Optimization of cogging torque of tangential magnetizing parallel structure hybrid excitation synchronous motor based on PSO and entropy TOPSIS method

Abstract

In order to solve the problem that the cogging torque of tangential magnetizing parallel structure hybrid excitation synchronous motor (TMPS-HESM) rises sharply when the excitation current increases. Maxwell & Workbench & optiSLong co-simulation tool was used for research. Firstly, the length and width of the permanent-magnet, rotor eccentricity, and stator parameters are parameterized. The multi-objective optimization is carried out by using particle swarm optimization (PSO). Secondly, through sensitivity analysis, the influence degree of different input parameters on cogging torque, average torque and torque pulsation is discussed. Then, the quadratic regression model of high sensitivity parameters and cogging torque, average torque and torque pulsation is established respectively. Finally, entropy technique for order preference by similarity to an ideal solution (TOPSIS) method is used to select the optimal solution from pareto solution set. The results show that the cogging torque and torque ripple of the optimized motor are suppressed obviously, and the average torque is also improved.

Keywords

Tangential focus type hybrid excitation cogging torque PSO entropy TOPSIS method

1. Introduction

The Hybrid excitation motor is a new type of motor that organically combines permanent-magnet and electric excitation. It has the advantages of high power density and high efficiency. There are many classification methods of hybrid excitation motors. It is generally classified according to magnetic coupling relationship. They can be divided into series magnetic circuit hybrid excitation motors, parallel magnetic circuit hybrid excitation motors and series and parallel magnetic circuit mixed excitation motor [1–6]. TMPS-HESM is a kind of parallel magnetic circuit hybrid excitation motor. The excitation winding is placed in the rotor slot by centralized winding, and the permanent-magnet are embedded between two adjacent rotor teeth.

Cogging torque is an inherent property of hybrid excitation machines. The cogging torque will cause torque ripple, vibration, and noise of the motor. Thus the stability of the motor operation is affected [7]. Reducing the cogging torque of the motor has always been a hot research topic of scholars today. In [8], the pole-arc coefficient and permanent-magnet thickness of the permanent-magnet dual-rotor motor for electric vehicles are optimized and analyzed, and the goal is to optimize the cogging torque and electromagnetic coupling. In [9], the three-phase 12/10-pole single-stator/dual-rotor AF-HESM motor is taken as the research object, and the stator slot shape, permanent-magnet shape factor and rotor tooth bevel angle are optimized by optimizing. Thereby increasing the induced electromotive force and reducing the cogging torque. In [10], a method of opening auxiliary slots on the top of the rotor teeth of the motor is adopted, to weaken the cogging torque by changing the number and type of auxiliary slots. In [11], for the surface-mounted permanent-magnet synchronous motor, the influence of the permanent-magnet width on the cogging torque and torque ripple was explored. In [12–14], asymmetrical V-shaped rotor structure, eccentric rotor, and stepped rotor are adopted respectively to suppress the cogging torque. Although the above methods have achieved relatively suitable optimization results, because the variables considered in the optimization design are relatively single, while optimizing the target variables, other electromagnetic characteristics of the motor are often affected.

Although the above methods have achieved more suitable optimization results, but because of the single variable in the optimization design, the optimization of the target variable often affects other electromagnetic characteristics of the motor. To solve the above problems, this paper takes TMPS-HESM as the research object. Based on the multi-objective optimization method of Maxwell & Workbench & optiSLong combined optimization analysis, the length and width of permanent-magnet, rotor eccentricity and stator parameters of motor were parameterized with the optimization objectives of cogging torque, average torque and torque ripple. According to the sensitivity analysis, the influence degree of different input parameters on the torque, average torque and torque ripple of the motor was determined. Finally, from the pareto diagram, through the entropy TOPSIS method to screen out the objective of the optimal solution, thus generation into the finite element analysis of motor air gap flux density, no-load electromotive force and efficiency map graph. This paper compares and analyzes confirming that the multi-objective optimization method can suppress the TMPS-HESM cogging torque.

2. Theoretical analysis of motor structure and cogging torque

2.1. Motor topology analysis

The research object of this study is an 8-pole and 48-slot TMPS-HESM, which is an electromotor. The permanent-magnet magnetic circuit and the electric excitation are parallel structures. The magnetic circuits of the two magneto-motive force sources provide the working magnetic flux in parallel, which together constitute the main magnetic field of the air gap. The magnetic potential of the excitation winding is dependent on the size of the main magnetic flux. The basic parameters of the motor are shown in Table 1, and the motor structure is shown in Fig. 1.

Fig. 1.

The structure diagram of TMPS-HESM.

Table 1

Basic motor parameters

Parameters	Value	Parameters	Value
Power (kW)	2.5	Rated voltage (V)	40
Rated speed (r/min)	3500	Length of the core (mm)	40
Number of slots/poles	48/8	Inner diameter of the stator (mm)	94.5
Number of turns of the excitation	40	Outer diameter of the stator (mm)	150

2.2. Theoretical analysis of cogging torque

The co-energy method was used to derive the cogging torque, which stipulates that when the rotor of the motor rotates, the energy in the permanent-magnet and the iron core does not change, and only the energy stored in the air gap changes [15].

By simplifying the calculation, the internal magnetic energy of the motor is obtained as: $\begin{eqnarray}\displaystyle W_{\text{energy}}=\frac{1}{2}Li^{2}+\frac{1}{2}(R+R_{m}){\varphi}^{2}+Ni{\varphi}. & & \displaystyle\end{eqnarray}$ (1)

Each term of the above items corresponds to the self-induction, permanent-magnet magnetic energy, and mutual inductance magnetic energy of the motor respectively; R and R _m are the air-gap reluctance and stator core reluctance; φ is the magnetic flux formed by the permanent-magnet and the field winding and the armature winding turn chain; i is the armature current size; L is the armature inductance; N is the number of turns of the excitation coil. $\begin{eqnarray}\displaystyle T_{e}=\frac{\partial W_{\text{energy}}}{\partial {\alpha}} & & \displaystyle\end{eqnarray}$ (2) where T _e is the motor torque. 𝛼 is the relative position angle of the stator and rotor.

Since the armature current is constant, the cogging torque can be approximately generated by the reluctance change of the motor air gap and the interaction of the magnetic field, and the corresponding cogging torque is expressed as: $\begin{eqnarray}\displaystyle T_{\text{cog}}=-\frac{1}{2}{\varphi}_{g}^{2}\frac{dR}{d{\alpha}}. & & \displaystyle\end{eqnarray}$ (3) The cogging torque of the motor is related to the air gap flux φ_g and the rate of change of the reluctance with the change of the rotor. The cogging torque is also obtained as: $\begin{eqnarray}\displaystyle T_{\text{cog}}({\alpha})=-\frac{\partial W_{\text{energy}}}{\partial {\alpha}}. & & \displaystyle\end{eqnarray}$ (4)

Since the magnetic permeability of the motor core is much larger than that of the permanent-magnet and the air, the magnetic field energy in the motor is mainly concentrated between the permanent-magnet and the air gap. The energy in the permanent-magnet can be considered as a constant, and the energy stored in the magnetic field is approximately the energy in the air gap. Therefore, the expression for the motor cogging torque is: $\begin{eqnarray}\displaystyle T_{\text{cog}}({\alpha})=-\frac{\partial W_{\text{energy}}}{\partial {\alpha}}=-\frac{\partial W_{PM}+\partial W_{\text{airgap}}}{\partial {\alpha}}\approx -\frac{\partial W_{\text{airgap}}}{\partial {\alpha}} & & \displaystyle\end{eqnarray}$ (5) where W _PM is the magnetic field energy in the permanent-magnet; W _airgap is the magnetic co-energy in the air gap. $\begin{eqnarray}\displaystyle W_{\text{airgap}}=\frac{1}{2{\mu}_{0}}\int _{V}B^{2}({\theta},i_{f})dV=\frac{1}{2{\mu}_{0}}\int _{V}B_{r}^{2}({\theta},i_{f})G^{2}({\theta},{\alpha})dV & & \displaystyle\end{eqnarray}$ (6) where G (θ, 𝛼) = h∕(h + δ), μ₀ is the air-gap permeability, B _r(θ, i _f) is the permanent-magnet remanence distribution V is the volume, B (θ, i _f) is the air-gap magnetic field distribution, i _f is the DC excitation current, h, δ are respectively magnetizing direction distribution of permanent-magnets and effective air gap length.

The Fourier expansion of $B_{r}^{2}({\theta},i_{f})$ is: $\begin{eqnarray}\displaystyle B_{r}^{2}({\theta},i_{f})=B_{r0}^{2}(i_{f})+\sum _{m=1}^{\infty }B_{rm}(i_{f})\cos (mN_{s}{\theta}) & & \displaystyle\end{eqnarray}$ (7) where $B_{r0}^{2}={\alpha}_{s}B_{res}^{2},B_{rm}=2B_{res}^{2}\sin (m{\alpha}_{s}{\pi})/m{\pi}$ , B _res is the residual magnetic density of the permanent-magnet; 𝛼_s is the rotor pole arc coefficient; N _s is the number of stator poles. Without considering local magnetic saturation and flux leakage, the air gap length corresponding to the rotor teeth is δ, and the air gap length corresponding to the rotor slot is approximately infinite, so the Fourier expansion of G ²(θ, 𝛼) is: $\begin{eqnarray}\displaystyle G^{2}({\theta},{\alpha})=G_{0}+\sum _{n=1}^{\infty }G_{n}\cos (nN_{r}{\theta}) & & \displaystyle\end{eqnarray}$ (8) where N _r is the number of rotor stages. When considering stator and rotor motion states, it is equivalent to increasing the initial phase of formula (8). At this time, the corresponding Fourier expansion is: $\begin{eqnarray}\displaystyle G^{2}({\theta},{\alpha})=G_{0}+\sum _{n=1}^{\infty }G_{n}\cos [nN_{r}({\theta}+{\alpha})] & & \displaystyle\end{eqnarray}$ (9) where G ₀ = 𝛼_s G ²(θ, 𝛼), G _n = 2G ²(θ, 𝛼)sin(n𝛼_sπ)∕nπ.

According to the above derivation, formula (6)–(9) is substituted into formula (5) to obtain the cogging torque expression: $\begin{eqnarray}\displaystyle T_{\text{cog}}({\alpha})=\frac{{\pi}_{r}N_{r}L_{a}}{4{\mu}_{0}}(R_{r}^{2}-R_{s}^{2})\sum _{n=1}^{\infty }nG_{n}B_{rnN_{L}}(i_{f})\sin (nN_{r}{\alpha}) & & \displaystyle\end{eqnarray}$ (10) where N _L = N _r∕LCM (N _s, N _r); L _a is the axial length of the motor; R _s and R _r are the inner diameter of the stator and the outer diameter of the rotor; i _f is the excitation current of the motor; G _n and B _rnNL are the air-gap flux density function and fourier decomposition coefficients of the air-gap permeability function respectively.

According to (10), changing R _s, G _n and B _rnNL can weaken the motor’s cogging torque, and the sizes of G _n and B _rnNL are related to the length and width of permanent-magnets, stator tooth structure and magnetic pole eccentricity. In addition, when the mixed excitation motor enters different excitation currents, the air gap flux density of the motor will be affected, and thus the cogging torque will be affected.

3. Co-simulation based on Maxwell & Workbench & optiSLong and PSO

3.1. Parametric model establishment

The stator and rotor parts of the motor are parameterized respectively. Changing the stator groove depth and slot body height can affect the magnetic saturation coefficient and harmonic magnetic density amplitude. Changing the stator slot width and stator inner diameter can affect the air gap permeability and thus affect the harmonic flux density amplitude. The cogging torque can be optimized by improving harmonic magnetic density amplitude. The rotor eccentricity was designed to suppress the cogging torque by improving the air-gap flux density waveform. Meanwhile, the length and width of permanent-magnets were taken as input variables to set parameters, and the motor parametric modeling was established, as shown in Fig. 2. The inner diameter of the setting is R, the slot width is B _s0, the height of the stator tooth slot is H _s0, the depth of the stator slot is H _s1, the width of the axial permanent-magnet is b, the length of the permanent-magnet is l, the rotor eccentricity is H, the radius of the eccentric rotor is R _d and the cutting thickness is h.

Fig. 2.

Permanent-magnet parameter structure diagram.

Within a reasonable size range, the initial values and value ranges of the seven design variables are determined as shown in Table 2.

Table 2

Initial value and variation range of motor optimization variables

Parameters	The initial value	Value range
l (mm)	7.6	[6.8, 8.5]
b (mm)	6.5	[6, 7]
R (mm)	94.5	[94.5, 95.4]
H (mm)	0	[0, 12]
B_s0 (mm)	2.5	[1.5, 2.5]
H_s0 (mm)	1.2	[0.8, 1.3]
H_s1 (mm)	16.8	[16.2, 18]

3.2. Finite element simulation of exciting current on initial motor cogging torque

In maxwell, the excitation current is parametrically scanned. The excitation current range is from 0 A to 12 A, and the step size is set to 4 A, so that other parameters remain unchanged. The finite element simulation results are shown in Fig. 3(a). When the excitation current rises from 0 to 12 A, the cogging torque rises sharply, as shown in Fig. 3(b). When the excitation current is 12 A, the maximum value of cogging torque is 2.027 Nm. When the excitation current is small, the cogging torque is already very minimal, and its further suppression has little effect on the motor. Therefore, it is proposed to select the excitation current of 8 A ∼ 12 A to conduct the suppression analysis of the cogging torque. Therefore, Optimization was carried out when the excitation current was 12 A.

Fig. 3.

(a) Initial cogging torque at different excitation currents; (b)Variation trend of cogging torque with increasing excitation current.

3.3. Multi-objective PSO

(1) Setting constraints

OptiSLong is a robust analysis and optimization software that integrates efficient and reliable multi-objective optimization processes. The multi-objective optimization algorithm adopted in this paper is PSO, which has the characteristics of cooperative search and can use personal local information and group global information to guide the search [16,17]. The optimal individuals make full use of their own experience and group experience to adjust their state, and the individuals are sorted according to the crowding distance, to maintain the diversity of particles in the population and make the Pareto surface evenly distributed. During each iteration, the position and velocity of particles are updated by local and global optimal advantages. The basic equation of PSO algorithm is: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}V_{id}^{k+1}={\omega}V_{id}^{k}+c_{1}r_{1}(P_{id}^{k}-X_{id}^{k})+c_{2}r_{2}(Q_{id}^{k}-X_{id}^{k})\\[6.0pt] X_{id}^{k+1}=X_{id}^{k}+V_{id}^{k+1}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (11)

Where V _id is the velocity of the i-th particle in the d-dimensional space, X _id is the position of the i-th particle in the d-dimensional space, ω is the inertial weight, K is the current iteration number, D is the size of particle search space, c ₁ and c ₂ are local learning factor and global learning factor respectively, r ₁ and r ₂ are random numbers between [0, 1], P and Q are local and global maxima respectively.

The weight coefficient of PSO is optimized by linear decreasing weight method, as shown in (12). The inertia weight coefficient ω is set to linearly decrease from ω_Max = 0.9 to ω_min = 0.4, and the local learning factor and global learning factor c ₁ and c ₂ are set in asynchronous time-varying mode, as shown in (13). The purpose of this setting is to enhance the global search at the initial stage of optimization, and to promote the convergence of particles to the global optimal solution during the search period. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\omega}={\omega}_{\max }-({\omega}_{\max }-{\omega}_{\min })\frac{k}{N_{\text{cmax}}}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \left\{\begin{array}{@{}l@{}}c_{1}=c_{1\text{max}}-(c_{1\text{max}}-c_{1\text{min}})\frac{k}{N_{\text{cmax}}}\\[6.0pt] \displaystyle c_{2}=c_{2\text{min}}+(c_{2\text{max}}-c_{2\text{min}})\frac{k}{N_{\text{cmax}}}.\end{array}\right.\end{eqnarray}$ (13)

Where, N _cmax is the maximum number of iterations, c _1max = c _2max = 0. 9, c _1min = c _2min = 0.1.

Couple optiSLong to Maxwell, set the optimization algorithm to PSO. According to the number of output variables and input variables, the initial population is 10, the maximum population size of PSO is 400, the maximum number of iterations is 20, and the mutation rate is set to 30%.

In the Workbench, the optimization goals are set to minimum torque ripple, cogging torque, and maximum average torque. The specific objective function and constraint conditions are shown in (14). $\begin{eqnarray}\displaystyle f(x)=[\min (T_{\text{cog}}),\min (T_{\text{rip}}),\max (T_{\text{avg}})],\quad T_{\text{cog}}<T_{\text{cog}0},T_{\text{rip}}<T_{\text{rip}0},T_{\text{avg}}>T_{\text{avg}0} & & \displaystyle\end{eqnarray}$ (14) where f (x) is a multi-objective function. T _cog0, T _rip0 and T _avg0 are the minimum required values of motor cogging torque, torque ripple and average torque, respectively.

(2) Sensitivity analysis

In order to obtain the functional relationship between the optimization objective and the optimization variables, seven motor size parameters in Table 2 were selected for sensitivity analysis, optimization variables with significant sensitivity values were selected to build the response surface mathematical model, and finally the functional relationship between the optimization objective and the optimization variables was obtained. The sensitivity analysis results are shown in Fig. 4.

In Fig. 4, the stator inner diameter, slot width, and stator slot depth have a relatively significant influence on output variables. The length and width of permanent-magnet only have a certain effect on the average torque and the cogging torque, but have minor effect on the torque ripple. The stator slot opening width has little effect on the average torque, but it has some significance for the optimization of the cogging torque and torque ripple. To simplify the analysis process, the influences of stator inner diameter R and slot width B _s0 on average torque and cogging torque, as well as groove depth H _s1 and stator inner diameter R on torque ripple are discussed by pareto diagram.

Fig. 4.

Parameter sensitivity analysis.

Fig. 5.

Pareto frontier graph.

(3) Pareto optimal solution set

The seven motor structural parameters in Table 2 were used as optimization variables for PSO iteration, and the optimization results are represented in the form of pareto frontier graphs, as shown in Fig. 5. The Pareto solution means that improving any objective function is bound to weaken the solution of at least one other objective function. The pareto frontier graph is the optimal solution set formed by the calculated optimal solution in space, which is represented as a 3D surface formed by the connection line of the red point.

According to the results, the quadratic response surface method was used to find the quantitative relationship between the optimization variables and the high sensitivity factors, so as to obtain the optimal response performance of the parameter combination. The quadratic response surface model is shown in (15), where 𝜆₀ is a constant and 𝜆_i is a first-order coefficient. 𝜆_ii is the second-order coefficient; 𝜆_ij is the second-order interaction coefficient; δ is the error. $\begin{eqnarray}\displaystyle Y={\lambda}_{0}+\sum _{i=1}^{k}{\lambda}_{i}x_{i}+\sum _{i=1}^{k}{\lambda}_{ii}x_{i}^{2}+\mathop{\sum }_{i<j}{\lambda}_{ij}x_{i}x_{j}+{\delta}. & & \displaystyle\end{eqnarray}$ (15)

Since the stator inner diameter and slot width have a great influence on the cogging torque, the response surface model of the fitted stator inner diameter and slot width on the cogging torque can be obtained according to the quadratic response surface method, as shown in (16). $\begin{eqnarray}\displaystyle T_{\text{cog}}=34880-725.5R-404.8B_{s0}+3.773R^{2}+4.142RB_{s0}+3.038B_{s0}^{2}. & & \displaystyle\end{eqnarray}$ (16)

According to Pareto diagram, the response surface curve of tooth cogging torque can be obtained as shown in Fig. 6, where the depth of chromatographic color represents the size of optimization target value. With the increase of stator inner diameter, the amplitude of cog torque decreases. With the increase of the width of the notch, the amplitude of the torque decreases, then increases.

Fig. 6.

Cogging torque response surface.

Similarly, the response surface model of stator inner diameter and slot width with respect to average torque is: $\begin{eqnarray}\displaystyle T_{\text{avg}}=41580-864.2R-520.3B_{s0}+4.491R^{2}+5.337RB_{s0}+2.66B_{s0}^{2}. & & \displaystyle\end{eqnarray}$ (17)

The response surface model of stator groove depth and stator inner diameter on torque pulsation is: $\begin{eqnarray}\displaystyle T_{\text{rip}}=42270-829.6R-285.1H_{s1}+4.319R^{2}+0.173RH_{s1}+7.423H_{s1}^{2}. & & \displaystyle\end{eqnarray}$ (18)

Figures 7 and 8 respectively represent the response surface curves of average torque and torque ripple. The amplitude of average torque decreases with the increase of stator inner diameter. As the width of the groove increases, the amplitude of the average torque decreases first and then increases. When the inner diameter of the stator decreases, the torque ripple increases gradually, and with the deepening of the stator groove, the torque ripple decreases gradually. The above response surface models all have high gradients, which proves that the response surface model has a better range of design variables. According to the above analysis, pareto optimal solution set is finally obtained with 18 solutions, the specific results are shown in Table 3.

Fig. 7.

Average torque response surface.

Fig. 8.

Torque ripple response surface.

Table 3

Pareto optimal solution set

Serial number	Cogging torque/(N⋅m)	Average torque/(N⋅m)	Torque ripple/(%)
1	2.915	9.072	23.11
2	1.355	8.986	10.25
3	1.330	8.938	8.75
…	…	…	…
16	0.568	8.686	9.28
17	1.136	8.689	9.17
18	2.027	8.649	24.99

4. Selecting the global optimal solution based on entropy TOPSIS method

The Entropy TOPSIS method is used to study the distance between the evaluation object and the ideal solution. This method can objectively analyze the pros and disadvantages of Pareto solution under the condition of clear data, to help select the optimal solution. The specific algorithm flow is as follows:

For the existing n target objects, m evaluation indicators. Write each index value in matrix form, as shown in (19), then the initial matrix can be expressed as: X = [X ₁ X ₂…X _j]. $\begin{eqnarray}\displaystyle X_{j}=[x_{1j},x_{2j},\ldots x_{nj}]^{\text{T}},\quad (j=1,2,\ldots ). & & \displaystyle\end{eqnarray}$ (19)

The reciprocal method is adopted to transform low-expectation indicators into high-expectation indicators, promoting the positive trend of evaluation indicators: $x_{ij}^{\prime }=1/x_{ij}(x_{ij}>0)$ .

The data are squared and normalized to eliminate dimensionality so that each index value is shown in (20). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}Z_{j}=[z_{1j},z_{2j},\ldots z_{nj}]^{\text{T}},\quad (j=1,2,\ldots m)\\[6.0pt] z_{ij}=x_{ij}^{^{\prime }}/\sqrt{\sum _{i=1}^{n}(x_{ij}^{\prime })^{2}},\quad (i=1,2,\ldots n;j=1,2,\ldots m).\end{array} & & \displaystyle\end{eqnarray}$ (20) The normalized matrix can be expressed as: Z = [Z ₁ Z ₂…Z _j].

The new matrix is obtained by multiplying the weights by the data after calculating the weights by entropy method. The positive and negative ideal vectors z ⁺ and z ⁻ are obtained as shown in (21). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}z^{+}=[z_{1}^{+},\quad z_{2}^{+},\ldots ,\quad z_{m}^{+}];\quad z^{-}=[z_{1}^{-},\quad z_{2}^{-},\ldots ,\quad z_{m}^{-}]\\ z_{j}^{+}=\max \{z_{1j},z_{2j},\ldots ,z_{nj}\},(j=1,2\ldots m);z_{j}^{-}=\min \{z_{1j},z_{2j},\ldots ,z_{nj}\},(j=1,2\ldots m).\end{array} & & \displaystyle\end{eqnarray}$ (21) The calculation of positive and negative ideal solution distances is shown in (22). $\begin{eqnarray}\displaystyle D_{i}^{+}=\sqrt{\mathop{\sum }_{j=1}^{m}(z_{j}^{+}-z_{ij})^{2}};D_{i}^{-}=\sqrt{\mathop{\sum }_{j=1}^{m}(z_{j}^{-}-z_{ij})^{2}}. & & \displaystyle\end{eqnarray}$ (22)

The relative proximity calculated according to the ideal distance is shown in (23). $\begin{eqnarray}\displaystyle C_{i}=\frac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}},\quad (0<C_{i}<1). & & \displaystyle\end{eqnarray}$ (23)

In order of relative proximity, the greater the relative closeness, the better the evaluation object.

According to the above process, the global optimal solution is taken as the target object, the average torque and torque ripple are taken as the evaluation indexes, and the trend and square sum of the data in Table 3 are normalized. Then the entropy method is used to calculate the weight results, as shown in Table 4.

Table 4

Calculating weight

Evaluation indicators	Weight
Cogging torque	60.56%
Torque ripple	39.32%
Average torque	0.12%

The new matrix is obtained by multiplying the calculated weights with the normalized matrix. The maximum and minimum values of the cogging torque, average torque and torque ripple are found as the positive, and negative ideal solutions. The results are shown in Table 5. The positive and negative ideal solution distance and relative proximity were calculated, and the relative proximity was sorted, as shown in Table 6.

Table 5

Positive and negative ideal solutions

Evaluation indicators	Positive ideal solution z+	Negative ideal solution z−
Cogging torque	0.280	0.055
Torque ripple	0.108	0.035
Average torque	0.000	0.000

Table 6

Evaluation calculation results

Serial number	Positive ideal solution distance D+	Negative ideal solution distance D−	Relative proximity C	Sort results
1	0.236	0.003	0.012	18
2	0.164	0.081	0.330	16
3	0.161	0.092	0.365	15
…	…	…	…	…
16	0.013	0.233	0.949	1
17	0.141	0.105	0.427	7
18	0.214	0.024	0.100	17

Table 7

Pareto optimal solution set

Structural parameters	Before optimization	After optimization
l (mm)	7.6	8.3
b (mm)	6.5	6.37
R (mm)	94.5	94.51
H (mm)	0	10.95
B_s0 (mm)	2.5	2.39
H_s0 (mm)	1.2	0.97
H_s1 (mm)	16.8	18.13

According to the sorting results, the 16th solution is selected as the global optimal solution, which is located in the red marked position in the Pareto graph of Fig. 5. The motor structure parameters before and after optimization are shown in Table 7, and the motor structure diagram after optimization is shown in Fig. 9.

Fig. 9.

Optimized model of TMPS-HESM.

5. Validation of optimization results

(1) Comparison before and after optimization of cogging torque under different excitation current.

Set the excitation current range to 0–12 A and the simulation step size to 4 A to compare the cogging torque before and after optimization. After optimization, the cogging torque waveform is shown in Fig. 10. Figure 11 shows the comparison of the peak torque of the cogging before and after optimization.

Fig. 10.

Cogging torque at different excitation currents.

Fig. 11.

Comparison of cogging torque peak values.

Fig. 12.

Output torque before and after optimization.

Fig. 13.

Air gap magnetic density comparison.

The peak torque of the tooth groove is significantly suppressed compared with that before optimization. When the excitation current is 12 A, the peak value of cogging torque decreases from 2.027 N⋅m before optimization to 0.567 N⋅m, decreasing by 72%. When the excitation current is 4 A, the peak value of cogging torque decreases from 0.463 N⋅m before optimization to 0.2065 N⋅m, with a decreased amplitude of 55%. It shows that under different excitation current conditions, the cogging torque is inhibited to different degrees after optimization by this method, and with the increase of excitation current, the inhibition effect becomes more obvious.

(2) Average torque and torque ripple

The comparison of output torque before and after optimization is shown in Fig. 12. The average value of output torque increases from 8.64 N⋅m before optimization to 8.68 N⋅m. Meanwhile, the amplitude of torque waveform oscillation is greatly reduced, and the torque ripple decreases from 25% to 9.28%.

(3) Air gap magnetic density and no-load back EMF

The air gap flux density and no-load back potential before and after optimization are compared, as shown in Figs 13 and 14. It can be found that the waveform of air-gap flux density after optimization tends to be more sinusoidal, and the motor no-load back electromotive force tends to be flat slip.

Fig. 14.

No-load back EMF comparison.

6. Conclusion

A 2.5 kW tangential poly-magnetic hybrid exciter motor is analyzed in this paper. Aiming at the problem that cogging torque rises sharply when excitation current increases, a mathematical model of cogging torque of hybrid excitation motor was built by energy method. Particle swarm optimization algorithm was used to analyze the parametric model and a quadratic regression model between optimization parameters and optimization variables was obtained by sensitivity analysis. Finally, the optimal parameters are selected by entropy weight TOPSIS method. The results are satisfactory and the following conclusions are drawn:

According to the sensitivity analysis, the stator inner diameter and groove width of the motor have great influence on the cogging torque and the average torque, and the torque ripple is mainly related to the stator inner diameter and groove depth. Meanwhile, the quadratic response surface model obtained by Pareto diagram and sensitivity analysis has a high gradient, which also proves that the value range of the optimized variable tool is more appropriate.

Through co-simulation, the entropy weight TOPSIS method was used to screen the data, and the optimization result was obtained: when the excitation current was 12A, the coaming torque was reduced from 2.027 N⋅m before optimization to 0.567 N⋅m, a reduction of 72%, which solved the problem that the coaming torque increased sharply with the rise of the excitation current.

Not only the cogging torque is suppressed, but also the torque ripple, average torque and air gap magnetic density of the motor are optimized to different degrees, which achieves the effect of multi-objective optimization of the motor and improves the overall performance of the motor.

Footnotes

Acknowledgements

This work was partially supported by National Natural Science Foundation of China (51677089).

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