8] combined the second type of FPSM with nondominated sorting genetic algorithm II (NSGA-II) to optimize Vernier machines.The first type of FPSM is more accurate than the second type of FPSM when studying magnetic saturation and optimizing parameters [9].However, to our knowledge, the first type of FPSM has not been combined with any optimization algorithm.
In this article, an accurate multiobjective optimization strategy for surface-mounted PM machines (SMPMMs) is presented, which is based on combining the first type of FPSM with NSGA-II.A nonlinear iterative algorithm (NIA) is introduced to consider the magnetic saturation effect.Based on statistical analysis, this article compares the convergence speed of two different iterative solving methods (ISMs): the bisection method (BM) and the relaxation method (RM).To our knowledge, this is the first time these two ISMs are compared for the first type of FPSM, which can help designers find the best ISM for them and accelerate the process of obtaining the optimal designs.The optimal objectives in this study are average torque, torque ripple, and PM usage.Finally, an optimal case is selected from the Pareto front and validated by FEM.

II. STUDIED SMPMM AND NONLINEAR FPSM A. Studied SMPMM
The structure of the studied SMPMM is shown in Fig. 1, and the fixed parameters during optimization are given in Table I.As shown in Fig. 1, the studied machine is divided into six types of SDs.SD I to VI represent the rotor yoke, the PMs, the air gap, the stator slots, the stator teeth, and the stator yoke, respectively.The radii R 1 , R 2 , R 3 , R 4 , R 5 , and R 6 are the

TABLE I FIXED PARAMETERS DURING OPTIMIZATION
rotor inner radius, the PM inner radius, the PM outer radius, the air gap outer radius, the slot outer radius, and the stator outer radius, respectively.The stator opening angle is marked as δ.The analytical model in this article is formulated in the 2-D polar coordinate system based on the assumptions shown in [9].However, contrary to [9], the permeability of the rotor yoke is finite, and the magnetic saturation effect of this SD is considered.

B. Introduction of FPSM
The six types of SDs are shown in Fig. 2. The governing equations of magnetic vector potential (MVP) derived from the magnetostatic Maxwell's equations are formulated for different SDs based on the material properties of each SD.The general solution of MVP for each SD is obtained by using the Fourier series and the separation variables in polar coordinates.The general solutions of MVP for SD II to VI are given in [9] and [10].In this article, the magnetic field of the rotor yoke (SD I) is considered.The general solution of MVP in the rotor yoke is given by where A 10 , A 2n , and A 3n are SD I's integration constants.
The boundary conditions (BCs) are used to connect these general solutions of MVP to build a linear system.The six types of SDs in this SMPMM can be divided into two types: 1) periodic SDs, such as SD I, II, III, and VI; and 2) nonperiodic SDs, such as SD IV and V.There are two types of BCs considered in this model.One BC type is over angle intervals for a given radius (θ-edges), and the other is  over radius intervals for given angles (r -edges) [3].These two types of BCs are shown in Fig. 2. Both θ -edges and r -edges BCs are considered in nonperiodic SDs, but only θ -edges BCs need to be considered in periodic SDs.All BCs are given in [3] and [9].Finally, the constants for each SD are determined by solving the linear system obtained from the BCs.

C. Nonlinear Iterative Algorithm
The nonlinear ferromagnetic material M-19 steel is used in SD I, V, and VI.The µ r − B curve is shown in Fig. 3.
The flowchart of the NIA is shown in Fig. 4. Some candidate points (CPs) are selected (Fig. 2) to consider the nonlinearity.First, the relative permeabilities of all soft magnetic material CPs are set to an initial value µ c .Then, the magnetic flux Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.density of each CP is obtained by solving the matrix of the linear system, and the calculated relative permeability for all CPs (µ cal ) is obtained based on the curve µ r (B) shown in Fig. 3.The calculated relative permeability in the SDs I and VI requires selecting the minimum value µ calm among all CPs in each SD as the representative value for the whole SD.Next, all CPs' relative permeability is iteratively updated all using the ISM.The BM, shown in Fig. 5, is used as the ISM to take the nonlinearity into account.When the relative errors are smaller than the allowable errors ξ , the algorithm is finished.

III. OPTIMIZATION PROCESS
There are many objectives during the motor design, such as high average torque and low torque ripple.However, these objectives are usually in conflict with one another.Many algorithms are developed to find the optimal design, such as genetic algorithm, particle swarm optimization, and NSGA-II, that can be used for multiobjective optimization.In this article, NSGA-II is adopted as an optimization algorithm and combined with the nonlinear FPSM mentioned above to search for the optimal design for SMPMMs.The flowchart of the proposed multiobjective optimization strategy is illustrated in Fig. 6, and the average torque, torque ripple, and PM volume represent the individual fitness.
The fixed parameters and the design variables with the corresponding range during optimization are provided in Tables I and II, respectively.We aim to maximize the

TABLE III DESIGN PARAMETERS OF OPTIMIZATION CASES
torque T ave in a constrained motor volume and minimize the torque ripple T ripple to reduce the vibration and noise.We also aim to reduce the PM volume V PM to lower manufacturing costs.Therefore, the optimization objectives are set as average torque (T ave ), torque ripple (T rip ), and PM volume (V PM ), and the objective functions and constraint are listed as follows: Functions: Max(T ave ), Min T rip , Min(V PM ) Constraint: R 1 > 0. ( The maximum number of generations is set at 200, and the population for each generation is set at 20.The optimization results with a clear 3-D Pareto front and corresponding 2-D projections are presented in Fig. 7.An optimal case on the Pareto front is selected for validation, and the parameters are shown in Table III.The time to solve the matrix generated by the FPSM once is about 0.4 s.For each calculation point, multiple rotor positions need to be calculated, and each rotor position requires several matrix calculations.

IV. FEM VALIDATION
In order to validate the accuracy of the proposed optimization strategy, the electromagnetic parameters calculated by nonlinear FPSM for the selected optimal case, such as magnetic flux density, electromagnetic torque, and cogging torque, are compared with FEM and with the traditional SD model.
The on-load radial and tangential magnetic flux density distribution in the middle of the air gap and the corresponding harmonic spectrum are shown in Figs. 8 and 9, respectively.The relative errors of radial flux density for the nonlinear FPSM and traditional SD model are 3.5% and 12.6%, respectively.The electromagnetic torque calculated by the nonlinear FPSM is compared with the FEM and the traditional SD model in Fig. 10.The relative errors of the electromagnetic torque for the nonlinear FPSM and traditional SD model are 2.1% and 18.2%, respectively.The results predicted by the nonlinear FPSM match well with FEM results, but there are some errors in the traditional SD model because it cannot consider magnetic saturation.The cogging torque predicted by the nonlinear FPSM is also compared with FEM and traditional SD model in Fig. 11, and there is excellent agreement between these three waveforms because the machine has a low saturation level in this situation.
Both the FPSM and the FEM involve the resolution of a linear system.For a matrix of 2401 in size, the FPSM gives an average error of 2.43% in a torque cycle.For the FEM with a matrix of 2442 in size, the average error is 6.48%.Therefore, for this similar small matrix size, the FPSM gives less error.

V. COMPARISON OF TWO ISMS
Different ISMs can be used to update the relative permeability of the CPs.Those methods have different convergence speeds.Two of them, namely BM (Fig. 5) and RM (Fig. 12), are compared for their convergence speeds using the same  NIA mentioned above.The range of coefficient α in the RM is from 0 to 1.
We randomly selected 500 machines for the studied SMPMM.We used two iterative methods to obtain the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.average number of iterations required to calculate several rotor positions within a torque cycle for each machine.There are 11 rotor positions selected in a torque cycle for calculation.Therefore, for each ISM, 5500 calculations are executed.
Fig. 13(a) shows the distribution of the number of machines under different ISMs.For the RM, most machines have less than ten iterations as α increases.For the BM, most machines need more than ten iterations, which requires more computation time than RM.Overall, the convergence speed of the RM is faster than BM when α is bigger or equal to 0.6.
The average iteration number of these results is taken as the reference value for convergence speed [Fig.13(b)].For the RM, as α increases, the iteration number required by the RM gradually decreases.Therefore, a larger α can speed up the computation for this optimization strategy.The BM needs more iterations than the RM when α > 0.6, but the BM does not have a coefficient that needs to be determined.When α = 1, the model of the SMPMM converges very quickly, in two iterations in our case, with the allowable error ξ = 0.2.The RM with α = 1 is the best approach in our situation.

VI. CONCLUSION
This article presents an accurate multiobjective optimization strategy, which combines the nonlinear FPSM with the NSGA-II algorithm to find a Pareto front that helps design SMPMMs.The comparison results between this strategy and FEM for one optimal case confirm the effectiveness of the presented optimization strategy.Also, two different ISMs are compared for their convergence speed based on statistical analysis.The RM has a faster convergence speed than the BM when using α ≥ 0.6.With our type of SMPMM, the RM with α = 1 is the best approach as it converges very quickly, in two iterations only.

Fig. 8 .
Fig. 8. On-load radial magnetic flux density in the middle air gap of an optimal case.(a) Flux density distribution.(b) Harmonic spectrum.

Fig. 9 .
Fig. 9. On-load tangential magnetic flux density in the middle air gap of an optimal case.(a) Flux density distribution.(b) Harmonic spectrum.

Fig. 13 .
Fig. 13.Statistical analysis.(a) Distribution of the number of machines under different ISMs for ξ = 0.2 (N 1 represents the average iteration number for each machine).(b) Comparison of iteration numbers for different ISMs under two different allowable errors.

TABLE II VALUE
RANGE OF DESIGN PARAMETERS DURING OPTIMIZATION