ANN-based optimization approach devoted to the sizing of arbitrary rotor pole geometries of permanent magnet motors for electric motorcycle

Abstract

Various design approaches have been studied for performance improvement of permanent magnet motors, especially for electric vehicles application. This paper deals with an effective concept that is based on adopting non-traditional geometries for permanent magnet poles. The study focuses on a general methodology for optimal sizing of the rotor poles’ pre-defined geometrical parameters considering certain objectives. For this purpose, the artificial neural network is employed for creating an accurate and simple model to be used in a multi-objective optimization procedure. An interior crescent-shaped permanent magnet motor for an electric motorcycle is studied as a typical case study to prove the performance of the proposed method. Finite element models are developed to create the required dataset for the modeling stage as well as to verify the results.

Keywords

Permanent magnet motors design optimization permanent magnet shaping ANN model torque profile electric motorcycle

1. Introduction

Permanent magnet (PM) machines have gained significant attention in high-performance applications. This is mainly due to their outstanding advantages in terms of high efficiency and high power density [1,2]. Nowadays, they are used in most of the commercial electric and hybrid electric vehicles, especially in light duty designs [3]. In such applications, it is usually desired to obtain excellent torque profile (i.e. large mean torque and low pulsating torque) [4–6], which determines several performance characteristics of the system, including its output power capability as well as mechanical vibrations and life endurance. Therefore, it can find several studies on design modifications of PM machines considering their torque characteristics modifications [7,8]. Among the various approaches, PMs design can be an effective solution, which includes their optimal geometry definition, location, and sizing. This is a challenging item of PM machines design since it determines not only the machine performance but also a significant portion of the total price. Conventionally, simple basis geometries are assumed for PMs, such as the rectangular shape that is widely used in interior PM rotors. In such designs, most of the efforts are focused on the optimization of the PMs arrangements and/or their location [9–11]. The main advantage of such simple PM shapes is their relatively easy and cheap manufacturing. However, better performance characteristics can be achieved by defining the more complex geometries [12]. The additional cost of the new PM geometries can be justified (1) in high-performance applications, where excellent performance is of great importance, and (2) in mass production cases by reducing the total PMs mass thanks to the potentials of the new pole shapes. Both of the above merits can be achieved if proper design optimization is adopted for the new machine (i.e. the machine with new PMs geometry).

PMs design procedure includes two main steps: (1) defining a proper basis geometry for PMs, (2) optimally defining the geometrical parameters of the pre-defined PMs geometry or optimally sizing the PMs. This paper will address the second step, which is an optimization problem. However, due to the complexity of the alternated PMs geometry, their design optimization is challenging. This is mainly due to the difficulties associated with developing an analytical model, which can consider all the geometrical features of a complex PM shape. On the other hand, finite element (FE) model, which is known for its high accuracy and high flexibility in the modeling of arbitrary geometries, is too much time consuming to be used within a search-based optimization procedure.

Considering the mentioned issues, an efficient approach is used in this paper to optimally size the PMs with arbitrary complex basis geometries. The paper utilizes a mathematics-based model instead of the mentioned physics-based ones. The modeling procedure can be considered as a regression process over a set of design samples that can be collected via Design Of Experiments (DOE) techniques. Various basis functions could be used for the function approximation task, such as Response Surface Methodology [13,14], Kriging model [15,16] and Artificial Neural Network (ANN) [17].

Considering an electric motorcycle application, the motor specification is determined as the first step in this paper, then FE model is utilized to represent the experiment and to collect the initial dataset. Next, ANNs are used as the basis function of the approximated model. Finally, a multi-objective intelligent optimization method is adopted to solve the PM sizing optimization problem. In order to prove the methodology, a small power PM synchronous motor with crescent-shaped PMs is analyzed as the case study.

2. Specifications of motorcycle

The required rated power (P_r) to drive a motorcycle is determined via the equation of motion, considering the vehicle speed (V ) and propulsion force (F) [18]: $\begin{eqnarray}\displaystyle P_{r}\text{} & =\text{} & \displaystyle FV\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle (M∼a+0.5\times {\rho}C_{D}A_{f}V^{2}+{\mu}_{r}M∼g)V\end{eqnarray}$ (1) where M is the total mass (i.e. the motorcycle + passenger), a is the desired acceleration, g is the gravitational acceleration, and μ_r is the tire-road friction coefficient Also, the second term of the force corresponds to the drag force exerted to the motorcycle, in which 𝜌 is the air density, C_D is drag coefficient, and A_f stands for the vehicle frontal area. Table 1 illustrates the motorcycle specifications along with the required electric motor rated parameters.

Table 1

Electric motorcycle and its electrical motor specifications

Parameter	Value	Unit
Speed (V )	80	km/h
Mass (M)	190	kg
Rolling friction coefficient (μ_r)	0.02	-
Drag coefficient (C_D)	0.4	-
Frontal area (A_f)	0.6	m²
Electric Motor:
Rated power (P_r)	5	kW
Rated speed (n)	3500	rpm
Rated current (I_r)	14.8	A _rms
Pole numbers	8	-
Phase numbers	3	-

3. Initial design

Figure 1 illustrates the case study that is an interior PM (IPM) synchronous motor. The crescent shape of the PMs is considered as a typical example of a non-conventional PM geometry, which has the potential to reduce the space field harmonics and increase the machine’s saliency ratio. The initial design is performed by using a simple set of the analytical sizing equations. First, the main dimensions of the machine (i.e. rotor diameter and stack length) should be calculated [19]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{R}=\sqrt[3]{{\displaystyle \frac{P_{r}}{{\eta}{\pi}^{2}K_{D}K_{W}{\sigma}_{t}n/60}}}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L=K_{D}D_{R}\end{eqnarray}$ (3) where D_R is the rotor diameter, and L is the stack length. Also K_D and K_W denote the diameter ratio and fundamental winding factor, respectively, n is the rotor speed in revolutions per second, and 𝜂 is an approximation of the motor efficiency. The tangential Maxwell stress tensor (σ_t) is obtained via fundamental magnetic loading $((\hat{B}_{g1}))$ and electrical loading (A) [19]: $\begin{eqnarray}\displaystyle {\sigma}_{t}={\displaystyle \frac{1}{2}}\hat{B}_{g1}A. & & \displaystyle\end{eqnarray}$ (4) The slot area is determined considering the required space for the coil that is: $\begin{eqnarray}\displaystyle S_{slot}={\displaystyle \frac{2N_{c}I_{ph}K_{fill}}{J}} & & \displaystyle\end{eqnarray}$ (5) in which K_fill and J are respectively the slot fill factor and conductors’ current density. Also, each coil turn number can be approximated as (for single-layer winding): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle N_{c}={\displaystyle \frac{{\varepsilon}V_{ph}\times \sqrt{2}m}{{\pi}^{2}K_{W}\overline{B}_{g}D_{R}LQn/120}}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\varepsilon}=E_{ph}/V_{ph}.\end{eqnarray}$ (7) In the Eq. (6), V_ph and E_ph are phase voltage and electromotive force (EMF), respectively, m is the phase number, B _g represents the air-gap mean flux density, and Q is the number of slots.

Fig. 1.

2D view of the case study: crescent shaped IPM synchronous motor.

A complete PM shape can be defined by the variables illustrated in Fig. 2.

Fig. 2.

Geometrical variables of the crescent shaped PM.

Table 2

Calculated parameters of the initial design

Parameters	Values (mm)
Stator outer diameter	200
Rotor diameter	123
Stack length	74
R ₁	30
R ₂	50
D _m	6.5
W _m	20
L _g	0.75

Among all the existing geometrical variables, three variables are selected as the optimization variables in this study, while the others are assumed to be constant. The three optimization variables are D_m, W_m, and L_g. As the first step, an initial size for PMs should be defined, and to do so a simplified approach is employed in this paper. To make an initial sizing for the PMs it is assumed that the crescent-shaped PMs are simplified to an almost rectangular-shaped pieces, the equivalent width and height of which are respectively W _PM and h _PM. Now, the relation between the actual PMs dimensions (see Fig. 2) and the equivalent width and height should be computed. The following equation is used for initial PMs sizing, considering their desired working point (B_m): $\begin{eqnarray} \displaystyle \hspace{-25.0pt}\overline{h}_{PM}\text{} & =\text{} & \displaystyle {\displaystyle \frac{1}{\sqrt{R_{1}^{2}-(d+R_{1}-R_{2}-D_{m})^{2}}}}\nonumber\\ \displaystyle \hspace{-10.0pt}\text{} & \text{} & \displaystyle \times \left[\displaystyle \int _{0}^{\sqrt{R_{1}^{2}-(d+R_{1}-R_{2}-D_{m})^{2}}}\left(\sqrt{R_{1}^{2}-x^{2}}-(d+R_{1}-R_{2}-D_{m})\right)\mathrm{d}x\right.\nonumber\\ \displaystyle & & \displaystyle \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \left.-\displaystyle \int _{0}^{W_{m}/2}\sqrt{R_{2}^{2}-x^{2}}-d\right] \end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle d\text{} & =\text{} & \displaystyle R_{2}\cos \left(\sin ^{-1}\left({\displaystyle \frac{W_{m}}{2R_{1}}}\right)\right)\nonumber\\ \displaystyle \overline{h}_{PM}\text{} & =\text{} & \displaystyle {\displaystyle \frac{{\mu}_{rec}B_{m}g_{e}}{B_{r}-B_{m}}}\end{eqnarray}$ (9) where μ_rec and B_r are PMs recoil permeability and residual flux density, respectively. The equivalent air-gap (g_e), includes the effects of the slots and saturation respectively through the carter factor (K_C) and saturation factor (K_S): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle g_{e}=K_{c}K_{s}g\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{c}={\displaystyle \frac{({\pi}D_{R}/Q)}{({\pi}D_{R}/Q)+{\xi}W_{os}}}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\xi}={\displaystyle \frac{2}{{\pi}}}\left(\tan ^{-1}{\displaystyle \frac{W_{os}}{2g}}-{\displaystyle \frac{2g}{W_{os}}}\times \ln \left(\sqrt{1+\left({\displaystyle \frac{W_{os}}{2g}}\right)^{2}}\right)\right).\end{eqnarray}$ (12)W_os stands for stator open slot width. Also, to approximate the rotor pole width, the following equations can be used for its average width ( W _PM): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{W}_{PM}=R_{1}\sin ^{-1}\left({\displaystyle \frac{\sqrt{R_{1}^{2}-(d+R_{1}-R_{2}-D_{m})^{2}}}{R_{1}}}\right)+R_{2}\sin \left({\displaystyle \frac{W_{m}}{2R_{2}}}\right)\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{W}_{PM}={\displaystyle \frac{\overline{B}_{g}}{{\sigma}\times B_{m}}}\times {\displaystyle \frac{P}{{\pi}D_{R}}}\end{eqnarray}$ (14) σ indicates the PMs leakage factor that could be considered within 0.85–0.95. Finally, the initial design parameters are illustrated in Table 2.

4. ANN-based optimal design

As was mentioned earlier, the paper focuses on the optimal sizing of the PMs with non-conventional basis geometries. To deal with an optimization problem, three steps must be performed. First, a mathematical representation of the optimization problem should be determined, in which the objective function(s) and design variables as well as the possible constraints are formulized. Then, a proper modeling approach is employed to compute the objective(s). Finally, the optimum points are searched via an appropriate methodology. These steps are going to be presented in the following.

Fig. 3.

Sequential procedure of the design optimization method.

Fig. 4.

Partial one pole FE model of a typical sample of the studied motor: (a) Calculated mesh, (b) computed Flux lines.

4.1. Optimization problem

PMs design affects various characteristics of an electrical machine. Remarkable portion of the total cost of a PM machine depends on the PM usage. Hence it is important from the cost point of view that is always a crucial perspective of small electric vehicles.

Moreover, the air-gap flux density distribution, which determines various performance characteristics of the machine strongly depends on the PMs design. To more clarify this point, it is good to use the Eq. (15) that describes the temporal-space distribution of the PMs field (excitation field) in the air-gap of a PM machine: $\begin{eqnarray}\displaystyle B_{PM}({\theta},t)=F_{PM}({\theta},t).{\Lambda}({\theta},t) & & \displaystyle\end{eqnarray}$ (15) where B_PM is the air-gap flux density, F_PM is the magnetomotive force of the PMs and 𝛬 stands for the specific permeance of the air-gap (the ratio of the magnetic permeability to the length of the flux pass in the air-gap), and θ and t indicate the spatial position and time, respectively. Since B_PM directly contributes to the torque production, it is a determining factor for both mean torque and pulsating torque characteristics. It should be noted that in an interior PM rotor the PM shape design influences both F_PM and 𝛬 functions: 𝛬 is affected by both the stator and rotor cores structure, and the PMs shape and magnetization determine F_PM The mean torque is the result of the “fundamental F_PM” and the “average 𝛬” products, while the other harmonics components of F_PM and 𝛬 produce pulsating torque. Therefore, one can clearly realize the importance of PMs design for Torque profile improvement in an electrical machine In this study, the torque profile is characterized by two items: (1) mean torque, and (2) ripple rate. The first one directly determines the torque density, and the second item affects the level of mechanical vibrations and acoustic noise, which all are important for an electric vehicle drive application.

Fig. 5.

Developed multi-layer perceptron architecture.

Fig. 6.

Training and testing results of three networks, evaluated by MSE (Mean Square Error) criterion, (a) for mean torque (b) and torque ripple.

Taking all the mentioned requirements of an electric motorcycle three characteristics are considered in this paper to define the optimization problem, which are mean torque T_mean, torque percent ripple rate (T_ripple), and PMs mass (M_PM). In this regard, two-objective formulization is assumed in this study: $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{}l@{}} \mathit{Max}\,{:}∼T_{\mathit{mean}}/M_{PM}\\ \mathit{Min}\,{:}∼T_{\mathit{ripple}}\% \end{array} \right. & & \displaystyle \end{eqnarray}$ (16)

4.2. Modeling procedure

Assuming a basis geometry for PMs that is defined by a general set of the parameters $\{x_{i}\},∼i=1,\ldots ,n$ , a set of samples must be collected and analyzed via experiments to create a dataset. Usually, FE model is used instead of the laboratory experiments, since it can provide arbitrary precision. Then a basis mathematical function must be selected, in order to fit the collected samples. ANNs are good candidate for this step since they are capable to fit various complex functions. This is thanks to the flexibility in improving their architecture (i.e. neuron activation functions and connections). The fitting process is equivalent to the well-known ANN learning methods, which can be carried out through various supervised learning approaches.

Table 3
Comparison of objective functions for initial and optimal design

Characteristic Initial design Optimal design Unit

Mean torque/PM mass 14.6∕0.7738 = 18.86 14.7∕0.7326 = 20.06 N.m/kg

Torque ripple 30.5 17.1 %

Characteristic	Initial design	Optimal design	Unit
Mean torque/PM mass	14.6∕0.7738 = 18.86	14.7∕0.7326 = 20.06	N.m/kg
Torque ripple	30.5	17.1	%

Table 4

Comparison of the predicted values (via ANN) and computed values (FEA) of the optimal design

Characteristic	Predicted by ANN	Resulted from FE model	Unit
Mean torque	14.68	14.67	N.m
Torque ripple	16.99 %	17.1 %	%

Fig. 7.

Calculated Pareto front.

Fig. 8.

Initial and optimal designs (a) EMF, and (b) Full load torque waveform.

The sequential modeling procedure can be seen within the flowchart of Fig. 3, which illustrates the entire design steps of this study.

In this study, a factorial method is considered to collect the initial dataset and 847 samples are required to cover the variables space. 2-dimensional timestepping FE model was used to calculate all samples with high precision. Figure 4 shows a typical sample that is modeled via FE method. Due to the existing symmetry in the machine geometry, onepole partialmodel is developed to reduce the modeling cost. Next, a multi-layer perceptron ANN is selected as the basis function to fit the optimization objectives, and the back-propagation method is used for its training. Separated ANNs are employed for each objective function, i.e. percent torque ripple, and mean torque per PM mass.

Two hidden layers are used for each ANN, as shown in Fig. 5, while the neuron numbers in each layer can be selected via try and error.

The activation function of the hidden layers is “hyperbolic tangent”, therefore the relation between ANN’s output and inputs can be formulated as: $\begin{eqnarray}\displaystyle Y=\displaystyle \mathop{\sum }_{i=1}^{N_{2}}\left[W_{i}\left(\tanh \left(b_{i}+\displaystyle \mathop{\sum }_{j=1}^{N_{1}}W_{ji}\left(\tanh \left(b_{j}+\displaystyle \mathop{\sum }_{p=1}^{3}W_{pj}\times U_{p}\right)\right)\right)\right)\right] & & \displaystyle\end{eqnarray}$ (17) Where U, W, b and Y are inputs, weights, biases and outputs, respectively. Also, N₁ and N₂ indicate the number of neurons in the first and second hidden layers. The ANNs are trained by 95% of the samples while the rest of dataset are used for the testing level. Figure 6 shows the results of training and testing the networks corresponding to mean torque and torque ripple, in order to verify the regression accuracy. It should be noted that the dataset is normalized within the range (−0.6, 0.6).

4.3. Optimization

After developing a proper model, the only remained task is to find the best solution for the design variables. This is usually done within an optimization procedure, which tries to find the extreme corresponding to the certain objective(s). As was seen previously, a multi-objective optimization problem is introduced in this study. From the mathematical point of view, there are several approaches to represent and solve a multi-objective optimization problem [20]. Among all, Pareto optimization is a popular method that gives a set of optimal solutions. This approach can be utilized within various search-based optimization methods such as evolutionary algorithms. Non-dominated Sorting Genetic Algorithm (NSGA II) is employed in this study to find the optimal Pareto solutions. Figure 7 illustrates the resulted Pareto front that contains the non-dominated set of the optimal solutions. The final solution is usually defined manually by the expert, and the knee point can be a good selection. The selected design is defined in Fig. 7 that corresponds to [D_m, W_m, L_g] = [6.42, 17.54, 0.4]. The selected design is analyzed by FE model, to evaluate the performance of the approximated model. Figure 8 shows the torque waveform and induced voltage of the selected final design in comparison with the initial design. The detailed information is also presented in Table 3. Moreover, the comparison of FE and ANN predictions shown in Table 4, confirm the precision of the proposed algorithm.

5. Conclusion

An efficient and accurate framework was presented in order to optimally design the unconventional basis geometries of the PMs in PM machines. A surrogate model based on ANN was developed to replace the traditional physicsbased models, which can efficiently consider all the geometrical features of an arbitrary PM shape. Hence, a search-based optimization algorithm was easily employed to find the optimal design. The proposed approach can be adopted to achieve high performance designs with high flexibility. An interior crescent-shaped PM synchronous motor was proposed as an example of an unconventional PM geometry and the optimization methodology was successfully applied to the case study.

References

Gieras

J.F.

Wang

R.-J.

and Kamper

M.J.

, Axial Flux Permanent Magnet Brushless Machines, Springer Science & Business Media, 2008.

Schofield

Giraud-Audine

Powell

and Howe

, Optimal design of permanent magnet brushless AC machines for electric vehicle applications, International Journal of Applied Electromagnetics and Mechanics15(1-4) (2002), 143–148.

El-Refaie

A.M.

, Motors/generators for traction/propulsion applications: A review, IEEE Vehicular Technology Magazine8(1) (2013), 90–99.

Zhu

Liu

and Wang

, Electromagnetic performance analysis of interior PM machines for electric vehicle applications, IEEE Transactions on Energy Conversion33(1) (2017), 199–208.

Zhao

Lipo

T.A.

and Kwon

B.-I.

, Comparative study on novel dual stator radial flux and axial flux permanent magnet motors with ferrite magnets for traction application, IEEE Transactions on Magnetics50(11) (2014), 1–4.

Zhu

Chen

Xie

Liu

and Wang

, Electromagnetic performance comparison of multi-layered interior permanent magnet machines for EV traction applications, IEEE Transactions on Magnetics54(11) (2018), 1–5.

Yokoi

and Higuchi

, Stator design of alternate slot winding for reducing torque pulsation with magnet designs in surface-mounted permanent magnet motors, IEEE Transactions on Magnetics51(6) (2015), 1–11.

Zhu

Kemp

Moule

and Williams

, Optimal step-skew methods for cogging torque reduction accounting for three-dimensional effect of interior permanent magnet machines, IEEE Transactions on Energy Conversion32(1) (2016), 222–232.

Putri

A.K.

Hombitzer

Franck

and Hameyer

, Comparison of the characteristics of cost-oriented designed high-speed low-power interior PMSM, IEEE Transactions on Industry Applications53(6) (2017), 5262–5271.

10.

Hwang

M.-H.

Han

J.-H.

Kim

D.-H.

and Cha

H.-R.

, Design and analysis of rotor shapes for IPM motors in EV power traction platforms, Energies11(10) (2018), 2601.

11.

Di Barba

Mognaschi

M.E.

Rezaei

Lowther

D.A.

and Rahman

, Many-objective shape optimisation of IPM motors for electric vehicle traction, International Journal of Applied Electromagnetics and Mechanics (2019), s1–s14.

12.

Hong

and Yoo

, Shape design of the surface mounted permanent magnet in a synchronous machine, IEEE Transactions on Magnetics47(8) (2011), 2109–2117.

13.

Moghaddam

H.A.

Vahedi

and Ebrahimi

S.H.

, Design optimization of transversely laminated synchronous reluctance machine for flywheel energy storage system using response surface methodology, IEEE Transactions on Industrial Electronics64(12) (2017), 9748–9757.

14.

Xiang

Zhu

Quan

Zhang

and Fan

, Multilevel design optimization and operation of a brushless double mechanical port flux-switching permanent-magnet motor, IEEE Transactions on Industrial Electronics63(10) (2016), 6042–6054.

15.

Lei

Zhu

Guo

Shao

and Xu

, Multiobjective sequential design optimization of PM-SMC motors for six sigma quality manufacturing, IEEE Transactions on Magnetics50(2) (2014), 717–720.

16.

Nabeta

S.I.

Kriging models and torque improvements of a special switched reluctance motor, in: 2007 IEEE International Electric Machines & Drives Conference, Vol. 1, IEEE, 2007, pp. 559–563.

17.

Meo

Zohoori

and Vahedi

, Optimal design of permanent magnet flux switching generator for wind applications via artificial neural network and multi-objective particle swarm optimization hybrid approach, Energy Conversion and Management110 (2016), 230–239.

18.

Nair

S.S.

Nalakath

Santhi

Dhinagar

S.J.

Deshpande

Y.B.

and Toliyat

H.A.

, Design aspects of high torque density-low speed permanent magnet motor for electric two wheeler applications, in: 2013 International Electric Machines & Drives Conference, IEEE, 2013, pp. 768–774.

19.

Pyrhonen

Jokinen

and Hrabovcova

, Design of Rotating Electrical Machines, John Wiley & Sons, 2013.

20.

Chiandussi

Codegone

Ferrero

and Varesio

F.E.

, Comparison of multi-objective optimization methodologies for engineering applications, Computers & Mathematics with Applications63(5) (2012), 912–942.