Optimizing Weight and Torque of Squirrel-Cage Induction Motors Using Genetic Algorithms

This paper presents a multiobjective fuzzy genetic algorithm optimization approach to design the submersible induction motor with two objective functions: the full load torque and the manufacturing cost. A multiobjective fuzzy optimization problem is formulated and solved using a genetic algorithm. The optimally designed motor is compared with an industrial motor having the same ratings. The results of optimal design show the reduction in the manufacturing cost, and the improvement in the full load torque of the motor.


INTRODUCTION
In recent years, Genetic Algorithm (GA) has been used as potent tools in design optimization of electrical machinery [1,2].Unlike the standard Non-Linear Programming techniques (NLP), the GAs are able to find the global minimum instead of a local minimum.It does not require the use of the derivative of the function.When dealing with real measurements involving noisy data, the derivative of function is not always easily obtainable or may not even exist [3].
Optimization of electric machines should be realized by making trade-off between different objectives.For example, size of the machine should be small, it should be inexpensive, its efficiency and power factor should be good, etc. Taking note of these, the importance of multiobjective optimization is understood in this field.Several multiobjective approaches for the electrical machines design have been proposed.Kim et al [4] presented the multiobjective optimal design of induction motor for electric vehicle using a modified evolution strategy, and used the weighting method among the multiobjective optimization methods.Mirzaeian et al [5] proposed a novel multiobjective method for optimal design of a switched reluctance motor with two objective functions.
A previous study shows the application of GA technique to optimize the 75hp three-phase induction motors.The objective functions are separately optimized by using GA [6].In this paper, an approach for multiobjective design optimization of submersible induction motor with two objective functions, i.e., the manufacturing cost and the full load torque utilizing the concept of fuzzy sets, convex fuzzy decision-making and a genetic algorithm having feature of a unique search [7] are presented.In the method, the objective functions are combined by fuzzy memberships so that the chromosomes with best performances for all objective functions have more chances to be chosen for participation in the next generation.The multiobjective fuzzy genetic algorithm (MFGA) optimization results show that the proposed approach is viable and reliable.

MULTIOBJECTIVE FUZZY OPTIMIZATION
A multiobjective optimization problem can be defined as follows [11]: ] is a vector objective function, h i (X) and g j (X) are equality and inequality constraint functions, respectively.u k X and l k X are the upper and lower bounds of X k , respectively.
It needs the membership function of each fuzzy objective function.Most applications that involve fuzzy set theory tend to be independent of the specific shape of the membership functions [12].The fuzzy objective function can be stated using the following membership function representation [11].
where X * is the solution for each of the objective functions.The fuzzy constraints membership function is defined as follows [11]: where ) X ( measures the degree of satisfaction for any n X ℜ ∈ in jth fuzzy constraint.The degree of satisfaction of jth constraint varies in between 0 and 1.For each fuzzy constraint, the allowable tolerances are given by d j .

Fuzzy decision-making
Using membership function the objective functions and constraints are defined as fuzzy subsets.The optimal decision is implemented by selecting the best alternative from the fuzzy decision space (D).This is given as follows: [11] ) . One of the three fuzzy decisions can selected: intersection decision, convex decision and product decision.In the present study Convex decision-making principles are utilized.The convex decision [13] providing a framework to incorporate the relative importance of all the objectives and constraints uses the arithmetic mean.This can be defined as follows [11]: where α and β are weighting factors, which satisfy The weights i α and j β can be obtained from a linear weighted sum as follows; Thus for the convex decision the membership function can be stated as: where i α and j β satisfy m ,....., 2 , 1 j 0 n ,....., 2 , 1 i 0 The multiobjective fuzzy optimization problem can be converted into single-objective optimization problems as follows:

IMPLEMENTATION OF THE OPTIMAL DESIGN PROCEDURE
A submersible induction motor, having specifications shown in Table 1, was chosen for optimization.The motor's equivalent circuit model shown in Fig. 1 was used.This model is basically a per phase representation of a balanced poly-phase induction machine in the frequency domain [9,10].The objective functions have to be defined to evaluate each motor design.In this study, the manufacturing cost and full load torque are selected as the objective function.(13) The punching cost C p is estimated as 20% of the total cost.Thus the total cost or also the objective function is expressed as follows: Second objective function: It should be noted that the full load torque as second the objective function is defined as: where the τ 1 is the correction factor, the n 1 is the synchronous speed.
As shown in eqn (14) and eqn (15), it has been already selected two objective functions.Accordingly, the membership functions of manufacturing cost and full load torque for the submersible induction motor corresponding to the fuzzy objective functions are defined as:  In the present study, it is assumed that fuzzy constraints ( 0 j = β ) is zero in eqn (10).Then, the multiobjective fuzzy optimization is defined as: Which maximize

≥
In this point, the Genetic algorithms (GAs) were used to solve this problem.Table 2 shows the practicable domains and the resolution for the design parameters.Here, it is assumed that the stator exterior diameter for the submersible induction motor is fixing.In the proposed design, each individual is classified by a string of 100 digits, composed by ten substrings, each of them representing an independent variable as reported in Table 2.The flowchart of the design optimization procedure for the MFGA optimization is depicted in Fig. 2.An explanation of each step is given in the following: i) Initial Execution of the program starts with the performance specifications such as the initial motor design variables.The initial population is randomly generated, each of length l.Each individual represents a possible solution to the problem.The stator and rotor parameters are separately calculated.ii) Multiobjective fuzzy optimization function To compute the degree of satisfaction of objective function and to check whether any of the constraints exceeded, the design calculations are transferred into the multiobjective fuzzy optimization.The degree of satisfaction values of each individual is transferred for evaluating in the fitness function.
iii) Fitness function The performances of population members are evaluated by using fitness function.Then the fitness values of each individual can be calculated as follows: where Di µ is the satisfaction degree of i. the individual, and ) x ( P is the penalty function.The constrained problems are converted to unconstrained problems By means of penalty function.According to the constraints, penalty function is defined as following: Where r is the number of total constraints, C N is the number of constraint exceeded.In the Eqn.( 19), it is worth noting that the penalty function becomes inactive when the constraint inequality is satisfactory, and that the penalty function becomes active when the constraint inequality is unsatisfied.

iv) Selection strategy
The selection of parents to produce successive generations plays an important role in the GA.The goal is to allow the fittest individuals to be selected to reproduce process.In this work, the roulette wheel selection strategy is used.
v) Crossover operation Crossover is a mixing operator that combines genetic material from selected parents.In this case, single-point crossover is used to perform crossover operation.
vi) Mutation operation This is a common genetic manipulation operator, and it involves, the random alteration of genes during the process of copying a chromosome from one generation to the next.Raising the ratio of mutations increases the algorithm's freedom to search outside of the current region of parameter space.Mutation changes from a "1" to a "0" or vice versa.In this case, bit mutation, and elitism is used.
vii) Reinsert offspring At this point a new population is achieved: all individuals are evaluated as described in step 2 and the subsequent steps are repeated.The procedure is stopped ('test convergence' in Fig. 2) after a prefixed number of generations.

THE RESULTS
The MFGA optimization procedure has been successfully applied to optimize the manufacturing cost and the full load torque of submersible induction motor.The performance results of submersible induction motor obtained from MFGA optimization are found to be quite satisfactory.Particular attention has to be given to choice of the GA parameters.The population size greatly affects the quality of the result and computation time.It has been observed that small populations exhibit large fluctuations both in the average and best fitness while great populations cause premature convergence.However, while with a high P c produces many new strings in the new population, deteriorating the successive generations, on the other hand, a low P c causes a contracted search that can be ineffective.In the same manner, while a high P m can compromise the convergence of the procedure, a low P m inhibits the search toward new zones.The best results are achieved with a medium population size (N=100-200), and by selecting the crossover and mutation probabilities in the ranges P c =0.5-0.9 and P m =0.01-0.05,respectively.Moreover, the highest achievable degree of satisfaction (degree of membership) for the given constraints and objectives is 0.9369, i.e. the best compromise solution due to the competing objectives.The results of the industrial motor and the MFGA optimization for submersible induction motor are given in Table 3 and 4. According to Table 4, while achieving performance improvements, the manufacturing cost of the motor is reduced by 7%.On the other hand, the starting torque and the full load torque are desirably increased by 15% and 9%, respectively, which are a remarkable increase.An essential remark here is that temperature rise of the motor is not known initially.Therefore, a fix value is given to the program.In view of the results, it is concluded that the MFGA optimization is suitable and can reach successful designs with lower cost and higher torque compared with the industrial motor while satisfying almost every constraint.Also, it was shown that MFGA optimization concludes with a good performance regarding the cost of different components and their dependencies on region, manufacturer and time.However, it is important to notice that while the performance improved, the efficiency and the power factor of the motor increased which shows additional capabilities of the optimization process.
Fig. 3 depicts examples of performance characteristics of an industrial motor and the MFGA optimization as a function of the speed.The figures show a remarkable performance improvement on the optimized motor with respect to the industrial motor.

CONCLUSIONS
In this paper, an approach for multiobjective design optimization of the submersible induction motor utilizing the concept of fuzzy sets, convex fuzzy decisionmaking and a genetic algorithm has been presented.The multiobjective fuzzy optimization technique based on GA has been successfully applied for the optimal design of the submersible induction motor.The computer simulation results have shown the effectiveness of the proposed method.While the manufacturing cost decreased by 7%, the full load torque increased by 9% that shows remarkable results.

Figure 1 .
Figure 1.The per-phase equivalent circuit model of a submersible induction motor

Figure 2 .
Figure 2. The flowchart of the design procedure for MFGA optimization

Table 1 .
Specifications of submersible induction motor

Table 2 .
Design parameters and their limit values

Table 3 .
The design parameter values obtained from the MFGA optimization

Table 4 .
Comparisons of the different designs (Slip=0.05)