## 5 Development of Accurate Oil Temperature Prediction Model Based on CAM
### Design Variable Selection and Sensitivity Analysis  
Theoretical analysis indicates that the ORFR, main circuit flow rate, working load, and cooling power all exert a certain influence on the steady-state oil temperature of the HCS. Furthermore, CAM modeling has initially identified the effective heat-dissipation area of the oil tank, the total efficiency of the hydraulic pump and motor, the roughness of the hydraulic pipe, and the pressure of the oil refill system as key design variables. The larger heat-dissipation area of the oil tank indicates a stronger heat-dissipation ability, resulting in lower hydraulic oil temperatures. The total efficiency of the hydraulic pump and motor comprises mechanical and volumetric efficiencies. The higher total efficiency means better energy conversion ability and less heat generation in the HCS. The roughness of hydraulic pipes significantly influences the friction coefficient of flowing hydraulic oil, with an increased surface roughness leading to greater heat generation. The pressure of the oil-refill system acts as back pressure, indirectly influencing HCS efficiency. The information of the designed variables is listed in Table 2.  
In the development of CAM, a lower dimension leads to a reduced number of elements required in the sample space and increased model fitting accuracy. The influence on steady-state temperature varies among the initially selected nine design variables, necessitating the elimination of some design variables with lower impact to reduce the dimension of the sample space. Sensitivity analysis is commonly employed as a method for dimension reduction, and Sobol method, based on variance, provides a quantitative consideration of variable interactions [23]. The Sobol method decomposes the objective function into different sub-functions according to Formula (18) [24]. \[f(x)=f_{0}+\sum_{i=1}^{k}f_{i}(x_{i})+\sum_{1\leqslant i<j\leqslant k}f\big{(} x_{i},x_{j}\big{)}+\cdots+f_{1,2,\cdots,k}(x_{1},x_{2},\cdots,x_{k}) \tag{18}\]  
where \(x\) is a vector of \(k\) design variables, and \(f_{0}\) is a constant. By individually calculating the variances of each term in Formula (18), the sensitivity of each variable can be determined. The first-order sensitivity coefficients obtained using \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable Name** & **Code Name** & **Range** \\ \hline ORFR (L/min) & A & [8, 16] \\ main circuit flow rate (L/min) & B & [48, 64] \\ working load (N-m) & C & [160, 320] \\ cooling power/kW & D & [4.80, 6.00] \\ Effective heat-dissipation area of the tank/m\({}^{2}\) & E & [0.80, 1.20] \\ Total efficiency of hydraulic pump & F & [0.75, 0.90] \\ Total efficiency of hydraulic motor & G & [0.75, 0.90] \\ Hydraulic pipe roughness/mm & H & [0.02, 0.06] \\ Pressure of oil-refill system/MPa & I & [1.20, 2.40] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design variables information. the Sobol method represent the sensitivity without accounting for interactions between variables, while the \(k\)-order sensitivity coefficients reflect the sensitivity when considering interactions among \(k\) variables. The global sensitivity coefficient is derived from considering interactions among all variables. Given the significant influence of the steady-state temperature of HPP on the performance of HCS, this study focuses solely on examining the sensitivity of the nine design variables to the steady-state temperature. The findings are presented in Figure 11.  
The factors with a sensitivity of less than 0.2 will be excluded. The results indicate that the sensitivity of the first seven factors exceeds 0.2, and factor C (working load) has the most significant impact on the steady-state temperature of HPP. Conversely, factors H (hydraulic pipe roughness) and I (replenishment pressure) exert relatively minimal influence. In the HCS, heat generation primarily stems from volumetric loss and mechanical loss in hydraulic components, with frictional losses due to hydraulic pipe roughness being comparatively low.