A Benchmark for a Mono and Multi Objective Optimization of the Brushless DC Wheel Motor

Mono-objective optimization problem

The problem contains 5 design variables (see Fig. 2). All parameters are continuous: Ds is the bore (stator) diameter, Be is the magnetic induction in the air gap, δ is the current density in the conductors, Bd is the magnetic induction in the teeth and Bcs is the magnetic induction in the stator back iron.

There are 6 inequality constraints applied on this problem that are the outputs of the constraints' function block in figure 2. The total mass of the active parts (Mtot) must not exceed 15 kg. The outer diameter (Dext) must be lower to 340mm so that the motor fits into the rim of a wheel. The inner diameter (Dint) must be superior to 76mm for mechanical reasons. The magnets must support a current in the phases (Imax) of 125A (five times the rated current) without demagnetisation. The temperature of the magnets (Ta) must be inferior to 120C. Finally, the determinant Discr(Ds, δ, Bd, Bs) used for the calculation of the slot height must be positive.

The objective is to maximize the efficiency of the motor, i.e. minimize () which is the output of the objective function block shown in figure 2.

The Matlab function to compute the objective and constraints functions are here.

The mono-objective optimization problem is as follow:

 Minimize     (-η)    

    with         150mm < Ds< 330mm , 0.9T < Bd < 1.8T , 2.0A/mm2 <δ < 5.0A/mm2, 0.5T < Be < 0.76T, 0.6T < Bcs< 1.6T

    s.t.          Mtot< 15kg , Dext< 340mm , Dint > 76mm , Imax > 125A , discr > 0 , Ta < 120C

where η, Mtot, Dext, Dint, Imax, and Ta are results of the analytical model and discr is the determinant used for the calculation of the slot height.

The optimal solution is:
Ds Be δ Bd Bcs
201.2 mm 0.648 T 2.04 A/mm2 1.8 T 0.89 T

 

You can download the Matlab script for the optimization here.

Fig.2– Optimization loop and functions' blocks.

Figure 2 shows how the objective and constraints functions are called by the Matlab fmincon solver. The Matlab function fmincon attempts to find a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.


Multi-objective Optimization problem

In the case of the multi-objective optimization problem, the constraint total mass Mtot is changed to a second objective to minimize. Thus, the objective functions are to maximize the efficiency and to minimize the total mass simultaneously.

The multi-objective optimization problem is as follow:

 Minimize     F=[-η , Mtot]   

    with         150mm < Ds< 330mm , 0.9T < Bd < 1.8T , 2.0A/mm2 <δ < 5.0A/mm2, 0.5T < Be < 0.76T, 0.6T < Bcs< 1.6T

s.t.           Dext< 340mm , Dint > 76mm , Imax > 125A , discr > 0 , Ta < 120C

 



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