Fast Design of Spoke-Type PM Motor With Auxiliary Notches Based on Lumped-Parameter Magnetic Equivalent Circuit Model and Hybrid Multiobjective Optimizer

The multi-objective design of PM motor is time-consuming. The accuracy complexity of the solver model and the efficiency of the optimizer affect the cycle of electromagnetic design. A fast design method of spoke-type PM motors with auxiliary notches based on lumped-parameter magnetic equivalent circuit (MEC) model and a hybrid multi-objective optimizer (HyMOO) are proposed in this article. The MEC model is established to quickly reflect the influence of design parameters on electromagnetic and torque performance in the account of auxiliary notch structure in the rotor lamination. Meanwhile, an HyMOO is proposed considering the Grey Wolves Optimization (GWO) model, to solve more complex multimode problems involving more parameters. The accuracy and high calculation speed of the proposed MEC are verified in comparison with the FE method. A benchmark test by general distance (GD) and inverted generational distance (IGD) proves the HyMOO with better converge speed and robustness. Based on the MEC model and HyMOO, a fast electromagnetic design is applied for the motor with requirements of 140Nm rated torque and 4.5% torque ripple. The optimal solutions are validated by FE analyses, and the best design are chosen, manufactured as prototype, and tested. Both the FE and experimental analyses verify the reliability of the fast design and the proposed motor.

design process shows high efficiency and high accuracy, and 98 solves the problem of rapid and stable optimization design 99 that meets the design requirements of the proposed machine 100 in the early stage of the project. 101 The rest of the article begins with a description of the 102 machine topology, design requirements, and performance 103 specifications in Section II. In Section III, an original FE 104 model, MEC and parametric chart, including special struc-105 tural designs, are developed. The baseline comparison results 106 with FE model are presented according to the requirement of 107 analyzing the electromagnetic performance characteristics of 108 machine fast and accurately. The development of EMOGWO 109 and HyMOO result in Section IV, as well as the verification 110 benchmarks of optimizer performance. Section V carries out 111 the multi-objective designing for machine's torque perfor-112 mance, including establishing parameter constraints based on 113 design requirements and the mechanical structure of machine 114 topology. The resultant solutions are testified by FE analysis 115 and prototype experiments in Section V. Section VI draws a 116 conclusion to the article. 118 The proposed FSCW spoke-type PM motor is a 16-pole/ 119 18-slot design, and its cross-section is shown in Fig. 1. 120 The tangential magnetic circuit design of rotor provides a 121 significantly enhanced magnetic concentration effect, while 122 FSCW has a lower slot-per-pole ratio than distributed wind-123 ing, which significantly improves the air-gap magnetic den-124 sity and increases the average torque. Compared to conven-125 tional spoke-type PM motors or derivations with eccentric 126 poles used by previous study [6], [17], [19], the machine has 127 auxiliary notches (ANs) to adjust the rotor magnetic circuit 128 and linearly changes part of the air-gap length along the 129 circumferential direction, equivalently adjusting the air-gap 130 magnetic density to make it more sinusoidal, thus reducing 131 harmonics ( Fig. 1(a)). Meanwhile, the notch structure opti-132 mizes the magnetic concentration effect and reduces man-133 ufacturing accuracy requirements compared with eccentric 134 poles. 135 At the same time, to further weaken the potentially 136 harmonic-related problems caused by FSCW, a four-layer 137 concentrated winding layout is applied. Compared with 138 double-layer winding, the four-layer winding, based on the 139 advantage of spatial misalignment, can reduce both the sub-140 and super-stator magneto-motive force (MMF) harmonics 141 by further stacking the number of winding layers per slot. 142 As shown in Fig. 1(b), each phase has three positive and three 143 negative sectors, and the back-EMF is proven to be more 144 sinusoidal with less total harmonic distortion (THD) [20]. 145 In addition, another combination of ANs is set at the inner 146 edge of the rotor lamination, right in the middle of two adja-147 cent PMs, to further modify the rotor magnetic circuit, reduce 148 flux leakage and maximize PM utilization.

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The specifications of pre-design are demoed in Table. 1. 150 The main stator/rotor lamination factors and winding speci-151 fications have been constrained by the space or mechanical 152 99422 VOLUME 10, 2022      To ensure the accuracy of the baseline FE model, the model 171 consists of 27266 cells, where the air gap section is divided 172 into 3 layers along tangential direction to capture the mas-173 sive air gap magnetic flux density distribution of the pro-174 posal spoke-type FSCW PM motor. Fig. 2 shows the rotor 175 configuration and design variables of the target parametric 176 model topology. 10 design parameters need to be rationally 177 considered and optimized. w m and h m are the thickness and 178 length of the PM, respectively, h mo1 and h mo2 are the length of 179 the PM outer and inner offset, α 1 is the electric angle between 180 the ends of two adjacent outer ANs, α 2 is the electric angle 181 between the ends of the inner AN, h n1 and h n2 are the length 182 of the outer and inner AN respectively. R ro is the rotor outer 183 radius obtained from the initial design specification.  (Fig. 4). 209 In the pictures, air linear permeances, PM linear permeances, 210 nonlinear permeances, PM flux sources and winding MMF 211 sources are represented by different symbols and adjacent 212 elements are connected by sequential nodes. All permeances 213 can be calculated using the following equation.
in which A i is the area flux passing through, L i is the length in 216 flux direction, µ 0 and µ i are the air permeability and relative 217 permeability, respectively.

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All elements are interconnected by nodes to form a nodal 219 network system, calculated by applying Kirchhoff's law. 220 VOLUME 10, 2022      the stator unit is divided into four main sections, namely the 228 stator yoke, teeth, tooth tips and slots.

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The yoke consists of one non-linear permeance element. 230 A tooth tip unit is subdivided into three non-linear perme-231 ance elements and two air linear permeance elements. The 232 elements form nodes at the junction with the air-gap, respec-233 tively, to capture flux paths and leakage accurately. The MMF generated by the armature winding can be cal-235 culated by, where, N w and I i are the number of turns and phase current, 238 respectively. The tooth is split into two parts so that each 239 winding of the four-layer winding form can be represented by 240 the corresponding MMF source and a non-linear permeance 241 element in series. Finally, the leakage in the slot is calculated 242 by a single air linear permeance element connected to the 243 central node of the adjacent tooth, respectively.

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To approach the actual situation, the non-linear flux elements 246 of the rotor core are divided into more parallel branches in 247 various shapes based on the spatial dimension of the flux path, 248 while the previously mentioned parameters to be optimized 249 need to be accurately expressed in these elements.

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According to the results shown in Fig   minimum of two nodes on the stator side, and the comple-287 mentary condition is proper for each stator-side node. Then, 288 the adjacent nodes in the middle of two radial elements are 289 connected using the permeance in circumferential direction, 290 respectively ( Fig. 7). This division requires the model to be 291 updated in real-time during the calculation process based on 292 the relative physical positions of the rotor and stator; the 293 increase in calculation time is inevitable but enables fine 294 approximation to be made.
The iterative calculation flowchart of MEC model is shown in 297 Fig. 8. First, the nodal system and element calculation func-298 tions are initialized according to the design specifications, 299 operating conditions, and parameter values to be optimized 300 for the machine. A structure array is created to track the MMF 301 of the node network and the relative permeability of each 302 permeance element [22]. In combination with this structural 303 array, the permeance and flux matrices are built and used to 304 calculate the MMF matrix at each iteration. After that, the 305 relative permeability fit values of the non-linear permeability 306 elements can be calculated by interpolation of the BH curve 307 of materials and basic equations below: where µ r and µ rf represent the former and fit value of ele-310 ment permeability. The nonlinear iterative calculation contin-311 ues until µ r converges to within a set tolerance, after which 312 the relevant results are recorded, and the model is reinitialised 313 to start the next set of calculations.          Table. 2 shows the comparison of processing time in the 366 prementioned subsections by the proposal MEC and FE 367 model, respectively. In single-step calculation, the MEC 368 model requires less than 13% of processing time compared 369 with FE. In 60-step calculation for one electrical cycle, the 370 final time is still significantly ahead of the FE model with 371 calculation speed that is about 22 times faster than FE, even 372 though the discrete calculation approach and data processing 373 causes the parallel MEC process to lose some time.   Thus, the MEC model is proven to be a good combination 387 of computational speed and computational accuracy.

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The significance of multi-objective optimization in engineer- , and the influence intensity is adjusted by 450 the weight coefficient σ , which is expressed as follows: where X pbest represents indivisuals' local best position. 454 In anticipation, the introduction of local best is more feasible 455 for enhancing the individual exploration ability and adjusting 456 the head guidance strength according to the pre-set value.

458
In the original MOGWO algorithm, the control parameter 459 increases linearly with the iteration, and the exploration inten-460 sity of the algorithm decreases. The application of nonlin-461 ear control parameter a in the GWO algorithm is discussed 462 in [29], and it is also applicable in multi-objective algorithms. 463 Under the premise of lack of exploration ability, a is adjusted 464 to power function form, expressed as: where t max is the maximum iteration value, t f is the compen-467 sation iteration value in [0, 0.5], and k is a rational number 468 greater than 1. Specifically, t f is used to adjust the response 469 delay of function a to t, and k is the exponent of the power 470 function, which two jointly affect the convergence speed and 471 exploration ability of the algorithm. The larger values of t f 472 and k can improve the searching ability of the algorithm in 473 the pre-and mid-iteration period.
where pop represents the population of all individuals, X tol 496 is the tolerant difference array, for X j satisfying (21) will 497 be reset before the objective function is calculated as the 498 descriptions below: where dim is the dimension of X, r is a random value in

509
The main process of the HyMOO is shown in Fig. 16.  where S is the resultant solutions of optimization algorithm, 524 P s is the real Pareto front of objective function. GD accurately 525 reflects the convergence of the optimization results, with 526 lower GD values representing more reliable results, which 527 is suitable for verifying the accuracy of a multi-objective 528 optimization algorithm that needs to be applied in practical 529 engineering.

541
The testing environment was chosen to be Python-3.9, and  Table. 3.   significant GD advantage, however, the IGD values are very 573 poor, indicating that the algorithm may have trapped into 574 local optimum several times during convergence, as verified 575 in Fig. 17(a), which illustrates the distributions of resultant 576 solutions from 20 tests of the three algorithms.

577
With the relatively more complex UF4 problem, both 578 GD and IGD of HyMOO outperform the MOPSO and the 579 MOGWO algorithm. The distributions of the resultant solu-580 tions are shown in Fig. 17(b), where the HyMOO results have 581   The design parameters are calculated based on the require-596 ments shown in Table 1 and the boundaries shown in Table 6 597 considering spatial constraints, pre-design calculations, and   According to the calculated results in the present study 608 average torque and torque ripple, the proposed HyMOO is 609 used to find the optimal solution sets. The calculation results 610 are shown in Fig. 18, where the Pareto solution sets are 611 marked in red and are considered as the selection range for 612 the optimal designs.

613
In addition, a computational comparison of the MOPSO 614 algorithm under the same simulation environment and initial 615 conditions needs to be mentioned in Fig. 19. It is easy to 616 see that the conventional MOPSO algorithm runs into local 617 optimal after the same converge iterations, which illustrate 618 that MOPSO is no longer able to meet the requirements at 619 VOLUME 10, 2022  silicon steel material. Hence, the structural strength of final 644 design meets the requirements of the machine's operating 645 requirements.

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A prototype FSCW spoke-type PM motor was built for 647 experimental verification. The prototype and the testing rig 648 are shown in Fig. 22.   Fig. 24 shows the measured output torque curve from the 658 prototype under 140Arms condition at 3000 rpm. According 659 to the calculations, the average output torque of the prototype 660 is 143.7Nm, which meets the design requirement of 140Nm; 661 the torque ripple is 2.90%, which is less than the design 662 requirement of 4.5%. Minor degradation in torque perfor-663 mance occurred, but the MEC model calculations proved to 664 still be an accurate reference in the fast design, while the 665 rated average torque and torque ripple of the prototype meets 666 the design requirements. Besides, the measured efficiency 667 topology. Even if the MEC model is modular designed to 707 adapt to different pole-slot combinations, how to establish 708 a more general analytical model for PM motor is still the 709 primary task of future research. 710