# Contact Force Surrogate Model and Its Application in Pantograph-Catenary Parameter Optimization
## 1 Introduction  
The problem of the pantograph-catenary relationship becomes more prominent, and the design of pantograph-catenary faces more technical challenges with further increases in train speed [1; 2; 3]. The contact force between the panhead strip and the contact wire is a direct reflection of the pantograph-catenary interaction. The mean contact force and the standard deviation of contact force (SDCF) are the main evaluation indicators of current collection quality [4; 5; 6]. A good pantograph-catenary performance refers to minimizing the SDCF as much as possible while satisfying the requirements for the average contact force. Therefore, to improve the current collection quality, it is common to optimize the pantograph-catenary parameters to minimize the objective function, i.e., the SDCF. To reduce the computational complexity and improve optimization efficiency in parameter optimization, it is effective to conduct a sensitivity analysis of the SDCF with respect to pantograph-catenary parameters first. For example, the first-order sensitivities of nine pantograph parameters for a GPU pantograph and a simple suspension catenary at 100-400 km/h [7], and a DSA380 pantograph and a stitched suspension catenary at 350 km/h [8]; the first-order sensitivities of four catenary parameters and nine pantograph parameters for a simple suspension catenary at 90-150 km/h [9]; the first- and second-order sensitivities of four catenary parameters for a CX-type pantograph and a simple suspension catenary at 300 km/h [10]. Existing reports indicate that the sensitivity to the equivalent mass of the panhead vertical vibration is the highest among the pantograph parameters, while other parameters show significant differences in sensitivity rating. The sensitivitiesto the span and contact wire density are higher than those to the pantograph parameters, and the sensitivities to parameters such as the pre-tensions of the contact wire and messenger wire are relatively high among the catenary parameters. However, differences in the sensitivity analysis results of pantograph-catenary parameters arise from variations in the operating speed, catenary type, and baseline parameters considered in different reports. Additionally, as pantograph-catenary parameters involve multiple parameters with strong nonlinear coupling, it is essential to investigate higher-order or total sensitivities that characterize the interaction effects between pantograph-catenary parameters. Typically, the objective of the joint optimization of pantograph-catenary parameters is to minimize the SDCF. In terms of single-parameter optimization, the control variable method can be used effectively [11]. In terms of multi-parameter optimization, the robust design technique [12] and optimization algorithms [13; 14] can be used to search for the optimal solutions. As for optimization algorithms, the optimization iteration process involves a significant number of dynamic calculations for pantograph-catenary samples. The more parameters to be optimized, the greater the number of samples required. Classical pantograph-catenary finite element models consume approximately 10 core hours per sample, resulting in substantial computational costs. To mitigate computational resource consumption, it is a viable approach to use the pantograph-catenary finite element method to precompute a sufficient number of pantograph-catenary interaction samples and then employ data fitting or machine learning algorithms to establish the functional relationship between the SDCF and pantograph-catenary parameters. This approach can significantly alleviate the computational burden in optimization iterations. Based on a multi-layer feedforward deep residual neural network, a surrogate model of the SDCF with respect to 11 catenary parameters was established [15].