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Design Optimization Of Switched Reluctance Machine Engineering Essay

This paper presents Particle Swarm Optimization (PSO) based design optimization of Switched Reluctance Machine (SRM).The SRM design is formulated as multiobjective constrained optimization problem. The objective functions for obtaining desired design are maximization of average torque and minimization of torque ripple with stator and rotor pole arc as design variables. The optimization procedure is tested on 8/6, four-phase, 5 HP, 1500 rpm SRM.The results are compared and investigated with those obtained from Genetic Algorithm (GA) technique and FEA simulation. The results demonstrate that the proposed method is effective and outperforms GA in terms of solution quality, accuracy, constraint handling and computational time.

Keywords:Average Torque Genetic Algorithm,Particle Swarm Optimization,Switched Reluctance Machine, Torque ripple.

I.INTRODUCTION

Simple and robust structure, high efficiency and fault tolerability of Switched Reluctance Machine (SRM) are good reasons for its selection in variable speed applications [1]. The output power of an SRM is higher than that of a comparable induction motor and the torque-inertia ratio is also higher due to the absence of rotor windings [2]. Each stator pole has a simple winding, which is usually concentric. Suitable windings are connected together to form the motor phases. This simplicity gives the SRM the possibility of operating at very high speeds with a high output and a better mechanical acceleration [3, 4]. The main disadvantage of SRM is higher torque ripple which contributes to acoustic noise and vibration. The torque pulsation in SRM is due to highly non-linear and discrete nature of torque production mechanism [2]. In [5], an approach to determine optimum geometry of SRM to minimize torque ripple is discussed. Generalized regression neural network based optimization of SRM to maximize average torque and minimize torque ripple is discussed in[6]. In [7] an optimum design approach for a two-phase switched reluctance motor (SRM) drive using GA is proposed. Optimization algorithms such as Genetic Algorithm (GA) have been used in the optimal design of SRM [8, 9] to minimize torque ripple. From the literature it is evident that computational intelligence techniques like genetic algorithm and artificial neural network have been successfully applied for design optimization of SRM. However the relevance of modern evolutionary algorithms and their performance analysis with respect to design optimization of SRM is yet to be explored .Hence in this work an attempt has been made to apply Particle Swarm Optimization (PSO), for design optimization of SRM. The focus of this work is pole arc optimization of SRM with the objective of maximizing average torque and minimizing torque ripple using PSO approach. The PSO [18-25] algorithm is one of the modern evolutionary algorithms. This algorithm was first proposed by Kennedy and Eberhart. PSO is a population-based search algorithm characterized as conceptually simple, easy to implement and computationally efficient.The performance of PSO algorithm is compared with GA based optimization. The results show that PSO based approach performs better in terms of solution quality, accuracy and convergence time.

II. Design optimization of SRM- Problem

Formulation

The optimal design problem can be formulated as the following multiobjective nonlinear optimization problem:

(1)

where ,,and F is the feasible set of problem(1) which is described by the inequalities as follows

(2)

Where is called the constraint function .We denote the vector made up of all objective functions, that is

(3)

A.Pareto Optimal Solution

An ideal solution of (1) would be a point such that

(4)

The point seldom exists, therefore(1) turns into finding some or all the pareto optimal solutions. A point is a pareto optimal solution of (1) if there does not exist any feasible point such that

(5)

and (6)

for at least one index

There exists a wide variety of methods that can be used to compute Pareto optimal solutions. A widely used technique consists of reducing the multiobjective problem given by equation (1) to a single objective one by means of “scalarization” procedure. The “scalarization” procedure in this work consists of assigning each objective function a cost coefficient and then minimizing the function obtained by summing up all the objective functions scaled by their cost coefficients [10], that is

(7)

The global solution of the problem is affected by the coefficient .The cost coefficient is determined by the following equation

(8)

B.. Design Variables

The structure of 8/6 SRM is presented in Fig1. The torque characteristics of SRM depend on number of poles, number of phases, stator- rotor pole overlap angle and pole geometry [11]. From the literature [12, 13] it is evident that torque output and torque ripple are sensitive to stator and rotor pole arcs and their selection is a vital part of SRM design process. The choice of pole arcs depends on the application and there is no distinct value that is suitable for all applications. Further optimum pole arcs are a compromise between various conflicting requirements [3]. Hence in this study the two most influential parameters on average torque and torque ripple are taken as design variables.

Rotor Pole arc ()

Stator pole arc. ()

The remaining design parameters are treated as fixed for the optimization process.

Fig 1 Schematic diagram of 8/6 SRM

C. Objective Function

The multiobjective problem formulation is given by

(9)

Where

=Maximization of average torque.

= Maximization of inductance ratio(to minimize

torque ripple).

The constants A and B are determined using the “scalarization” procedure. In view of the fact that the average torque and inductance ratio of the motor is to be maximized, minus sign is introduced in the fitness function .

D.Calculation of average torque

Several methods are reported for the analysis of the SR motor, such as finite-element method (FEM) [14], [15], magnetic equivalent circuit (MEC) method [16], and piecewise linear model [17].In this work analytical method described by [3] is used to evaluate the performance of the machine.

The average torque is given by

(10)

(11)

Where

A comprehensive program is written in Matlab to compute the difference of co energies at aligned and unaligned position.The aligned coenergy is calculated with trapezoidal integration algorithm. Once is determined, the average torque is calculated using equation (10).

Figure 2 Flux Linkage vs Current Characteristics

The results obtained are compared and validated with FEA model. The design data for validating the analytical model is given in appendix1. The flux linkage-current characteristics obtained by analytical model and FEM model are presented in Fig 2. Table 1 presents the torque value computed by two methods. The closeness of the results have confirmed and validated the analytical model.

Table 2 Comparison of average torque

Analytical Value

FEM Calculation

Average Torque

23.14 Nm

22.9 Nm

For optimal design of the SR motor by evolutionary algorithms such as PSO, a large number of performance evaluations are required and the computational time by analytical method would be very large. Therefore, in this study, an Artificial Neural Network(ANN) based expert system for performance prediction of the SR motor has been developed. The ANN expert system can be used as a very fast performance prediction tool in design optimizing programs. A feed forward neural network as shown in fig 3 is trained with discrete points in the design domain considering the rules of feasible triangle [1].

Fig 3 Neural Network Structure

The stator and rotor pole arc form the input to the network and the torque obtained by analytical computation form the output. Backpropoagation algorithm is used to train the neural network. The results of the network are satisfactory when tested with data which were not subjected to training. This network is incorporated in the optimization routine to compute the value of average torque.

E. Evaluation of Torque Ripple

Torque ripple expected from SRM is evaluated from the torque dips in T-i-θ characteristics. Torque dip is the difference between the peak torque of a phase and the torque at an angle where two overlapping phases produce equal torque at equal levels of current. This is due to the deficiency of the incoming phase in supplying the necessary torque in those rotor positions[11] . Fig4 shows the torque dip present in the initial design. The effect of pole arc variation on mean torque and torque dip can be evaluated from Inductance overlap ratio given by equation(18) .Inductance overlap ratio gives a direct measure of torque overlap of adjacent phases.

(12)

From equation(18) it is evident that by widening the stator and rotor poles, torque overlap can be increased. The higher the , the lower will be the torque dip and the higher will be the mean torque as well. Since the inductance ratio has to be maximized the fitness function is taken as minus of.

Fig 4 Torque dip characteristics

F Design Constraints

The following are the constraints are imposed on the design optimization problem according to the rules of feasible triangle[1].

(13) (14)

(15)

To have a practically feasible and acceptable final design the following performance constraints are imposed.

(i) Average torque should be greater than 21 N-m.

(ii) Clearance space between the tips of windings should be greater than 5 mm.

The constraints are taken into account by penalizing the fitness proportionally to the constraint violations.

iii. Overview of PSO

Recently, several heuristic optimization techniques such as genetic algorithm (GA), ant colony algorithm (ACO), PSO and recently biogeography-based optimization (BBO), are developed to solve a variety of complex engineering problems that are difficult to be solved using traditional optimization methods. PSO is developed by Kennedy and Eberhart [18]. It was found to be reliable in solving non-linear problems with multiple optima. In PSO, a number of particles form a ‘‘swarm” that evolve or fly throughout the feasible hyperspace to search for fruitful regions in which optimal solution may exist. Each particle has two vectors associated with it, the position (Xi) and velocity (Vi) vectors. In N-dimensional search space, Xi = [xi1, xi2, . . ., xiN] and Vi = [vi1, vi2, . . ., viN] are the two vectors associated with each particle i. During their search, members of the swarm interact with each others in a certain way to optimize their search experience. There are different variants of particle swarm paradigms but the most commonly used one is the gbest model where the whole population is considered as a single neighborhood throughout the flying experience [38]-[39]. In each iteration, particle with the best solution shares its position coordinates (gbest) information with the rest of the swarm. Each particle updates its coordinates based on its own best search experience (pbest) and gbest according to the following equations:

(16)

(17)

where c1 and c2 are two positive acceleration constants, they keep balance between the particle’s individual and social behavior when they are set equal; rand1 and rand2 are two randomly generated numbers with a range of [0, 1] added in the model to introduce stochastic nature in particle’s movement; and w is the inertia weight and it keeps a balance between exploration and exploitation. In our case, it is a linearly decreasing function of the iteration index:

(18)

where itermax is the maximum number of iteration, iter is the current iteration number, wmax is the initial weight and wmin is the final weight. In conclusion, an initial value of w around 1, with a gradual decline toward 0 is considered as a proper choice. The most important factor that governs the PSO performance in its search for optimal solution is to maintain a balance between exploration and exploitation. Exploration is the PSO ability to cover and explore different areas in the feasible search space while exploitation is the ability to concentrate only on promising areas in the search space and to enhance the quality of potential solution in the fruitful region. Exploration requires bigger step sizes at the beginning of the optimization process to determine the most promising areas then the step size is reduced to focus only on that area. This balanced is usually achieved through proper tuning of PSO key parameters. Recently, PSO developments and applications have been widely explored in engineering and science mainly due to its distinct favorable characteristics [24]. Just like in the case of other evolutionary algorithms, PSO has many key features that attracted many researchers to employ it in different applications in which conventional optimization algorithms might fail such as:

It only requires a fitness function to measure the ‘‘quality” of a solution instead of complex mathematical operations like gradient, Hessian, or matrix inversion. This reduces the computational complexity and relieves some of the restrictions that are usually imposed on the objective function like differentiability, continuity, or convexity.

It is less sensitive to a good initial solution since it is a population based method.

It can be easily incorporated with other optimization tools to form hybrid ones.

It has the ability to escape local minima since it follows probabilistic transition rules.

More interesting PSO advantages can be emphasized when compared to other members of evolutionary algorithms like:

It can be easily programmed and modified with basic mathematical and logic operations. It is inexpensive in terms of computation time and memory.

It requires less parameter tuning.

It works with direct real valued numbers that eliminates the need to do binary conversion of classical canonical genetic algorithm.

iv. Implementation of PSO for Optimal

Design of SRM

In this design, PSO is used to find a set of design variables which ensure that the function F(x) has a minimum value and all the constraints are satisfied. The algorithm for design optimization process is given below

Read specifications and performance indices of motor

Initialize PSO parameters such as wmax, wmin, c1, c2 and Itermax

Generate initial population of N particles (design variables) with random positions and velocities.

Compute objective value and performance indices of motor

Evaluate the fitness of each particle

Update personal best: Compare the fitness value of each particle with its pbests. If the current value is better than pbest, then set pbest value to the current value;

Update global best: Compare the fitness value of each particle with gbest. If the current value is better than gbest, set gbest to the current particle’s value;

Update velocities: Calculate velocities Vk+1 using Equation (16)

Update positions: Calculate positions Xk+1 using Equation (17)

10.Return to step (4) until the current iteration

reaches the maximum iteration number

11.Output the optimal design variables.

v. Results and Discussion

The performance of the proposed method is tested on a 5HP motor. The specifications of the sample motors are given in Appendix 1. The results of the proposed method are also compared with the GA based design method. These methods are coded in Matlab and executed using a Pentium IV based PC as the test platform. During the process the following parameter setting is used in PSO: Number of particles=30, initial inertia weightW max=0.9, final inertia weight Wmin=0.4, acceleration factor C1 = C2 = 1.5 maximum iteration Itermax = 100. The results obtained from PSO and GA based design methods are given in Table 2. The convergence characteristics of GA and PSO methods for the sample motor are shown in Fig5. From the results it is seen that the PSO converges quickly and explore higher quality solution than the GA. To test the robustness of GA and PSO, 20 independent trials were carried out. The obtained results by each method are given in Table 3. From the table, it is clear that the minimum, average and standard deviations obtained by PSO are better than GA.

Table 2 Results of Optimal Design

Initial Design

Optimal design

PSO

GA

Stator Pole arc

18 deg

21.91 deg

22.77deg

Rotor Pole arc

22 deg

22.91 deg

21.74deg

Average Torque

22.98 Nm

29.67 Nm

29.40 Nm

Inductance ratio

0.1667

0.315

0.310

Torque dip

8.84 Nm

5.77 Nm

6 Nm

Table 3 Comparison of PSO and GA methods

GA

PSO

Best Solution

-1.99685

-1.9978

Worst Solution

-0.97673

-1.6237

Mean

-1.65346

-1.98719

Standard Deviation

0.341875

0.03512

CPU Time

49 seconds

47 seconds

Figure5 Convergence characteristics of GA and PSO based methods

VI. Conclusion

This paper describes the design optimization procedure of SRM using PSO with the objective of maximizing average torque and minimizing torque ripple. The proposed approach is useful in the initial design stage of the machine to determine optimal pole arc. ANN is used for faster performance prediction during the optimization process. The results obtained by this approach show improvement in average torque and reduction in torque dip. Average torque of 29.19 Nm is obtained when the geometry is investigated by FEA simulation. The results show PSO algorithm is superior to GA in terms of global exploration, robustness, fast convergence and statistical accuracy.

APPENDIX 1

Design Data of the machine

Machine configuration

8/6

Power output

5 hp

Stator pole arc

18 degrees

Rotor pole arc

22 degrees

Air gap length

0.5 mm

Outer stator diameter

190 mm

Bore diameter

100.6 mm

Stack length

200 mm

Shaft diameter

28 mm

Speed

1500 rpm

Height of stator pole

32.7 mm

Height of rotor pole

19.8 mm

Turns per phase

154

Rated current

13 A

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