## 3 Multi-Objective Optimization
### Multi-Objective Optimization Design  
#### 3.2.1 Orthogonal Experimental Design  
The multi-objective optimization was carried out for the AFPMBLDC with arc-shaped magnets and the following three objectives were selected as indicators: flux density, no-load back EMF, and output torque. The optimized flux density is expected to be stronger and the total harmonic distortion rate of no-load back EMF is expected to be smaller. The optimized parameters are selected as \"A\", representing the pole arc coefficient; \"B\" representing permanent magnet thickness; \"C\" representing permanent magnet arc radius; and \"D\" representing air-gap length.  
The level value of the optimized parameters is selected according to the experience parameters of the motor design and the actual processing technology. The parameter ranges of the pole arc coefficient, magnet thickness, magnet arc radius, and air-gap length are shown in Table 2, and the four values of horizontal values are shown in Table 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Polar arc coefficient & 0.6–0.9 \\ Permanent magnet thickness & 2.0–3.5 \\ permanent magnet arc radius & 18–24 \\ Air-gap length & 0.5–1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the model.  
Figure 4: Flow chart of the [PERSON] method.  
Based on the four variable optimization parameters selected above, a range of four level factors were determined for each parameter, an orthogonal table was established. If the traditional single-variable optimization method was used; 4\({}^{4}\) = 256 experiments are required, while only 4 \(\times\) 4 = 16 experiments are required to achieve the multivariable and multi-objective optimization design of the motor using [PERSON]'s method. The orthogonal table for establishing experiment L16(4 \(\times\) 4) is shown in Table 4. Among them, 1, 2, 3 and 4 respectively correspond to horizontal influence factor 1, horizontal influence factor 2, horizontal influence factor 3 and horizontal influence factor 4 in Table 3.  
Using the time-step finite element method, the three-dimensional simulation model of the AFPMBLDC with arc-shaped magnets is constructed, and the three optimization objectives of each group of experiments are analyzed and calculated by using the transient field solver after the winding is intense and the mesh is dissected. The Mag-B represents the flux density, \"E\" represents the no-load back EMF, and \"T\" represents the output torque.  
Based on the data in the orthogonal table, the motor model is simulated. The specific values of flux density, no-load back EMF, and output torque are shown in Table 5. For example, the data result of No. 1 in Table 5 is the result of motor parameter simulation based on the data of No. 1 in Table 4; that is, under the condition of a motor pole arc coefficient of 0.6: a permanent magnet thickness of 2.0 mm, a permanent magnet arc radius of 18 mm, and air-gap length of 0.5 mm, the flux density is 0.59 T; the no-load back EMF is 8.56 V, and output torque is 0.52 V.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **A** & **B** & **C** & **D** \\ \hline
2 & 1 & 2 & 2 & 2 \\
3 & 1 & 3 & 3 & 3 \\
4 & 1 & 4 & 4 & 4 \\
5 & 2 & 1 & 2 & 3 \\
6 & 2 & 2 & 1 & 4 \\
7 & 2 & 3 & 4 & 1 \\
8 & 2 & 4 & 3 & 2 \\
9 & 3 & 1 & 3 & 4 \\
10 & 3 & 2 & 4 & 3 \\
11 & 3 & 3 & 1 & 2 \\
12 & 3 & 4 & 2 & 1 \\
13 & 4 & 1 & 4 & 2 \\
14 & 4 & 2 & 3 & 1 \\
15 & 4 & 3 & 2 & 4 \\
16 & 4 & 4 & 1 & 3 \\ \hline \end{tabular}
\end{table}
Table 4: Orthogonal table.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Optimization Parameters** & **A** & **B** & **C** & **D** \\ \hline Horizontal influence factor 1 & 0.6 & 2.0 & 18 & 0.5 \\ Horizontal influence factor 2 & 0.7 & 2.5 & 20 & 0.8 \\ Horizontal influence factor 3 & 0.8 & 3.0 & 22 & 1.2 \\ Horizontal influence factor 4 & 0.9 & 3.5 & 24 & 1.5 \\ \hline \end{tabular}
\end{table}
Table 3: Parameters of the model.