With the wide application of interior permanent magnet synchronous machines (IPMSM) in electric vehicles, the performance of IPMSM, such as high efficiency, high torque, low torque ripple, high flux-weakening capability and less magnet mass, to meet the requirements of electric vehicles, is important [1, 2, 8]. Therefore, the optimal design of IPMSM should be conducted to obtain better performance.
The traditional method of structure optimization for motors is optimizing each parameter one by one. However, the mutual restriction between structure parameters and their complex coupling with electromagnetic performance, makes the method, which independently optimizes each single structure parameter, not only complicated and time consuming but it also produces a poor optimization result. With the development of optimal design for motor and computer techniques, some quick optimization, which combines optimization algorithms with computer software, has successively appeared.
The Taguchi method [2, 3, 4, 5, 6, 7] is a robust local optimization algorithm. It can analyze the best combination of each structure parameter using a minimum number of “experiments” by implementing an orthogonal experiment array to perform a multi-objective optimization for the motor. But its defect in common with other local optimization algorithms is that its optimization result depends on the selection of the initial point, which means it only searches the adjacent range of the initial point and its global searching capability is poor.
On the other hand, the genetic algorithm (GA) [8, 9, 10] is an intelligent stochastic search algorithm, which has robust global searching ability. This could fill the gap of the Taguchi method in the optimal design of motors. But the poor local search capability of GA makes it easily trapped in a local extremum at the later evolution stages.
This paper proposes a hybrid genetic algorithm (HGA), which is a combination of GA and the Taguchi method. The HGA not only can overcome the defect of the poor global searching capability for the Taguchi method, but can also cover the poor local searching ability of GA. The HGA will be used to optimize the rotor structure parameters of an IPMSM for an electric vehicle application considering maximum torque and efficiency, minimum torque ripple and iron loss as the optimization objective. The optimization results of the HGA design are compared with the initial and GA design results.
2 Model of the IPMSM
The initial design of the IPMSM is a 15-slot, 8-pole, 1.5-kW, 1000-r/min machine with V-shaped barriers in the rotor as shown in Figure 1(a). The rotor inside radius Rri is fixed at 22.5 mm and the other structure parameters of the rotor shown in Figure 1(b) are optimized by the HGA. The IPMSM design with V-shaped rotor barriers is widely used in electric vehicles because of its wide flux-weakening capability .
Consider the average torque Tavg, efficiency η, torque ripple coefficient Kmb and core loss PFe as the optimization objectives. For the first two, a larger value is better; and for the last two, a smaller value is better.
The torque ripple coefficient Kmb is the quantity which measures the degree of torque ripple for a machine, and it is defined as follows: (1)
where Tmax and Tmin is the maximum and minimum instantaneous torque under steady state, respectively.
The calculation formula of efficiency η is shown as: (2)
where ω is the rated angular speed and Pcu is the copper loss.
The implementation process for the HGA is: first, perform adequate global searching for the initial design of the IPMSM using the GA optimal scheme; then, take advantage of the local searching capability of the Taguchi method by taking the GA optimized design and optimizing it further by the Taguchi method.
3 Global searching by GA
Finite element (FE) software is used to analyze the performance of the designs as part of the optimization process. The GA optometric of the software is applied to conduct GA optimization for the IPMSM. The optimization considered nine structural parameters of the rotor shown in Figure 1, these are: outer radius of rotor barriers Rb, minimum distance between side magnets Dm, magnetic bridge length Hrib, bottom width of rotor barriers O1, depth of rotor yokeO2, width of magnetic bridge between two adjacent poles Rib, thickness of magnet Th, width of magnet Wid and rotor outside radius Rro. The range of the variables above is shown in Table 1.
3.1 The Establishment of the Objective Function and its Standardization
In the GA optimization, the objective optimization is realized using a defined cost function, and the minimum point of the cost function corresponds to the optimum point. For a multi-objective optimization, a weight coefficient is assigned to each objective according to its degree of importance, wi (the objective function with larger wi is more important). The sum of the products of each objective value and their weighting degree wi to calculate the total cost function. The constraint formula of the objective functions is shown as (3), and total cost function formula shown as (4). (3)
where gi(x) is the i-th objective function expression; conditioni is the corresponding conditional operator “ < ”, “=” and “>” and Gi is the given goal value of the i th objective function. (4)
where N is the number of objective functions; εi is the residual target and its value reflects the degree that the simulated response values of the objective function deviate from the given goal value range. As shown in Figure 2, if the response value is within the goal value range, then εi = 0, if not, εi is the difference of the response value and the goal value.
In this paper, four objectives of the machine are used for optimization, that is the average torque Tavg, torque ripple coefficient Kmb, iron loss PFe and efficiency η. Due to the significant difference between these four objective function values, they are standardized to a goal value between 1 and 10.
For instance, the optimization goal value of the average torque Tavg is 16 Nm and the value range is 12 Nm ~ 16 Nm, thus its objective function g1 is defined as: (5)
It can be seen from formula (5) that the value of g1 is 1 when the average torque Tavg is 16 Nm, and g1 is 10 when the average torque Tavg is 12 Nm.
Similarly, the optimization goal value of the torque ripple coefficient Kmb is 1.6% and value range is 1.6%~2.2%, and its objective function g2 is defined in formula (6); set the optimization goal value of the iron loss PFe to 28 W and value range is 28 W ~ 35 W, defined its objective function g3 as formula (7); set the optimization goal value of efficiency η to 96% and value range is 90%~96%, defined its objective function g4 as formula (8).
In this optimal design, the conditioni of the formula is “<”, thus the corresponding constraint formula of the four objective functions is shown as: (9)
Then, set the importance degree of each optimization objective. The importance degree of all four objective functions is set to 1 in this paper. Combining Figure 2(a) with the total cost function formula (4) gives the total cost function of the average torque Tavg, torque ripple coefficient Kmb, iron loss PFe and efficiency η as follows: (10)
3.2 Simulation results of GA design
After the optimization variables and objective function are defined, the values of the five major parameters of GA optimization are set: population size popsize = 20, selection pressure Sp = 10, crossover probability Pc = 0.75, mutation probability Pm = 0.01 and iteration termination generation N = 500, with the other parameters chosen as their default values. The GA optimization results are shown in Table 2 and Figure 3. They indicate that the average torque Tavg and efficiency η of the GA design are increased by 6.7% and 5.5%, respectively, compared with the initial design, the torque ripple coefficient Kmb and iron loss PFe is respectively reduced by 47.6% and 4.9%, as well. In addition, the PM mass of the GA design is reduced from 337.5 g to 306.5 g and reduced 9.2% compared with the initial design.
4 Local searching by taguchi method
The performance of the IPMSM is significantly improved by GA optimization. However, local searching by the Taguchi method can effectively overcome the defect of GA, which is easily trapped in a local extremum during the later evolution stages. The conducted Taguchi optimization for GA scheme can further optimize the performance of the machine.
In the applied Taguchi method for optimizing the structure parameters of the machine, an orthogonal array should be established according to the number of optimization variables and their levels. In the orthogonal array, the value taken by each parameter at each experiment is imported into optometric of the FE software and analyzed to calculate the performance of the machine.
4.1 Established orthogonal experiment array
Starting from the GA optimal design, take six structure parameters of the rotor: the minimum distance between side magnets Dm, magnetic bridge length Hrib, depth of rotor yoke O2, width of magnetic bridge between two adjacent poles Rib, thickness of magnet Th and width of magnet Wid shown in Figure 1 and use them as optimization factors for the Taguchi optimization design, because these six parameters have greater effect on the performance of IPMSM. The values of the other parameters are maintained at the same value as the GA optimal design. These 6 optimization parameters are represented as A, B, C, D, E and F, respectively. Every optimization factor can take 5 levels and its values are shown in Table 3. The orthogonal array L25(56), which is established according to the number of optimization variables and its levels, is shown in Table 4.
4.2 Finite element simulation
FE simulation is conducted according to the value of each level taken by each optimization factor for each experiment in Table 4 and used to obtain the result for the average torque Tavg, torque ripple coefficient Kmb, iron loss PFe and efficiency η. The calculated results for each experiment are shown in Table 4.
4.3 Analysis of mean
Then the mean of one performance for each optimization factor under each level was calculated. For example, the mean of average torque for factor A under level 1 is calculated as: (12)
where Tavg1~Tavg5 is the average torque of the 1st up to 5th experiments for factor A under level 1.
Similarly, the mean of average torque Tavg, torque ripple coefficient Kmb, iron loss PFe and efficiency η for the other 5 factors under each level can be calculated respectively, and the calculation results are shown in Table 5. It can be seen from Table 5, that the combination of the levels taken by each factor to make the average torque Tavg largest, the torque ripple coefficient Kmb smallest, the iron loss PFe smallest and the efficiency η highest, respectively, is A(1)B(1)C(4)D(5)E(5)F(5), A(1)B(5)C(1)D(1)E(1)F(2), A(2)B(1)C(5)D(1)E(1)F(1) and A(1)B(1)C(4)D(5)E(5)F(5). Obviously, the level combination of each factor that makes each performance optimal, respectively, is different. Thus the analysis of the variance was conducted to analyze the relative importance of the effects of each optimization variable on each performance to get the optimal result.
4.4 Analysis of variance and determined optimization scheme
By analyzing the variance of the experimental results, the relative importance of the effects of each optimization variable on each performance can be obtained. The calculation formula of variance is shown as: (13)
where SA is the variance of one performance under factor A; Q is the level number of each factor and Q = 5; mA(i) is the mean of one performance for factor A at level 1 in Table 5; m is the overall mean of one performance in Table 4. Thus the variance of the average torque Tavg, torque ripple coefficient Kmb, iron loss PFe and efficiency η under other factors can also be calculated from equation (13). The calculation results are shown in Table ??.
It can be seen from Table 6 that the change of factor A, D and E has the largest effects on the torque ripple coefficient Kmb, the change of factor B and C has the largest effects on iron loss PFe, and the change of factor F has largest effects on the average torque Tavg and efficiency η. Therefore, the levels selected for factors A, D and E are to make the torque ripple coefficient Kmb smallest, the levels selected for factors B and C are to make the iron loss PFe smallest, and the level selected for factor F is to make the average torque Tavg and efficiency η largest. The final optimal scheme for the factor level combination is A(1)B(1)C(5)D(1)E(1)F(5) and the corresponding values are shown in Table 7.
4.5 Results comparison
The FE model was built and the simulations were conducted according to the value taken by each optimization variable for the final optimal scheme shown in Table 7. Table 8 listed the results comparisons of the initial, GA and HGA designs, and Figure 4 shows the torque and iron loss comparisons.
It can be observed that the average torque Tavg and efficiency η of the HGA design is increased by 7.6% and 6.57% respectively compared with the initial design, and the torque ripple coefficient Kmb and iron loss PFe is reduced by 48% and 5.5% respectively as well. In addition, the PM mass of theHGA design changed from 337.5 g to 303 g and is reduced by 10.2% compared with the initial design. Each performance of the HGA design has realized further optimization compared to the GA design. The degree of performance improvements of the HGA design compared to the GA design is smaller than the GA design compared to the initial design. This shows that the GA optimal design is already close to the optimum solution, and that the Taguchi method overcomes the defect of the GA, which is easily trapped in a local extremum at the later evolution stage.
Figure 5 is the torque-speed characteristic comparison for the initial, GA and HGA optimised designs. It indicates that the speed range has widened to 0 to 7000 rpm from 0 to 4000 rpm and that the flux-weakening capability has improved for both the GA and HGA designs, especially for the HGA design.
A hybrid genetic algorithm (HGA), which combined a genetic algorithm (GA) with the Taguchi method, is proposed to optimize the rotor structure of an IPMSM for an electric vehicle application, to maximize the torque and efficiency, and minimise the torque ripple and iron loss. The following conclusions can be drawn:
The proposed HGA not only could overcame the defect of poor global searching capability of the Taguchi method, but also could cover the poor local searching ability for the GA. The Taguchi method is conducted to further optimize the GA optimized design.
Not only the performance such as torque, efficiency, torque ripple and iron loss have improved by using the GA and HGA, but also the amount PM material is reduced 9.2% and 10.2%, respectively, which reduces the motor manufacturing cost. In addition, the flux-weakening capacity of the IPMSM, which is important for electric vehicle applications, is also improved by optimisation by GA and HGA, especially by the proposed HGA.
This study was supported by the Beijing Natural Science Foundation (3172037) and the Natural Science Foundation of Hebei Province (E2017502025).
Wang A., Jia Y., Soong W., Comparison of Five Topologies for an Interior Permanent-Magnet Machine for a Hybrid Electric Vehicle, IEEE Trans. Magn., 2011, 47, 10, 3606-3609. CrossrefWeb of ScienceGoogle Scholar
Xia C., Guo L., Zhang Z., et al., Optimal Designing of Permanent Magnet Cavity to Reduce Iron Loss of Interior Permanent Magnet Machine, IEEE Trans. Magn., vol. 51, no. 12, article. 8115409, Dec. 2015. Web of ScienceGoogle Scholar
Sung-II K., Ji-Young L., Young-Kyoun K., et al., Optimization for reduction of torque ripple in interior permanent magnet motor by using the Taguchi method, IEEE Trans. Magn., vol. 41, no. 5, pp. 1796-1799, May. 2005. CrossrefGoogle Scholar
Hwang C., Li P., Liu C., Optimal Design of a Permanent Magnet Linear Synchronous Motor with Low Cogging Force, IEEE Trans. Magn., vol. 48, no. 2, pp. 1039-1042, Feb. 2012. Web of ScienceCrossrefGoogle Scholar
Hwang C., Li P., Frazier C.Chuang, et al., Optimization for Reduction of Torque Ripple in an Axial Flux Permanent Magnet Machine, IEEE Trans. Magn., vol.45, no.3, pp. 1760-1763, Mar.2009. Web of ScienceCrossrefGoogle Scholar
Ki-Chan Kim, Ju Lee, Hee Jun Kim, et al., Multiobjective Optimal Design for Interior Permanent Magnet Synchronous Motor, IEEE Trans. Magn., vol.45, no.3, pp. 1780-1783, Mar.2009. CrossrefWeb of ScienceGoogle Scholar
Shi T., Qiao Z., Xia C., et al., Modeling, Analyzing, and Parameter Design of the Magnetic Field of a Segmented Halbach Cylinder, IEEE Trans. Magn., vol. 48, no. 5, pp. 1890-1898, May. 2012. CrossrefWeb of ScienceGoogle Scholar
Hwang C., Lyu L., Liu C., et al., Optimal Design of an SPM Motor Using Genetic Algorithms and Taguchi Method, IEEE Trans. Magn., vol. 44, no. 11, pp. 4325-4328, Nov. 2008. CrossrefWeb of ScienceGoogle Scholar
Mahmoudi A., Kahourzade S., Abd Rahim N., et al., Design and prototyping of an optimised axial-flux permanent-magnet synchronous machine, IET Electric Power Applications., vol. 7, no. 7, pp. 338-349, 2013. Web of ScienceCrossrefGoogle Scholar
Lee D., Lee S., Kim J.-W., et al., Intelligent Memetic Algorithm Using GA and Guided MADS for the Optimal Design of Interior PM SynchronousMachine, IEEE Trans.Magn., vol. 47, no. 5, pp. 1230-1233, May. 2011. CrossrefGoogle Scholar
About the article
Published Online: 2017-12-29
Citation Information: Open Physics, Volume 15, Issue 1, Pages 984–991, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0122.
© 2017 Aimeng Wang and Jiayu Guo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0