Design of Experiments Based Optimization of Synchronous and Switched Reluctance Machines

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1 IEEE PEDS 2017, Honolulu, USA December 2017 Design of Experiments Based Optimization of Synchronous and Switched Reluctance Machines Tobias Lange, Claude P. Weiss, Rik W. De Doncker Institute for Power Electronics and Electrical Drives (ISEA), RWTH Aachen University, Germany. Abstract During the electrical machine design process, here specifically for synchronous reluctance and switched reluctance machines, a large amount of design parameters interact with each other influencing the resulting geometry. A methodology to handle multi-parameter calculations for electrical machine design optimization is presented. With statistical methods the multi-parametric problem can be fitted and optimized by an analytic expression, containing all relevant interactions. The advantages of this approach is shown by finite element method results for a reference design of a 3 kw synchronous reluctance and a 1 kw switched reluctance machine. The benefit of this approach is shown for geometry optimization with different, partly contradicting, design criteria such as lowest torque ripple and highest torque density. Design of experiment is highly recommended to understand the interaction of parameters and especially in regard to design time reduction. calculation effort. Genetic algorithms try to reduce the number of calculations by determining new populations and to force the evolution towards its optimum [3]. The DOE provides a planned methodical strategy to determine the cause and effect of parameters within its system. In contrast to the OFAT optimization, the DOE varies multiple parameters in each run, reducing the amount of calculations considerably. The experiments are planned as factorial designs with two or more levels and are reduced by fractions. The level describes the amount of values used for each parameter, e.g. in a twolevel design only the minimum and maximum values are used, while in three-level, an additional center point value between the minimum and maximum is considered. I. INTRODUCTION TO STATISTICAL OPTIMIZATION Machine design not only consists of many input variables which determine the resulting cross-section geometry, but also multiple optimization criteria are possible. To reduce the complexity of the design process, often only one optimization criteria i.e. efficiency, average torque or torque ripple is chosen. However, the electrical drive normally has to fulfill multiple criteria (primary and secondary) to fit the target application. In order to handle the large number of design parameters the benefit of using a planned statistical approach compared to a one factor at a time (OFAT) variation is shown. The many parameters for machine design are divided into either geometric, categorical or electrical. As an example, the rotor pole geometry contains geometric parameters, the stator tooth and rotor pole pair configuration is a categorical parameter, while the number of turns is an electrical parameter. The multi-parametric optimization is based on the design of experiment (DOE) [1] and response surface method (RSM) [2]. The simplest approach to find a solution for multi-parametric problems is to calculate, within a certain step size, all possible parameter combinations. This leads to an enormous Fig. 1: Machine design process with DOE method /17/$ IEEE 1,024

2 An optimization of ten independent parameters with three-level is possible within 158 runs, while the same variation with OFAT would result in runs. A regression model of the outcome (effects) is used to find the optimum values of the input variables. To avoid inaccuracies and side effects, the chosen input variables (design parameters) have to be independent. Choosing the design parameters effectively for each machine type is discussed in section III following the design procedure in section II. II. DESIGN PROCEDURE The general design procedure and optimization steps of geometric and electric parameters applied to synchronous reluctance (SynRM) and switched reluctance machines (SRM) are presented in this section. The methodology shown in fig. 1 is identical for both machine types. The design process is structured in the following steps: 1) Analytic pre-design 2) Screening 3) 1 st Response surface analysis of the rotor 4) Response surface analysis of the stator 5) 2 nd Response surface analysis of the rotor 6) Response surface analysis of details First, a rough draft design of the machine is created. The pre-design is developed and analyzed by analytic equations and standard geometric parameters depending on the pole width and number of teeth of the motor. Thereby, the geometric model is built parameter-based, which allows a semi-automatic import of the DOE parameters to adapt the machine design. Thereafter, the parameters which cause the main effects are identified by a screening. The screening is a two-level investigation which allows a large number of variables with a minimum number of runs. The experimental plan is created as fractional factorial design (FFD) [4]. The fraction factorial design is based on orthogonal arrays. It enables the identification of variable interaction with significant impact on the main effects. The higher the fraction of the FFD, the higher interactions occur. Nevertheless, in physical systems usually an interaction of up to two parameters occurs. Thus, a fractional design with a resolution of IV can be performed to avoid aliasing effects as far as necessary. In table I the number of experiments/runs for a certain number of design variables and its resolution is shown. TABLE I: Overview of screening runs and variables No. of variables No. of runs Resolution 3 7 full 4 8 IV 5 16 V 6 16 IV 8 16 IV 9 32 IV IV After the runs are calculated the sensitivity of the main effects to the parameters is determined by a statistical analysis of variance (ANOVA). Afterwards, the number of parameters is reduced to the main-effect parameters. From the identified main-effect variables a response surface analysis with a center composed design (CCD) is generated with the software Minitab 17. A CCD is a 3-level analysis which adds one center point as 3 rd level. An example for a 3 factor CCD is shown in fig. 2. The amount of runs does not increase exponentially with the number of independent variables, because the plan can be reduced by fractional plans as shown in table II. The parameter variation is performed as a finite element analysis (FEA). The FEA results are fed back to calculate the regression equation. This regression contains all dependencies of the main-effect variables and enables an optimization with regard to the optimization criteria. Thereafter, the machine is adapted to the optimum parameters and simulated in the FEA software Jmag Designer 16. Design of experiments gives the machine designer all necessary understanding to find the best machine dependent on the design goals within only 158 simulation runs at ten independent parameters. The statistical method clearly shows cross-coupling between the parameters allowing a trade-off between contradicting optimization criteria. III. MULTI-PARAMETRIC OPTIMIZATION: CASE STUDY The following section shows exemplary design studies to demonstrate the effectiveness of DOE machine design for different machine types, as well as handling contradicting optimization criteria during the design process. 1,025

3 TABLE II: Overview of experimental runs and variables TABLE III: Overview of the pre-designed draft motor No. of variables No. of runs Plan Discription Value 3 20 full 4 30 full 5 32 half 6 53 half 7 88 half 8 90 quarter quarter eighth W sl /W th Stator slot to tooth ratio 1 D 2 /D 3 Bore to outer stator diameter 0.6 X abr Rotor barrier to iron ratio 0.5 N b No. of barriers 4 L pole Rotor pole depth 40 mm λ air Air-gap length 0.4 mm L stk Active stack length 50 mm D 3 Stator diameter 240 mm N w No. of turns 100 Fig. 2: Three factor area-centered composite design A. 3 kw SynRM An effective reduction of calculation time during the design process is very important for SynRMs due to the large amount of geometrical parameters. The complex rotor geometry can have up to 20 independent design parameters and the stator with the electrical winding over ten parameters. An OFAT optimization is not purposeful. In this case study a SynRM with distributed winding and 18 stator slots and 6 rotor poles is investigated. In a first step analytical equations for the pre-design are used to determine the draft design of the motor. The starting parameters for the design are given in table III. The barrier width and the tooth width of the rotor are equally distributed for the pre-design. With this draft design the following screening is performed. Each barrier width and length as well as the rotor tooth width is treated as an independent variable. The chamfers and width of the barrier tips are independent variables as well. For the stator, the copper area with a fill factor of 0.5, yoke and tooth width as well as the stator tooth tip dimensions are varied during the three screening processes. Overall, 41 geometrical and electrical values are screened to sort the variables by their contribution to average torque, torque ripple, efficiency, torque to loss ratio, power factor and weight. For the first surface analysis the barrier width, rotor tooth width, rotor barrier to iron ratio as well as the stator tooth and yoke width are chosen as main-effect variables. Within this iteration ten main-effect variables are calculated in an FEA in 158 runs. Second, the motor can be optimized for different individual criteria or multiple criteria at once. The different optimizations for a certain design criteria are performed solely in the software Minitab 17 as no recalculation in FEA is necessary. To illustrate different optimization results four rotor designs are plotted in fig. 3. It can be seen that different design requirements result in strongly differing geometries. The comprehensive knowledge about the behavior and interaction of all main-effect variables can be gained within just 158 simulation runs and formulated as regression equation. In the shown example, the evaluated motors are optimized for multiple criteria such as minimum torque ripple, maximum average torque and maximum torque per loss. The TABLE IV: SynRM design results Description T avg PF T avg /P Ironloss T ripple T ripple 27.0 Nm Nm /W 4.50 % T ripple, T avg 27.1 Nm Nm /W 4.53 % T avg 30.0 Nm Nm /W % T avg /P loss 27.7 Nm Nm /W % 1,026

4 shown rotor structures are just extreme examples to show that changing optimization criteria, or even weighting differently, does not require new FEA simulations. The geometric structures are not for common use, because it does not make sense just to minimize the torque ripple without the consideration of average torque and losses. The rotor structure for maximum torque results in an expected geometry as reported in literature [5]. The resulting rotor designs lead to the machine performance listed in table IV. The output characteristic of these motors are calculated in FEA. It can be seen that the regression formula meets its demand for the different optimization criteria. After the optimization of the first main-effect variables the basic geometry of the motor is updated and the next ten significant main-effect variables are considered in another response surface analysis. This procedure is repeated until all variables are varied and optimized. The authors recommend to repeat at minimum the analysis with the first ten main-effect variables after passing the entire optimization process with the three RSMs. This will ensure very good and accurate results for the motor in few runs. Hence, once the final machine design is determined the electrical design can be adapted in detail. In this case study the motor is designed to maximize torque with minimum ripple, high efficiency and high torque to loss ratio. The result of the final design is shown in table V. To design the motor three screening runs and five surface response analysis are performed. The biggest advantage of DOE is to gain an analytic regression equation to classify and understand interacting and cross-coupling parameters in SynRMs. TABLE V: Final design results of 18 slot 6 pole SynRM T avg PF T avg /P Ironloss T ripple η eff 37.8 Nm Nm /W 11 % 89.6 % B. 1 kw SRM The use of multi-objective mathematical approaches for the design of SRMs has become increasingly popular, however, the primary focus has been on only the geometric optimization, seldomly considering the used control [6]. The control type, either hysteresis current control, single pulse operation, direct instantaneous torque or force control, used during machine operation (a) Minimum T ripple (b) Minimum T ripple and maximum T avg (c) Maximum T avg (d) Maximum Tavg /P loss Fig. 3: Resulting rotor designs of different optimization criteria too determines the optimal geometric design. Another design approach presented previously in literature can be via a solution space based pre-design which enables selecting a fitting tooth configuration for a given application [7]. Through this pre-design approach a good machine configuration can be found, however, cross-coupling effects between the design parameters are not directly visible and understandable. Using the statistical design approach allows a comprehensive understanding and time reduction during the entire machine design process. In this case study the geometric optimization for a 1kW, 12/8 laboratory drive, operated at 48V and 1500rpm using hysteresis current and single pulse control is presented. The primary geometric- and control parameters varied during the design procedure are listed in table VI. Thereby, the main effect contributors to the machine design are R 1, β r and β s [8], while in regard to hysteresis current control, turn-on angle θ on, turn-off angle θ off, number of windings N w and reference peak current I pk are the main contributors. The control parameters are especially important for torque ripple and efficiency optimization. Following the main design criteria, there are also secondary design criteria which have to be fulfilled. The maximum current density J max, determined by the thermal limits of the machine may not be exceeded, as well as the winding slot area A slot defined by the manufacturing slot-fill factor S fill has to be met. These two criteria J max < 7 A /mm 2 and 1,027

5 S fill < 45%, as well as the outer diameter of 125 mm and rotor shaft diameter of 22 mm are used as boundary conditions for the optimization. For the SRM design the software PC-SRD from SPEED [9] is used in conjunction with Minitab 17. The SRM design should be optimized with regard to highest efficiency and lowest torque ripple. A main benefit of using the statistical design approach is the possibility to shift designs from one optimization criteria to another. An optimal SRM design for maximum efficiency, torque per active mass or minimum torque ripple are contradicting criteria resulting in different crosssections. However, with the use of DOE and the resulting regression analysis after only one design procedure, different optimal designs for each criteria are found. (a) Efficiency (b) Torque ripple (c) Torque/ weight Fig. 4: Different machine cross-section designs depending on the optimization criteria All ten design variables listed in table VI are varied simultaneously using the DOE approach. The results of the 158 machine simulations are imported into Minitab and a regression equation for each design criteria efficiency, torque ripple, machine weight, current density and output power is calculated. Thereby, efficiency only incorporates the ohmic losses and machine weight only incorporates the active iron parts of the machine and the winding copper. Torque ripple is calculated as a percentage of peak-to-peak torque to average output torque, while torque per mass is the fraction of average output torque to machine weight. In fig. 4 the exemplary designs are shown in respect to their optimization criteria. Fig. 5 shows the respective output torque wave- Current in A (a) Torque waveforms TABLE VI: Geometric- and electrical parameters varied during DOE Description Values R 0 Rotor-yoke radius mm R 1 Air-gap radius mm R 2 Stator yoke radius mm L stk Active stack length mm β r Rotor-tooth arc deg β s Stator-tooth arc deg N w No. of turns θ on Turn-on angle deg θ off Turn-off angle deg I pk Peak current A Rotor position in el (b) Current waveforms Fig. 5: Resulting waveforms of the different optimized SRM designs forms and optimal current trajectories. The difference in cross-section design is clearly visible resulting from the different optimization criteria. The machine with highest efficiency also has the shortest stack length of 60 mm, with the most windings (55 turns) and is operated in single-pulse as can be seen in fig. 5 minimizing the ohmic losses and rms current. Minimizing torque ripple causes the stator- and rotor teeth to be considerably wider, β r and β s are between 16 and 17 and an earlier turn-on angle compared to the other designs. The 1,028

TABLE VII: SRM design results η eff T ripple T/W t m active SRM eff 88.4 % 216 % 1.49 Nm /kg 4.2 kg SRM ripple 84 % 61 % 0.95 Nm /kg 6.6 kg SRM wt 85.6 % 204 % 1.63 Nm /kg 3.

The reduction in iron weight during torque per mass optimization results in quite thin stator and rotor teeth.

Furthermore, the effect of the control variables on the optimization process has been shown. IV.

Starting with an existing machine design, the machine geometry and its electrical parameters are optimized through DOE to improve the overall machine performance.

The effectiveness of statistical design has been shown on a synchronous reluctance machine and a switched reluctance machine.

6 TABLE VII: SRM design results η eff T ripple T/W t m active SRM eff 88.4 % 216 % 1.49 Nm /kg 4.2 kg SRM ripple 84 % 61 % 0.95 Nm /kg 6.6 kg SRM wt 85.6 % 204 % 1.63 Nm /kg 3.9 kg reduction of torque ripple is a combination of the correct control parameters and fitting geometrical cross-section. The reduction in iron weight during torque per mass optimization results in quite thin stator and rotor teeth. The results of the three optimization criteria are listed in table VII The design procedure shows how different optimization criteria result in different final designs. Furthermore, the effect of the control variables on the optimization process has been shown. IV. CONCLUSIONS The DOE methodology is a powerful tool to perform multi-parametric optimization, necessary for electrical machine design, without loosing accuracy or validity. Starting with an existing machine design, the machine geometry and its electrical parameters are optimized through DOE to improve the overall machine performance. The number of mandatory calculations is decreased significantly compared to common design procedures, by applying a very good statistically proven method. The effectiveness of statistical design has been shown on a synchronous reluctance machine and a switched reluctance machine. Furthermore, it has been shown how the design variables and optimization criteria determines the final cross-section and control strategy. The presented method allows a comparison of different machine configurations while enabling a comprehensive understanding of the cross-coupling of different design variables. [2] Y.-C. Choi and J.-H. Lee, Rotor stator design on torque ripple reduction for a synchronous reluctance motor with a concentrated winding using rsm, in 2007 International Conference on Electrical Machines and Systems (ICEMS), Oct 2007, pp [3] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: Nsga-ii, IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp , Apr [4] P. J. W. Mark J. Anderson, DOE Simplified, Practical Tools for Effective Experimentation. CRC Press, Taylor and Francis Group, [5] G. Pellegrino, T. M. Jahns, N. Bianchi, W. L. Soong, and F. Cupertino, The Rediscovery of Synchronous Reluctance and Ferrite Permanent Magnet Motors. Springer, [6] C. Ma and L. Qu, Multiobjective optimization of switched reluctance motors based on design of experiments and particle swarm optimization, IEEE Transactions on Energy Conversion, vol. 30, no. 3, pp , Sept [7] B. Burkhart, A. Mittelstedt, and R. De Doncker, Solution space based pre-design approach to compare and select configurations of switched reluctance machines, in 8th IET International Conference on Power Electronics, Machines and Drives (PEMD 2016), April 2016, pp [8] M. C. Costa, S. I. Nabeta, A. B. Dietrich, J. R. Cardoso, Y. Marechal, and J. L. Coulomb, Optimisation of a switched reluctance motor using experimental design method and diffuse elements response surface, IEE Proceedings - Science, Measurement and Technology, vol. 151, no. 6, pp , Nov [9] Maccon, User s Manual for PC-SRD Switched Reluctance Brushless Motor Design Simulation Software, Department of Electronics and Electrical Engineering, University of Glasgow, ACKNOWLEDGMENTS The research project Torque-Drive (03EFHNW135) which is part of the EXIST-program was funded by the Bundesministerium für Wirtschaft und Energie and the Europäischen Sozialfonds. REFERENCES [1] W. G. Cochran and G. M. Cox, Experimental Designs. 2. Auflage. Wiley. John Wiley & Sons, ,029

CHAPTER 5 STRUCTURAL OPTIMIZATION OF SWITCHED RELUCTANCE MACHINE

89 CHAPTER 5 STRUCTURAL OPTIMIZATION OF SWITCHED RELUCTANCE MACHINE 5.1 INTRODUCTION Nowadays a great attention has been devoted in the literature towards the main components of electric and hybrid electric