## 2 Static Performance Optimization of Solenoid Electromagnetic Actuator
### Multi-Objective Genetic Algorithm Optimization for Structure Parameters of the LPS  
LPS is used to generate the push force to SPV, its performance directly influences the valve control quality. The static push-force magnitude, output efficiency and constant performance at different air gap are the main factors of the LPS. Thus, static performance optimization design objects include average push force, the standard deviation of the push force in the working stroke and the average push force to moving mass ratio. To solve this optimization issue, genetic algorithm is employed to obtain the optimal shape parameters. The presented 2D model is used to calculate the push force at each 0.2-mm interval air gap in the working stroke (from 1-mm air gap to 3.6-mm air gap). It means that there fourteen 2D models with different air gap A1 in one group of the LPS design parameters. All these work are accomplished by ANSYS MAXWELL (ANSYS, Inc., Canonsburg, PA, USA, 2019, version: ANSYS Electronics suite 2019R2). The following Equation (9) shows the optimization fitness function:  
\[\mathit{fitness}=\frac{\mathrm{F_{ex}}-\left|F_{AVG}\right|}{\mathrm{F_{ex}}}+ \frac{\mathrm{D_{ex}}-D}{\mathrm{D_{ex}}}+\frac{\mathrm{K_{ex}}-\frac{\left|F _{AVG}\right|}{\mathrm{K_{ex}}}}{\mathrm{K_{ex}}} \tag{9}\]  
Figure 5: Magnetic property of pure iron. Equation (9) includes three normalized objects and each factor: average electromagnetic force-\(F_{AVG}\), standard deviation of the absolute value of the static electromagnetic force-\(D\) and the average electromagnetic force to mass ratio of the moving parts in SPV-\(F_{AVG}\)/\(m\). In this study, \(F_{AVG}\) is calculated by the average push forces of each sampling air gap in the working stroke and the expectation average force-F\({}_{\text{ex}}\) is set as 170 N. It is used to evaluate the excitation force performance in the coil rated current (direct current). The LPS generates electromagnetic force to overcome the spring force and flow force of SPV and provides acceleration for the spool of SPV. The maximum flow force of SPV is 25-N and the maximum spring force is 50 N. The armature of the LPS pushes the spool from the end position (3.6-mm air gap) to the start position (1-mm air gap) in the LPS working stroke. The spring is used to store energy and the push the spool and the armature back to the end position of the working stroke. Thus, the magnitude of \(F_{AVG}\) is the key performance of the LPS for static requirement (overcome flow force) and dynamic requirement (large force generation). By the consideration of deviation between 2D simulation result and the measurement result, this study chose 170N as the value of \(F_{AVG}\) to ensure that \(F_{AVG}\) is twice as much as the sum of the spring force and flow force. \(D\) is used to evaluate the standard deviation of the push force at each sampling position and the expectation standard deviation of the push force--\(\text{D}_{\text{ex}}\)--is set as 10 N. It means that the difference between maximum push force and minimum push force is 65-N in the working range, at worst situation. In the Equation (9), \(F_{AVG}\)/\(m\) represents the push force output efficiency and the expectation force to mass ratio--\(\text{K}_{\text{ex}}\)--is set as 1.8 N/g. A genetic algorithm--100 population size and 49 generations--was employed to solve this optimization problem and the optimal shape design parameters was obtained. The GA optimization details and flow chart were shown in Figure 6. In this study, we set the individual crossover probability as 0.5 and variable crossover probability as 0.2 and variable exchange probability as 0.1 to guarantee that enough new individuals were produced. We also set the uniform mutation probability as 0.3 and individual mutation probability as 0.2 and variable mutation probability as 0.1 to escape local optimal result by the large mutation probability.