CHAPTER 5 STRUCTURAL OPTIMIZATION OF SWITCHED RELUCTANCE MACHINE

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Cornelius Greer
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1 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 vehicles like energy storage systems, converters, electric motors and about the energy management of the whole vehicle. SRMs are preferred for EV applications (Fahimi et al 2004, Krishnamoorthy et al 2006) due to their numerous advantages, such as simple and rugged motor construction, low weight, potentially low production cost, easy cooling, excellent power-speed characteristics, high torque density, high operating efficiency and inherent fault tolerance (Miller 1993, Krishnan 2001). Researchers have focused on design (Chiba et al 2011, Rahman et al 2002) and control aspects (Paramasivam et al 2004, Inderka et al 2002, Islam et al 2003, and Omekanda et al 2003) to enhance the performance of SRM for EV applications. The optimal design of SRMs for EVs using evolutionary algorithm has also received considerable attention in recent years (Faiz et al 2005, Kano et al 2010). The desirable characteristics of electric motors for EV applications include average torque, torque density, and efficiency. Therefore, design of SRMs in EVs has to give consideration to the three above requirements. Further for EV applications, volume is an important criterion. The motor has to be designed with a fixed external diameter to leave a place for the motor housing. In order to maximize the torque density, the torque has to be

2 90 maximized for a fixed axial length of the motor (Raminosoa et al 2010) (i.e., to maximize the torque/axial-length ratio). Hence for EV applications, the stator pole arc, the rotor pole arc, and the bore radius are considered as design variables. Maximization of average torque, average torque per motor lamination volume and minimization of copper loss are considered as objectives. These three criterions imply torque, torque density and efficiency respectively. By taking into consideration these objectives and design variables, this chapter presents a design optimization approach using swarm intelligence and evolutionary computation techniques to determine the dimensions of stator and rotor laminations. The suitability of different techniques is illustrated on 8/6 SRM. The final design emerging from the optimization approach is verified with FEA. 5.2 CALCULATION OF PERFORMANCE PARAMETERS In this work analytical method is used to calculate the value of average torque. A comprehensive program (based on the procedure discussed in chapter 2) is written in MATLAB to compute the average torque from the difference in coenergies at aligned and unaligned position. Analytical method is preferred over FEA technique because there are three design variables involved and a change in one or the other variables requires an entire modeling and FEA computation which takes a considerable amount of time. The motor lamination volume is calculated as V V V (5.1) s r where V s represents the volume of stator lamination and V r represents the volume of rotor lamination. Consequently, the average torque per motor lamination volume is determined as

3 91 T T V ave v (5.2) The copper loss is computed as discussed in section CLASSICAL METHOD Objective Function Formulation method is given by The multi-objective problem formulation using weighted sum T T P F max(w ) max(w ) min(w ) (5.3) av v cu opt tave tv cu Tavb Tvb Pcub where Tavb max(t av ) T max(t ) vb p min(p ) cub v cu Wtave Wtvb Wcub 1 (5.4) In the above equation T av denotes average torque, T v denotes torque density and P cu represents copper loss. W tave, W tv, and W cu represent the weight factors of the average torque, torque density and copper loss respectively. T avb represent the base value of average torque, T vb represents the base value of torque density and P cub represents the base value of copper loss. From equation it is seen that the optimization with three objectives is simplified to an optimization function by using three weight factors. Various weight factors indicate the shares which are taken up by average torque, torque density and copper loss in the objective function. The constraints on the design parameters are given by the Equations (5.5) - (5.8)

4 92 (5.5) r s 2 r s (5.6) Nr s (5.7) D (5.8) D 0 Further to have a practically feasible and acceptable final design the clearance space between the tips of windings is imposed as a constraint. The constraints are taken into account by penalizing the fitness proportionally to the constraint violations Outline of the Proposed Approach The flowchart of the proposed optimization approach using PSO and DE techniques is given in Figure 5.1 and Figure 5.2 respectively. First, design variables stator pole arc, rotor pole arc and bore radius are generated as an initial population. Using these variables the remaining design parameters, such as ratio of rotor diameter and stator diameter, pole arc, yoke thickness, height of rotor and stator teeth are determined using the design equations given by Miller (1993). Subsequently, the magnetizing curves are calculated using nonlinear magnetic analysis in accordance with the lamination design parameters of an entered individual. Then, the fitness is evaluated considering the desired performances. After the iteration of the fitness evaluation for all individuals, a new population for the next generation is updated by the suitable operations (Mutation and crossover in case of DE and position and velocity updating in case of PSO) corresponding to the algorithm. This process is repeated till maximum number of generations is reached.

5 Figure 5.1 Design optimization approach using PSO 93

6 Figure 5.2 Design optimization approach using DE 94

7 Performance and Simulation Results The performance of the algorithm is tested on a sample 5HP motor. The outer diameter, air gap and axial length are fixed as given in Table 5.1. The algorithms are coded in MATLAB and executed using a Pentium IV based PC as the test platform. In this work the solutions are illustrated by considering the following weight factors W tave =0.35, W tv =0.4, and W cu =0.25. Table 5.1 Main Dimensions of the sample motor Machine configuration 8/6 Air gap length 0.5 mm Outer stator diameter 190 mm Stack length 200 mm Speed 1500 rpm Upon execution of the algorithm, an optimal structure with the configuration given in Table 5.2 is obtained. The results obtained are the best solution over 20 independent trials. The performance parameters of the optimal design are given in Table 5.3. Table 5.2 Results of optimal design Design Parameter Value Stator pole arc degrees Rotor pole arc degrees Bore diameter mm Shaft diameter 28 mm Height of stator pole 28.3 mm Height of rotor pole 21.1 mm Turns per phase 156

96 Table 5.3 Performance parameters of optimal design Performance Parameter Value Average Torque 30.4Nm Torque Density 1454Nm/m 3 Copper Loss 189 W 5.3.3.1 Characterization using FEA The flux linkage current characteristics of the optimal machine obtained by FEA and analytical results are shown in Figure 5.

8 96 Table 5.3 Performance parameters of optimal design Performance Parameter Value Average Torque 30.4Nm Torque Density 1454Nm/m 3 Copper Loss 189 W Characterization using FEA The flux linkage current characteristics of the optimal machine obtained by FEA and analytical results are shown in Figure 5.3. It is noted that the flux-linkage/current characteristics is the main key for SRM design and comparison of the analytically obtained flux-linkage characteristics with the corresponding characteristics by FEA method shows a good agreement and it is expected that the machine satisfies the necessary requirements. The closeness of the results validates the application of the proposed optimization approach. Figure 5.3 Flux linkage vs. current characteristics

9 Comparative studies To validate the results obtained with the proposed method, the same problem is solved using GA, PSO and DE. In order to verify the robustness of the proposed methodology, the simulations are carried out for 20 independent runs and the results are summarized in Table 5.4 respectively. From the table, it is clear that the average values obtained from the enhanced PSO and DE methods are better than those of the above given methods. From the results in Table 5.4 it is evident that the enhanced PSO (EPSO2) and DE (CDE1) methods are more robust than the classical methods as the standard deviation of the fitness values for 20 runs is very low in the proposed methods. As all the objectives are converted as maximization problem, a minus sign is included in the fitness function The performance of the optimization technique in terms of fitness value with PSO and DE for the best run, out of 20 trials is shown from Figure 5.4 and 5.5 respectively. Table 5.4 Comparison of solutions obtained upon execution of different algorithm for 20 independent trials Best Worst Average Standard Algorithm Solution Solution Value Deviation GA PSO EPSO EPSO DE CDE CDE From the figures it is clear that the enhanced PSO and DE approaches method converges earlier than the other methods. On examining the convergence of EPSO1 and CDE1 approaches, CDE1 converges ahead of EPSO1 approach as shown in Figure 5.6.

10 98 Figure 5.4 Convergence characteristics of PSO approaches Figure 5.5 Convergence characteristics of DE approaches

11 99 Figure 5.6 Convergence characteristics of enhanced PSO and DE approaches 5.4 NSGA-II METHOD Objective Function Formulation The problem of determining optimal pole arc is formulated to provide trade off solutions between average torque, torque density and copper loss.the three objectives are defined as Maximization of average torque f1 max(t av) (5.9) Maximization of torque density f2 max(t v ) (5.10)

12 100 Minimization of copper loss f3 min(p cu ) (5.11) The NSGA-II algorithm to determine optimum design is described below Step1: Step2: Step3: Step4: Step5: Step6: Step7: Step8: Step9: Step10: Step11: Step12: Specify the bounds of design variables, design requirements and constraints Select the parameters like number of population, maximum number of generation, crossover and mutation probabilities. Generate initial population For each input calculate the internal dimensions of the machine using basic design ratios and determine average torque using analytical method Evaluate torque density, Copper loss and winding design. Evaluate objective function for initial population Set generation count Perform simulated binary crossover and polynomial mutation for the set of individuals Perform non-dominated sorting. (i.e., sorting the population according to each objective function value in ascending order of magnitude) Calculate crowding distance between the solutions. Perform tournament selection Increment generation count and repeat the steps 8-11 until the count reaches the maximum number of generations

13 Results and Discussion The 3D representation of trade off between different objectives obtained using NSGA-II is shown in Figure 5.7. Further the optimized results are compared with optimization results of a single objective optimization based on weighting function using CDE1 technique. Tradeoffs between the different objectives are projected on two-objective planes in Figures 5.8, 5.9 and From the results it is evident that NSGA-II is capable of identifying the trade-off among the conflicting objectives thereby providing alternative useful designs for a designer. The closeness of the results validates the application of NSGA-II for determining the optimum design of SRM for EV applications. Figure 5.7 Pareto-optimal front obtained using NSGA-II and CDE1

14 102 Figure 5.8 Projected view of Pareto front in average torque-copper loss plane Figure 5.9 Projected view of Pareto front in torque density-copper loss plane

15 103 Figure 5.10 Projected view of Pareto front in average torque-torque density plane 5.5 CONCLUSION In this chapter a systematic design optimization procedure for SRM considering the design parameters and objective functions to satisfy the need for EV applications using evolutionary computation and swarm intelligence techniques is proposed. First the solutions are obtained by converting multiobjective design problem into a single objective problem with suitable weight factors. The results of optimal design are verified with FEA. The results indicate that the enhanced PSO and DE algorithms better in terms of solution quality, statistical accuracy and convergence characteristics. Further NSGA-II technique is applied to determine the Pareto-optimal solutions. The results obtained by NSGA-II are validated by solving the problem using weighted sum approach by applying CDE1 approach. The results indicate NSGA-II obtained well-distributed Pareto-optimal solutions, assisting designers in understanding the trade-off between different objectives which is illustrated by projecting the results in the two objective plane.

Structural Optimizations of a 12/8 Switched Reluctance Motor using a Genetic Algorithm

International Journal of Sustainable Transportation Technology Vol. 1, No. 1, April 2018, 30-34 30 Structural Optimizations of a 12/8 Switched Reluctance using a Genetic Algorithm Umar Sholahuddin 1*,