Èñòî÷íèê:http://www.engr.psu.edu/ce/Divisions/Hydro/Reed/Education/CE%20563%20Projects/Nanda.pdf

Jyotirmaya Nanda Industrial and Manufacturing Engineering Department Penn State University University Park, PA 16802

Abstract

In this paper multi-objective evolutionary algorithm is proposed to design a universal electric motor. Designing a product often incorporates multiple objectives. Designing a product family has an added tradeoff, between commonality and individual product performance. The presence of multiple objectives gives rise to a set of Pareto-optimal solutions for individual products as well as the product family. The multi-objective evolutionary algorithm is used to help design a universal electric motor with competing design requirements. This is a step towards designing a family of electric motors with an acceptable balance between commonality in the product family and desired performance of individual products.

1 INTRODUCTION

The consumer market today is more turbulent and varied. Robust economy but cautious consumers; increasing globalization; increasing competition, often using new technology; diversified society of better educated and informed individuals; and many other factors has created a demand for unique custom products and services. Consumers no longer prefer to get directed to purchase pre-built products; rather, they want to have products and services that meet their particular needs. According to Pine (1993), “Customers can no longer be lumped together in a huge homogeneous market, but are individuals whose individual wants and needs can be ascertained and fulfilled”.

product family, a manufacture is able to reduce the part inventory, reduce the procurement cost, be more capable of handling volatility in consumer demand, and increase diversity in product offerings. The only snag being that increased commonality has an associated trade-off in individual product performance. The argument being, a single product designed specifically for a set of requirements is capable of performing better than a product belonging to a family of products that shares components. The components designed to be shared among different products will not perform as good as a custom component. When designing a product family the focus is more on the fa mily as a whole rather than a particular member of the family. Consequently decision maker has to make a tradeoff between commonality and performance during product family design. The tradeoff between commonality and performance is generally captured using one or more of the many commonality indices for product family design (Jio and Tseng, 2000; Kota, et al., 2000; Martin and Ishii, 1997; Siddique, et al., 1998; Simpson, et. al., 2001b) based on the direct and indirect benefits of commonality. Numerous exa mples of successful product families can be found in the literature: Swiss army knives and Swatch Watches (Ulrich and Eppinger, 2000), Xerox (Paula, 1997) and Cannon (Yamanouchi, 1989) photocopiers, Dell Computers (Schonfeld, 1998), Hewlett Packards printers (Feitzinger and Lee, 1997), Kodak (Wheelwright and Clark, 1995), Volkswagen (Bremmer, 1999), and Sony Walkmans (Sanderson and Uzumeri, 1995).

Many design researchers have started to use multiobjective optimization to examine the trade-off between commonality and individual product performance. Gonzalez-Zugasti, et al. (1999), use real options concepts to help select the most appropriate product family design from a set of alternatives; they also investigate the use of multi-objective optimization to design modular product platforms (Gonzalez-Zugasti and Otto, 2000; Gonzalez- Zugasti et al., 2000). Simpson et al., (2001) proposed a formal method that facilitates the synthesis and exploration of a common Product Platform Concept that can be scaled into an appropriate family of products known as Product Platform Concept Exploration Method (PPCEM). The product platform is modeled as a compromise Decision Support Problem (DSP) to model the necessary constraints and goals for the product platform. The compromis e DSP is a multi-objective mathematical construct which is a hybrid formulation based on mathematical programming and goal programming (Mistree et al., 1993).

Genetic Algorithms are well suited for solving combinatorial problems. Li and Azarm (2002) present a 2 two stage approach that employs a multiobjective genetic algorithm for product line design selection under uncertainty and with competitive advantage. D'Souza and Simpson (2003) present a method of using non-dominated sorting genetic algorithm (NSGA) to design a family of General Aviation Aircraft while optimizing the performance of the individual products

In this paper a Universal Electric motor is optimized using NSGA -II (Deb et al., 2000) as a step towards designing a family of Universal motors. The designing of a Universal Electric motor involves simultaneous optimization of multiple objectives that are competing in nature. The aim of using NSGA -II is to find out the Pareto-optimal or noninferior solutions of designing the motor. The problem is also highly constrained. Genetic Algorithm is used to explore the design space so that it can direct the designers towards feasible design variables that can later be used in designing the product family. The problem will later be extended to find commonality between different motors.

2 THE UNIVERSAL MOTOR PROBLEM

Universal electric motors are so named for their capability to function on both direct current (DC) and alternating current (AC). Universal motors deliver more torque for a given current than any other kind of AC capable motor (Chapman, 1991). The high performance characteristics and flexibility of universal motors have led to a wide range of applications, especially in household use where they are found in, e.g. electric drills and saws, blenders, vacuum cleaners, and sewing machines (Veinott and Martin 1986).

According to Meyer and Lehnerd (1997), in the 1970s Black & Decker developed a family of universal motors for its power tools in response to a new safety regulation: double insulation. Prior to that, they used different motors in each of their 122 basic tools with hundreds of variations, from jig saws and grinders to edgers and hedge trimmers. Through redesign and standardization of the product line, they were able to produce all of their power tools using a line of motors that varied only in the stack length and the amount of copper wrapped within the motor. As a result, all of the motors could be produced on a single machine with stack lengths varying from 0.8 in to 1.75 in, and power output ranging from 60 to 650 W. By paying attention to standardization and exploiting platform scaling around the motor stack length, material costs dropped from $0.77 to $0.42 per motor while labor costs fell from $0.248 to $0.045 per motor, yielding an annual savings of $1.82 million per year. Tool costs decreased by as much as 62%, boosting sales, increasing production volumes, and further improving savings. Furthermore, new designs were developed using standardized components such as the redesigned motor, which allowed products to be introduced, exploited and retired with minimal expense related to product development. Our goal is to demonstrate the use of the Genetic Algorithms to design a family of universal motors in a similar manner, starting with designing a single motor.

A schematic of a universal motor is shown in Figure 1. As shown in the figure, a universal motor is composed of an armature and a field which are also referred to as the rotor and stator, respectively. The armature consists of a metal shaft and slats (armature poles) around which wire is wrapped longitudinally as many as thousands times. The field consists of a hollow metal cylinder within which the armature rotates. The field also has wire wrapped longitudinally around interior metal slats (field poles) as many as hundreds of times

In this example problem, the design variables of interest for the universal motor are: the wire cross-sectional areas and numbers of turns in both the field and the armature; the radius, thickness, and stack length of the motor; and the current drawn by the motor.

Several textbooks are available for analyzing the performance of universal motors (Shultz 1992; Nasar and Unnewehr 1983; Veinott and Martin 1986; Chapman 1991). Such texts make use of the performance equations in terms of variables and constants (such as the magnetic field strength, the magnetic flux, and the motor constant K) which vary with respect to the physical dimensions of the motor; however, they do not detail the specific relationships between the physical dimensions of the motor and the resulting performance parameters.

A more sophisticated approach is presented by Kawanda (1965) to analyze the design factors associated with universal motor. The approach is based upon measurements of an existing universal motor, and it is neither intended, nor applicable, to an original design problem. Similarly, Dickin-Zangger (1962) presents a method for estimating the distribution of energy in a universal motor based on purposeful testing, as a basis for analytical and comparison of existing universal motor designs. Again, this approach does not provide relationships between physical motor dimensions and performance before a motor is actually built. In summary, the design literature of universal motors, where available, does not include a model (of any complexity) which relates physical parameters to resulting performance as we seek to develop.

In design related work, Wijenayake et al. (1995) develop a model for design optimization of permanent magnet motors that is rather complex with 53 input variables and 36 output variables. It is not applicable to universal motor design because permanent magnet motors use permanent magnets instead of wire coils to create a magnetic field. Boules (1990) present a similar approach for the design optimization of permanent magnet motors. Therefore, in order to provide computer simulation of universal motors, a mathematical model needs to be developed with input of physical motor dimensions and output of motor performance measures. Such a model is derived next.

In order to minimize power losses within the core of the motor when operating on AC power, a universal motor is constructed with slightly thinner laminations in both the field and the armature and less field windings. However, the governing electromagnetic equations for the operation of a series DC motor and a universal motor running on DC current are identical (Chapman 1991). The performance at full-load torque of a universal motor running on AC current is only slightly less than the performance of the same motor running on DC current. This discrepancy in performance is due to losses caused by the inherent oscillation in alternating current; for an overview of the losses associated with AC operation (see Chapman 1991).

These extra losses incurred in AC operation of a universal motor are difficult, if not impossible, to model analytically; thus, complicated finite element analyses are becoming more popular for modeling motor behavior under AC current. Since such a detailed analysis is beyond the scope of this work, the derived model for the performance of the universal motor is for DC operation for which simple analytical expressions are known or can be derived. Moreover, several texts indicate that the performance of universal motors under AC and DC conditions is quite comparable up until full load torque (Shultz 1992; Nasar and Unnewehr 1983; Veinott and Martin 1986; Chapman 1991); Shultz states that "Universal motors ... will operate either on DC or AC up to 60 Hz. Their performance will be essentially the same when operated on DC or AC at 60 Hz." For this work, all motors are designed for operation at full-load torque. Thus, it is assumed that designing a universal motor for DC conditions yields satisfactory performance for AC conditions as well

The formulae used for calculation of motor outputs, such as, power, torque, mass, and efficiency are illustrated in the following sections.

2.1 POWER

The basic equation for power output of a motor is the input power minus losses, where the input power is the product of the voltage (V) and current (I).

P = Pin – Plosses = VI – Plosses

as they heat-up (copper losses), (ii) at the interface between the brushes and the armature (brush losses), (iii) in the core due to hysteresis and eddy currents (core losses), (iv) in mechanical friction in the bearings supporting the rotor (mechanical losses), (v) in heating up the core and copper which adversely effects the magnetic properties of the core (vi) and the current carrying ability of the wires (thermal losses). For this analysis thermal losses, core losses, mechanical losses, and stray losses are neglected. The combined effects of all the aforementioned neglected losses will, conversely, decrease the output power and efficiency fro m the predicted value from the model. However, the following equations serve as a sufficiently accurate model for the DC operation of a universal motor.

2.2 TORQUE

Te torque of a DC motor is given by the product of a motor constant, K, the magnetic flux, f, and the current, I.

T = K f I

2.3 MASS

The mass of the motor includes mass of the stator, armature, and windings. The motor is modeled as a solid steel cylinder with length L for the armature and a hollow steel cylinder with length L, outer radius ro, and inner radius (ro-t) for the stator.

2.4 EFFICIENCY

The basic equation for efficiency, expressed as a decimal and not a percentage, is given by,For more detailed motor schematics, operation and performance measure see Simpson et al. (1999, 2001).

3 EVOLUTIONARY ALGORITHMS AND NSGA II

4 PROBLEM REPRESENTATION AND PARAMETER SELECTION

There are eight design variables that are used to evaluate the motors. The following sections describe the design variables, as well as the constraints used in the design problem.

The terminal voltage, Vt , is fixed at 115 volts to correspond to standard household voltage, and the length of the air gap, lgap, is set to 0.7 mm which is considered the minimum possible air gap length. A minimum air gap length is always desired because it maximizes torque while minimizing mass.

The constraints for each motor are listed in Table 1. The constraint on magnetizing intensity ensures that the magnetic flux within each motor does not exceed the physical flux carrying capacity of the steel (Chapman 1991). The constraint on feasible geometry ensures that the thickness of the stator does not exceed the radius of the stator since the thickness is measured from the outside of the motor inward. The required output power is taken as 300 W. This equality constraint is handled by putting it 6 in the objective function of NSGA II, which gave better results than formulating it as constraints. The objective for power was multiplied by a large factor to direct the search towards the targeted power.

There are three goals for the motor: 1. Maximize Efficiency (?) 2. Minimize Mass (M) 3. Maximize Torque(T) As shown in Table 1, a lower bound for efficiency, and torque, and a upper bound for mass have been imposed for the motor. The equations to calculate the goals are derived from Chapman (1991) and Cogdell (1996) for DC electric motors unless otherwise noted. This genetic algorithm will later be extended to the family of motors and commonality between the products will be explored.

To determine the range of potential population sizes, in the first run a small population size was used and the population size was doubled with each successive run. The percentage change in number of nondominated individuals for two successive run is calculated for each successive run. The population size increase is no longer done when the percentage change in number of non dominated individual fell below a pre specified value (see equation 11).

Figure 4. Majority of the applications use probability of crossover Pc between 0.6 and 0.9. To keep the disruptive effect of crossover for diversification, the crossover probability is kept at 0.7. Probability of mutation is taken as inverse of population. The number of real coded variables in the problem is eight with no binary representation. NSGA II code was modified to keep the first two variables, Nc and Ns, as integers by rounding off to the next integer value.

5 RESULTS

The results of the NSGA II run gave good insight to the interdependencies of design variables. It was observed that there is a significant increase in the value of torque with increase in the mass of the motor. The Pareto front for only torque and mass is shown in figure 5. Figure 6 gives the 2D representation of the solution space for mass and torque. The efficiency of the motor doesn’t vary much over the solution space and it remains in the region of 0.7 to 0.96. As already mentioned in section 2, the efficiency of the motors will appear higher than the actual values due to the assumptions made in calculation of power loss.

Also, increase in efficiency results in increase in mass. But it can be concluded from the results shown in figure 5-8 that it is easier to achieve the desired efficiency in the family of motors for a given mass than achieving the torque.

To find the common design variables in the search space a product family penalty function has to be introduced in the GA formulation which will try to find common design variables while maintaining the feasibility and desired output performance.

Also, offline analysis of the results will yield better analysis of the solution space and the trade off between various design variables.

6 CONCLUSIONS

As seen from the results the outputs form a Pareto frontier while designing the universal motor. The complexity of the problem will rise with the introduction of Product Family Penalty Function (Messac et al., 2002) to design the product family of universal motors.

The formulation of the problem will also change. Some of the possible methods of going forward are:

1. Individually optimize the products and do an offline analysis of the results to figure out the product family

2. Attach an extra binary header to each generation which will decide on which variables to keep common and which ones not, and then let the Genetic Algorithm find the product family

3. Introduce a penalty function for each variable that will give preference to commonality in product variables in the phenotypic space.

The decision regarding the parameters for the NSGA II for such a large problem will have to be carefully given to get any feasible results.