Minimization of Torque Ripple for an IPMSM with a Notched Rotor Using the Particle Swarm Optimization Method

Abstract

This paper presents a method to minimize torque ripple of a V-type IPMSM using the PSO (Particle Swarm Optimization) method with FEM. The proposed algorithm includes one objective function and three design variables for a notch on the surface of a rotor. The simulation model of the V-type IPMSM has 3-phases, 8-poles and 48 slots with 2 notches on the one-pole rotor surface. The arc-angle, length and width of the notch are optimized to minimize the torque ripple of the motor. The cogging torque of the model is reduced by 55.6% and the torque ripple is decreased by 15.5 %. Also, the efficiency of the motor is increased by 15.5 %.

1. Introduction

An Interior Permanent Magnet Synchronous Motor (IPMSM) has several advantages compared with a Surface Permanent Magnet Synchronous Motor (SPMSM). The IPMSM has a mechanically stable structure due to permanent magnets embedded in the rotor. It also has a slightly higher torque density per unit volume than other motors since the reluctance torque that results from an inductance difference between the d-axis and q-axis is combined with the magnetic torque. In addition, it is highly efficient and has a wide range of velocity control for low magnetic field operations. Thus, the IPMSM appears to be an excellent candidate for vehicle propulsion systems.

Unfortunately, the IPMSM produces significantly large torque ripple due to magnetic air gaps of similar length and mechanical structure. Therefore, the reduction of torque ripple that may cause vibration and acoustic noises becomes an increasingly critical issue in IPMSM. Currently, there are many studies on reducing such torque ripples, as well as cogging torque [1-3]. There are also a number of suggestions made on how to achieve this such as employing skew, adjusting of the number of slots per pole, controlling of the magnetizing region of the permanent magnet, and regulating the input current [4-5].

In this study, a 50 kW V-type IPMSM is modeled to reduce torque ripple by introducing two notches in the one-pole rotor surface as shown in Fig. 2. In the proposed algorithm, the position and size of the notch are optimized by an optimization algorithm, which uses a Latin Hypercube Sampling (LSH) strategy with a Response Surface Method (RSM) based on the multiobjective Pareto Optimization procedure. For obtaining an effective sampling point, PSO is also employed in the algorithm. To verify the proposed method, the simulation model is applied to and analyzed by the FE program [6-7].

 

2. Simulation Model and Specifications

Table 1 describes the main specifications of the simulation model of the IPMSM. The capacity of the motor is 50 kW with a speed of 1200~1500 rpm, a rated voltage of 500 V, 3 phases, 8 poles and 48 slots. The current has a wide range according to its operational conditions. In the FE simulation, 200 A is used as the rated current. Fig. 1 shows a cross-section of the simulation model (one pole) and the FE meshes. The total number of meshes is approximately 7,400. A double-layer mesh is used in an air gap to calculate the force and torque of the model using a rotating air gap and external circuit. Fig. 2 shows the optimization model of the notch, which has two design variables (i.e., notch angular length, ln, and pitch angle between notches, θN ) and one constraint. The notch angular length, ln, is constraint between 0.5 and 3[°], the pitch angle between notches, θN, is constraint between 8 and 20 [°] while the notch depth, dn, is kept at 0.5[mm]. Fig. 2(a) shows an FEM model of the motor (one pole) while Fig. 2(b) shows the FE meshes.

Table 1.Specifications of the 50 kW IPMSM model

Fig. 1.Simulation model of a V-type IPMSM (one pole) with FE meshes

Fig. 2.Notch model of a rotor’s surface for a V-type IPMSM

Fig. 3(a) shows the flux density distribution of the optimized model around the PM and the notch. Fig. 3(b) shows the flux density profiles along the center line of the air gap with/without the notch on the rotor’s surface. The horizontal axis of Fig. 3(b) is ‘Rotor position’; that is one pole of PM(0 to 180 degree). In the case of the notch, the average flux density is slightly lower than for the case of a rounded surface, as expected.

Fig. 3.(a) Flux density distribution around the notch; (b) Flux density profiles along the center line of the air gap (the average flux density is 0.703[T])

 

3. Optimization Algorithm with PSO

3.1 Cogging torque calculation

Torque perturbations arise from the variations of magnetic energy associated to a change in airgap length, as the rotor rotates without load current. Since the energy change in a PM and iron is negligible compared to that of air, the magnetostatic energy is given as [5],

Where µ0, P(θ) and , Fm (θ, a) represent the permeability of air, the air gap permeance function, and the air gap MMF function, respectively. The cogging torque for a PM brushless motor can be expressed as

where Ls , Rs , Rm and GnNL , BnML denote the stack length, the stator bore radius, the magnet outer radius, and the corresponding Fourier coefficients of the relative air gap permeance and the flux density function, respectively [5]. Eq. (2) reveals that the cogging torque is first governed by NL , GnNL and BnNL with the fundamental period of 2π /NL , where NL is the least common multiple of number of stator and rotor poles.

3.2. LHS and RMS

Latin Hypercube Sampling is a “space filling” experimental design strategy [6]. The RSM with a multi-quadric radial basis function is widely used for global interpolations. It is also very efficient due to its smoothness and fitting ability when given a limited number of sampling points. As such,

where x is the design parameter vector, ci is the coefficient corresponding to the i-th sampling point xi , g(x) is the multi-quadric radial basis function, and r is so called “shape parameter” whose purpose is to control the curvature of the single basis function near the center point [7-8].

3.3. PSO (Particle Swarm Optimization Method)

Particle Swarm Optimization is a heuristic search technique that simulates the movements of a flock of birds which aim to find food. This method is advantageous in terms of relative simplicity, quick convergence requiring less computing, and effective locating of all local optimum points. These points in turn are motivated by the clustering PSO algorithm [9-10].

Fig. 4 shows the procedure of the proposed optimization program with PSO. The PSO algorithm has five steps to determine an optimal sampling point: 1) identify main particle movements, 2) locate main particles that form a couple, 3) isolate a non-stop couple, 4) eliminate near-by couples, and 5) assess movement stopping criterion. The algorithm ends when the iteration number is reached at a pre-set figure, when an optimized position is found, or when the criteria of the objective function satisfy a predefined value.

Fig. 4.Flow chart of PSO based optimization algorithm

 

4. Simulation and Results

Two design variables, ln , θN , of the notch are constrained as shown in Fig. 4. Using the proposed optimization algorithm with PSO, 125 sampling points are selected through 5 iterations. The objective function to minimize the torque ripple is described as Eq. (4).

where τi+1 is the torque ripple at (i+1)-th iteration and τi is the average torque at i-th iteration. Fig. 5 shows three cases of the notch shape at a different length and arc angle; Fig. 5(a) θN =20[°], Fig. 5(b) θN =8[°] and Fig. 5(c) at θN =14.75[°]. As observed, Fig. 5(c) is the optimized model. Table 2 describes the optimized notch length, ln , the arc angle of θN and the torque ripple at each step of the optimization program. It should be noted that as the step increases, the torque ripple converges to 96.3 N.m. Fig. 6 shows the average torque ripple of the optimized model as a function of the iteration steps. After the 5th iteration step, the optimized arc angle, θN , is 14.74[°] and the notch angle, ln , is 2.38[°].

Table 2.Optimized variable and torque ripple at each step

Fig. 5.Simulation model of the notch on the rotor’s surface

Fig. 6.Average torque ripple of the optimized model as a function of the iteration steps

As shown in Fig. 7, the optimized cogging torque is reduced by 56.4% from 7.30 [N.m] to 3.19 [N.m]. Also, the torque ripple is reduced by 4.41 % as seen in Fig. 8. With the notch, the optimized Tmax is 437.6[N. m], Tmin is 340.8 [N.m], and Tavg is 389.1 [N.m]. However, without the notch (i.e., for a rounded surface), Tmax is 448.9 [N.m], Tmin is 335.1 [N.m], and Tavg is 389.2 [N.m].

Fig. 7.Comparison of cogging torque between the round and the notched surface model(The final value of θN =14.74[°] and ln =2.38 [°])

Fig. 8.Comparison of torque profiles between the round and the notched surface model model(The final value of θN =14.74[°] and ln =2.38 [°])

The torque ripple ratio can be calculated as in Eq. (5),

where Trr , Tmax , Tmin , Tavg denote the torque ripple ratio, the maximal torque, the minimal torque, and the average torque, respectively.

 

5. Conclusions

In this study, an optimization algorithm is proposed to minimize the torque ripple of a V-type IPMSM using a particle swarm optimization method with FEM. In the algorithm, an LHS strategy is incorporated with RSM based on a multi-objective Pareto Optimization procedure. PSO is also used to obtain an effective sampling point.

In order to verify the effectiveness of the proposed algorithm, the notch shape on the rotor’s surface of a 50 kW IPMSM is optimally formed with 2 design parameters and several constraints for minimizing torque ripple. The simulation model of the V-type IPMSM has 3-phases, 8-poles and 48 slots with 2 notches on the one-pole rotor surface. The arc-angle, length and width of the notch are optimized to minimize the torque ripple of the motor. In so doing, the cogging torque of the optimized model is thus reduced by 56.4% and the torque ripple is decreased by 4.41 %. In addition, the efficiency of the motor is increased by 15.5 %.