In the GA optimization, the objective optimization is realized using a defined cost function, and the minimum point of the cost function corresponds to the optimum point. For a multi-objective optimization, a weight coefficient is assigned to each objective according to its degree of importance, *w*_{i} (the objective function with larger *w*_{i} is more important). The sum of the products of each objective value and their weighting degree *w*_{i} to calculate the total cost function. The constraint formula of the objective functions is shown as (3), and total cost function formula shown as (4).
$$\begin{array}{}{\displaystyle {g}_{i}(x)conditio{n}_{i}{G}_{i}}\end{array}$$(3)

where *g*_{i}(*x*) is the *i*-th objective function expression; *condition*_{i} is the corresponding conditional operator “ < ”, “=” and “>” and *G*_{i} is the given goal value of the *i* th objective function.
$$\begin{array}{}\mathrm{Cos}t=\sum _{i=1}^{N}{w}_{i}{\epsilon}_{i}^{2}\end{array}$$(4)

where *N* is the number of objective functions; *ε*_{i} is the residual target and its value reflects the degree that the simulated response values of the objective function deviate from the given goal value range. As shown in Figure 2, if the response value is within the goal value range, then *ε*_{i} = 0, if not, *ε*_{i} is the difference of the response value and the goal value.

Figure 2 Diagram of the objective function residual for different *condition*_{i}

In this paper, four objectives of the machine are used for optimization, that is the average torque *T*_{avg}, torque ripple coefficient *K*_{mb}, iron loss *P*_{Fe} and efficiency *η*. Due to the significant difference between these four objective function values, they are standardized to a goal value between 1 and 10.

For instance, the optimization goal value of the average torque *T*_{avg} is 16 Nm and the value range is 12 Nm ~ 16 Nm, thus its objective function *g*_{1} is defined as:
$$\begin{array}{}{\displaystyle {g}_{1}=1+({T}_{avg}-16)\times 9/(-4)}\end{array}$$(5)

It can be seen from formula (5) that the value of *g*_{1} is 1 when the average torque *T*_{avg} is 16 Nm, and *g*_{1} is 10 when the average torque *T*_{avg} is 12 Nm.

Similarly, the optimization goal value of the torque ripple coefficient *K*_{mb} is 1.6% and value range is 1.6%~2.2%, and its objective function *g*_{2} is defined in formula (6); set the optimization goal value of the iron loss *P*_{Fe} to 28 W and value range is 28 W ~ 35 W, defined its objective function *g*_{3} as formula (7); set the optimization goal value of efficiency *η* to 96% and value range is 90%~96%, defined its objective function *g*_{4} as formula (8).

$$\begin{array}{}{\displaystyle {g}_{2}=1+({K}_{mb}-1.6)\times 9/0.6}\end{array}$$(6)

$$\begin{array}{}{\displaystyle {g}_{3}=1+({P}_{Fe}-28)\times 9/7}\end{array}$$(7)

$$\begin{array}{}{\displaystyle {g}_{4}=1+(\eta -96)\times 9/(-6)}\end{array}$$(8)

In this optimal design, the *condition*_{i} of the formula is “<”, thus the corresponding constraint formula of the four objective functions is shown as:
$$\begin{array}{}{\displaystyle {g}_{i}(x)<{G}_{i}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}i=1,2,3,4}\end{array}$$(9)

Then, set the importance degree of each optimization objective. The importance degree of all four objective functions is set to 1 in this paper. Combining Figure 2(a) with the total cost function formula (4) gives the total cost function of the average torque *T*_{avg}, torque ripple coefficient *K*_{mb}, iron loss *P*_{Fe} and efficiency *η* as follows:
$$\begin{array}{}{\displaystyle \mathrm{Cos}t=\sum _{i=1}^{4}{({g}_{i}-{G}_{i})}^{2}}\end{array}$$(10)

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