Abstract—This paper presents the optimum design of the brushless DC motor to minimize the material cost considering the fluctuation of the raw materials cost. The fluctuated cost data of the raw materials, which are assumed to be gathered some period, are reflected to the stochastic optimum design process. The constraint of the BEMF characteristics is also considered by the tolerance of the design variables. The BEMF Characteristics is analyzed by FEM. This optimum design result is compared to deterministic optimum design result. We get more robust optimization results by considering the fluctuation of design variables than of the each optimization results.

Index Terms—Motor, stochastic, Optimization

I. INTRODUCTION As the economic circumstance is getting globally

competitive, the price of the product as well as performance is most important sales point. To acquire the price competitiveness of the product, it is necessary that the optimum design of the material cost minimization. The deterministic optimization techniques such as Taguchi or Design of Experiment (DOE) are frequently used because of its convenience. But this method has a limit that can’t consider the tolerance or variation of the design variables because of being considered only single value optimization. The stochastic optimization methods have been developed for optimizing complex problems[1][2].

This paper presents the optimum design of the BLDC motor to minimize the material cost considering the fluctuation of the raw materials cost. At the same time, it is need to consider the characteristic variance of the motor according to tolerance of the design variables. We also consider the constraints which are influenced by the tolerance of the design variables in the optimization process.

The stochastic optimization is conducted by Crystal Ball that is software package adopted the scatter search method algorithm[3]. The optimized design results are compared with the deterministic optimum design results.

II. OPTIMIZATION MODEL

A. Analysis Model The Optimization model is the surface permanent magnet type BLDC motor, which is applied to home appliance.

The slot combinations of the motor are 32pole 36slots. Fig.1. shows the quadrisect analysis model and names of parts.

Magnet

Coil

Rotor case Stator core

Insulator

Fig. 1. Optimization model

B. Optimization Modeling The deterministic optimization is preceded by DOE

method that is simple and powerful. First of all, the objective function is defined to minimize material cost and the constraints are BEMF and winding fill factor. The constraint of BEMF is needed to guarantee the output power of motor and winding fill factor is needed to consider the winding possibility when the optimization is conducted.

Objective function =minimize (Material Cost) (1)

Constraints are BEMF ≥ 52 V (2)

Slot fill factor ≤ 57 % (3) The optimum design variables are selected which are most effective to the material cost and the characteristics of the motor.

TABLE I

THE OPTIMUM DESIGN VARIABLES Design variables Symbol LSL USL

Stator out-diameter SDO 250 mm 262 mm

Stator stack length STL 25 mm 30 mm

Turn/slot TUN 80 turn 100 turn

Magnet thickness MAT 5.5 mm 6.5 mm

Then, it is necessary to define the cost per unit weight of the raw materials because these are used to evaluate the

Optimization of the BLDC Motor Considering the Fluctuation of the Design Variables

Byoung-Ook Son*,**, Young-kwan Kim***, and Ju Lee* * Department of Electrical Engineering, Hanyang University, Seoul 133-71, Korea

** R&D Center, Daesung Electric co., Ansan 425-090, Korea *** R&D Team, Samsung Electric co., Suwon 443-742, Korea

material cost with the calculation of weight in the optimization procedure. We assume the raw materials cost for each part of the motor as the Table Ⅱ when the deterministic optimization is conducted.

TABLE Ⅱ THE COST PER WEIGHT OF THE RAW MATERIAL

Parts Raw Material Unit cost [$/kg] Density [g/㎤] Stator

core Si-steel 0.604 7.85

Coil Copper 6.190 8.90 Rotor

case CR 0.781 7.80

Magnet Ferrite 3.921 4.90 Insulator PET 3.941 1.63

The optimum design table is constructed with the 2-

level full factorial design. The material costs are evaluated from the multiplication of the raw material costs per weight and the calculated weights according to each design variable condition. The BEMFs are analyzed by FEM and the fill factors are calculated simultaneously.

TABLE Ⅲ THE FULL FACTORIAL DESIGN TABLE

No Design variables Calculated value SDO STL TUN MAT Cost BEMF F-factor

1 250 25 80 5.5 18.359 39.80 62.5 2 262 25 80 5.5 19.463 41.67 47.5 3 250 30 80 5.5 20.787 48.81 62.5 4 262 30 80 5.5 21.890 50.72 47.5 5 250 25 100 5.5 19.294 49.62 81.5 6 262 25 100 5.5 20.368 51.53 66.5 7 250 30 100 5.5 21.692 58.67 81.5 8 262 30 100 5.5 22.796 60.58 66.5 9 250 25 80 6.5 18.813 41.32 62.5

10 262 25 80 6.5 19.917 43.23 47.5 11 250 30 80 6.5 21.241 50.37 62.5 12 262 30 80 6.5 22.345 52.28 47.5 13 250 25 100 6.5 19.718 51.18 81.5 14 262 25 100 6.5 20.822 53.09 66.5 15 250 30 100 6.5 22.146 60.23 81.5 16 262 30 100 6.5 23.250 62.14 66.5

The optimum design results from DOE are as table Ⅳ.

This design results will be compared with the stochastic optimum design results.

TABLE Ⅳ THE DETERMINISTIC OPTIMUM DESIGN RESULTS

Design variables Value Material Cost BEMF Fill factor

Stator out-diameter 259.1 mm

21.80 $ 52.2 V 54.9 % Stator stack length 30.0 mm

Turn/slot 84 turn

Magnet thickness 5.5 mm

III. OPTIMIZATION BY STOCHASTIC METHOD

A. Stochastic optimization process The deterministic Optimization algorithms have limit

to solve the complex system. The stochastic optimizations that do not require a mathematical

formulation of the problem have been used for optimizing complex systems. The most popular stochastic optimization methods are genetic algorithms and scatter search method. We conduct the stochastic optimization using the commercial software package of the Crystal ball® which is the general-purpose optimizer implemented heuristic algorithms[4]. The stochastic optimization process of this paper is as below.

Define forecasts(Objective function, constraints )

Define forecasts(Objective function, constraints )

Define decision variables( Optimization design variables )

Define decision variables( Optimization design variables )

Define assumptions( Fluctuation of design variables )

Define assumptions( Fluctuation of design variables )

Derive regression equations( Forecasts , design variables )

Derive regression equations( Forecasts , design variables )

Optimization process( Stochastic algorithm )

Optimization process( Stochastic algorithm )

Optimized ?Optimized ?

Output Optimization results( Forecasts, decision variables )

Output Optimization results( Forecasts, decision variables )

Yes

No

Fig. 2. Stochastic optimization process

B. Optimization

The objective function and the constraints are defined same as the deterministic optimization. Then it is necessary to define the assumptions for objective function [i.e., to minimize material costs] and constraints [i.e., BEMF]. We construct the database of raw materials cost of design variables to reflect to the objective function. Databases have particular distributions as the tableⅤ which is assumed to gather cost data some period.

TABLE Ⅴ ASSUMPTION DATA FOR OBJECTIVE FUNCTION

Part Material (Symbol) Raw material cost per weight distribution

Stator core Si-steel (X1)

X = 0.686 , σ=0.053 Skewness= 0, Kurtosis= 1.8

Coil Copper (X2)

X = 7.101,σ=0.784 Skewness=-1.14, Kutosis= 5.4

Rotor case CR (X3)

X = 0.783, σ=0.008 Skewness=0.0, Kutosis= 2.4

Magnet Ferrite (X4)

X = 4.000 , σ=0.048 Skewness=0.0, Kutosis= 3

Insulator PET (X5)

X = 3.945, σ=0.046 Skewness=-1.15, Kutosis=5.4

And the BEMF is influenced by the tolerance of some

design variables. These tolerance data are also constructed as table Ⅵ. The distributions of assumption data are approximately defined as the average, standard deviation, skewness and kurtosis[4]. The skewness represents the symmetrical status of distribution. A skewness value greater than 1 or less than -1 indicates a highly skewed distribution and between -0.5 and 0.5 indicates a fairly symmetrical distribution. And the kurtosis refers to the peak of the distribution. A normal distribution has a kurtosis of 3 and the kurtosis value lower than 3 indicates the distribution flat.

TABLE Ⅵ ASSUMPTION DATA FOR CONSTRAINTS

Part Symbol Tolerance data distribution

Tolerance of SDO X6

X = 0, σ=0.08 Skewness= 0, Kurtosis= 2.4

Tolerance of STL X7

X =-0.05,σ=0.07 Skewness=-0.37,Kutosis= 2.4

Tolerance of MAT X8

X = 0.02, σ=0.05 Skewness=0.91, Kutosis= 2.4

After defining the assumption data, the constraints as well as the objective function must be built the regression equation for the optimum design variables to conduct the stochastic optimization. Equations (4)∼(8) show the regression equations of objective function about the each parts cost for design variables. The added up value of each parts cost is the total material cost. And the regression equation of BEMF is expressed as equation (9). Then stochastic optimization algorithms will search the optimum design variables according to the variation of the design variables and assumption data. We get the optimum design values by the crystal ball® after about 2,000 iterations. Stator core cost =(-4.9+0.019×SDO+0.087×STL)×X1 (4) Coil cost =(-0.5+0.018×STL+0.006×TUN)×X2 (5) Rotor Case cost=(-1.2+0.009×SDO+0.014×STL)×X3 (6) Magnet cost= (-1.8+0.005×SDO+0.1× STL

+0.02×MAT)×X4 (7) Insulator cost= (-0.53+0.003×SDO+0.008×STL)×X5 (8) BEMF = -93.3+0.159×(SDO+X6)+ 1.81×(STL+X7)

+1.56×(MAT+X8)+ 0.448×TUN (9)

TABLE Ⅶ THE STOCHASTIC OPTIMUM DESIGN RESULT

Design variables Value Material Cost BEMF Fill factor

Stator out-diameter 261.9 mm

21.6 $ 52.5 V 56.1 % Stator stack length 27.8 mm

Turn/slot 89 turn

Magnet thickness 6.37 mm

IV. OPTIMIZATION RESULTS COMPARISON To judge the validity of the each optimization results, it is necessary to evaluate the material cost and BEMF distribution according to the design variables variation. Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer[3]. So, the predicted cost and BEMF variation of the each optimization results can be simulated by the Monte Carlo method. The Crystal ball® also offers the solution of the Monte Carlo simulation. Table Ⅶ shows the comparison of the each optimization results.

TABLE Ⅷ THE COMPARISON OF THE EACH OPTIMUM DESIGN RESULTS

Design variables Deterministic results Stochastic results

Stator out-diameter 259.1 mm 261.9 mm

Stator stack length 30.0 mm 27.8 mm

Turn/slot 84 turn 89 turn

Magnet thickness 5.5 mm 6.37 mm

Predicted material cost 20.04∼23.97 $ 19.77 ∼23.67 $

Predicted average cost 22.806 $ 22.583 $

Predicted BEMF 51.6~52.6 V 52.0~52.8 V

Predicted fill factor 54.9 % 56.1 %

Fig.3. shows the predicted material cost distribution according to the fluctuation of the raw material costs for the each optimized design variables. The deterministic results is higher than the stochastic optimization results.

0

4

8

12

21 22 23 24

Material Cost [$]

Prob

abili

ty [%

]

Stochastic optimizationDeterministic optimization

Fig. 3. Predicted material cost distribution

Fig.4 shows the composition of the material cost for

each optimization results. In the deterministically optimized material cost composition, the sharing percentages of the stator core and the coil are higher than those of stochastic optimizations. It results from that the raw material cost of stator core and coil are relatively cheap when the deterministic optimization is conducted. But the raw material costs of the stator core and the coil are raised[i.e., fluctuated], its material cost has been increased sharply. Other wise, the stochastically optimized result that is considered the raw material cost fluctuations has the relative robustness than that of deterministic optimization.

19.0 18.8

15.4 16.8

14.9 15.0

22.2 22.0

28.5 27.3

Diterministic optimization Stochastic optimization

Stator coreCoilRotor CasemagnetInsulator

Fig. 4. Compositions of the material cost

The BEMF distribution of each optimization according to the tolerance of design variables is shown fig. 5.. We can see some data smaller than lower limit at the deterministic optimization results. It results from not considering the tolerance variation of design variables. On the other hand, all the data of stochastic optimization result are higher than lower limit because of considering the tolerance of design variables.

0

9

18

27

51 52 53

BEMF [V]

Prob

abili

ty [%

]

Stochastic optimizationDeterministic optimization

Fig. 5. Predicted BEMF distribution

V. RESULTS

This paper presents the optimum design of BLDC motor considering the fluctuation of design variables. To do this, it is necessary to construct the raw materials cost database that were gathered for some period and investigate the tolerance of the design variables. Then the regression equation between objective function and design variables is derived. Finally, the optimization is proceeded by the stochastic optimization method according to the raw material cost variation. We get more robust optimized results by considering the fluctuation of the design variables than deterministic optimization.

REFERENCES [1] Young-kyoun Kim, Jung-Pyo Hong, Jin Hur, “ Torque

Characteristic analysis Considering the Manufacturing Tolerance for Electric Machine by Stochastic Response Surface Method,” IEEE Transaction On Industry Applications, vol. 39, No.3, pp. 713-719, May/June 2003

[2] Dimitri N.Mavris, “ Economic uncertainty assessment using a combined design of experiment/Monte Carlo simulation approach with application to an HSCT,” 17th annual conference of the international society of parametric analysis, 1995

[3] Decisioneering Inc, Crystal Ball User Manual, Decisioneering Inc., 2005

[4] Manuel Laguna, “Optimization of complex system with Optquest” unpublished.