## 4 Multi-Objective Optimization  
### Significance Analysis of Influencing Factors  
There are many geometric parameters affecting the hydraulic performance of submersible pumps for wells, including the import and export placement angle, import and export diameter, outlet width, vane wrap angle, timing position, and other influencing factors. The impeller inlet and outlet diameters will have an impact on the assembly of the space guide vane and the radial dimensions of the submersible well pump, so they are not considered target parameters for this design optimization. The Plackett-Burman experimental design module of Design Expert 12.0 software was used to perform significance analysis on the impeller inlet and outlet placement angles, outlet width, blade wrap angle, and timing position. A multiple of 0.8 of the original model variable parameters was taken as the low level (\(-1\)), and a multiple of 1.2 of the original model variable parameters was taken as the high level (\(+1\)). For the temporal position, no rotation of the secondary impeller relative to the first impeller (i.e., 0\({}^{\circ}\) rotation of the secondary impeller) was taken as the low level (\(-1\)), and one-half rotation of the secondary impeller by one-half of the blade angle (i.e., 30\({}^{\circ}\) rotation of the secondary impeller) was taken as the high level (\(+1\)). Six sets of dummy factors, \(X_{6}\), \(X_{7}\), \(X_{8}\), \(X_{9}\), \(X_{10}\), and \(X_{11}\), were added as error analysis. Table 2 shows the names and level settings of the influencing factors. Twelve sets of tests were required, and the distribution of the high- and low-level arrangements of the influencing factors is shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline
**Variables** & **Parameter Name** & **Unit** & **Low Level** & **High Level** \\ \hline \(X_{1}\) & Imported placement angle \(\beta_{1}\) & \(/^{\circ}\) & 17 & 26 \\ \(X_{2}\) & Exit placement angle \(\beta_{2}\) & \(/^{\circ}\) & 21 & 32 \\ \(X_{3}\) & Outlet width \(b_{2}\) & /mm & 8 & 12 \\ \(X_{4}\) & Blade wrap angle \(\varphi\) & \(/^{\circ}\) & 77 & 116 \\ \(X_{5}\) & Impeller timing position & \(/^{\circ}\) & 0 & 30 \\ \(X_{6}\)–\(X_{11}\) & Virtual factors & — & — & — \\ \end{tabular}
\end{table} 
Table 2: Influencing factors and levels. Figure 6: Comparison of pump performance test results with numerical simulation results. According to the Plackett-Burman experimental design, the distribution order of the high and low levels of each influencing factor was arranged, and the specific values of the high and low levels corresponding to each influencing factor were substituted into Table 3. The 12 sets of structural parameters and a model of the 12 sets of submersible well pumps were created in CFturbo software. The model was imported into ANSYS19.2 CFX software for 12 sets of simulations, and the corresponding head and efficiency for each set of solutions were obtained as shown in Table 4.  
Table 5 demonstrates the degree of influence of the main geometric parameters on the head. As can be seen from the table, the percentages of the sum of squares of the outlet settling angle \(\beta_{2}\), outlet width \(b_{2}\), and blade wrap angle \(\varphi\) are 2.15%, 45.48%, and 9.36%, respectively, much greater than the percentages of the sum of squares of the inlet settling angle \(\beta_{1}\) and impeller timing position of 0.001% and 0.0777%, respectively. A positive coefficient assessment indicates that this influence is positively correlated with the results, while a negative coefficient assessment indicates that this influence is negatively correlated with the results. A higher coefficient assessment indicates that the factor has a greater influence on the results, while a lower coefficient assessment indicates that the factor has a smaller influence on the results.