## 4 Multi-Objective Optimization
### Multi-Objective Optimization Design  
From the previous section, it is clear that both the bypass magnetic bridges and the position of the permanent magnets have an effect on the demagnetization of the permanent magnets in the proposed model. To further understand the effect of other parameters on permanent magnet demagnetization, a multi-objective global optimization of the rotor structure of the proposed model is required. Torque performance is also particularly important for PMA-SynRMs. So, in this paper, the rotor structure of the proposed model will be optimized with average torque, torque ripple, and maximum demagnetization rate as the optimization objectives. This ensures good torque performance while enhancing anti-demagnetization performance to reduce the risk of permanent magnet demagnetization. Figure 8 shows the flowchart of the multi-objective optimization. For the optimization objectives maximum demagnetization rate, the average torque and torque ripple are defined by Equation (3):  
\[\left\{\begin{array}{l}D_{max}=f_{max}(\delta)\\ \\ T_{ave}=\dfrac{\sum\limits_{i=1}^{n}T_{i}}{n}\\ T_{rip}=\dfrac{T_{max}-T_{min}}{T_{ave}}\times 100\%\end{array}\right. \tag{3}\]  
where \(T_{ave}\), \(T_{rip}\), and \(D_{max}\) are the average torque, torque ripple, and maximum demagnetization rate, respectively; \(n\) is the number of steps of the motor in one operating period; \(T_{i}\) is the torque at step \(i\); \(T_{max}\) and \(T_{min}\) are the maximum torque and minimum torque of the motor in one operating period, respectively; and \(\delta\) is the demagnetization rate of the permanent magnets.  
The average torque, torque ripple, and maximum demagnetization rate are taken as the optimization objectives. The optimization objective function can be expressed by Equation (4):  
\[\left\{\begin{array}{l}O_{1}{=}\;max(T_{ave})\\ O_{2}{=}\;min(T_{rip})\\ O_{3}{=}\;min(D_{max})\end{array}\right. \tag{4}\]  
Figure 7: Effect of flux barrier angle C3 and permanent magnet drop height \(H\) on demagnetization.  
where \(O_{1}\), \(O_{2}\), and \(O_{3}\) are the average torque optimization objective, torque ripple optimization objective, and maximum demagnetization rate optimization objective, respectively.  
The parameters involved in this multi-objective optimization can be seen in Figure 4, and the constraint function in the multi-objective optimization can be expressed by Equation (5):  
\[s.t.\left\{\begin{array}{l}20\leq Ck\leq 25,k{=}1,2,3,4,5,6\\ 0\leq C7\leq 90;1\leq\omega\leq 2;0\leq H\leq 3\\ 0\leq b\leq 1;0\leq h\leq 1;0\leq q\leq 1;3.5\leq d\leq 4.5\\ N_{1}=T_{are}-83\geq 0\\ N_{2}=T_{rip}-0.15\leq 0\\ N_{3}=D_{max}-0.2\leq 0\end{array}\right. \tag{5}\]  
where \(Ck\) is the angle of the flux barrier; C7 is the deflection angle of the center permanent magnet; \(\omega\) is the width of the connecting flux barrier; \(b\) is the width of the bypass magnetic bridge; \(h\) is the height of the bypass magnetic bridge; \(q\) is the distance between the bypass magnetic bridges; \(d\) is the distance between the bypass magnetic bridge and the end of the flux barrier; \(H\) is the height of the descending permanent magnets; and \(N_{1}\), \(N_{2}\), and \(N_{3}\) are the constraint functions for the average torque, torque ripple, and maximum demagnetization rate, respectively.