## 5 Conclusions  
The effective vibration area (EVA) includes most of the catenary vibration caused by pantograph-catenary interactions and is the basis of the real-time catenary model for hardware-in-loop simulation. However, while the length of EVA can directly influence the calculation accuracy and efficiency of the real-time catenary model, the way to calculate the length of EVA and the influence of different key parameters on the length of EVA is not considered so far. In this work, the length of EVA in the pantograph-catenary system is investigated. Based on a pantograph-catenary interaction system, a new method is developed based on the energy distribution to obtain the length of the effective vibration area (EVA) of the catenary in the pantograph-catenary interaction system. Then, the length of EVA under the influence of different contact wire tensions and vehicle velocities is investigated. The appropriate length of EVA with respect to different vehicle velocities and contact wire tensions is obtained. The real-time catenary model is used to calculate the dynamic responses of the pantograph-catenary interaction under different lengths of EVA, and the results are compared with those from measurement data to validate the  
Figure 8: Time histories of the pantograph–catenary contact force at \(V=222\) km/h with respect to different lengths of EVA \(d_{e}\): (**a**) \(d_{e}=60\) m; (**b**) \(d_{e}=120\) m; (**c**) \(d_{e}=215\) m and (**d**) \(d_{e}=240\) m.  
Figure 9: Mean contact force and standard deviation of the pantograph–catenary contact force with respect to different lengths of EVA: (**a**) Mean contact force and (**b**) standard deviation. obtained appropriate length of EVA. The present research results enhance the theory of the new efficient pantograph-catenary interaction models based on EVA and can be used as a supplement to these models. Based on the investigation results, the following conclusions are drawn:  
1. The present methods can obtain the appropriate length of EVA in the pantograph-catenary system. The length of the EVA corresponding to the converged total energy obtains accurate results in the real-time catenary model. Therefore, it is suggested to use the present method to obtain an appropriate length of EVA before using the real-time catenary model and other similar methods with different kinds of pantograph-catenary interaction systems considered. 2. The length of the EVA is greatly influenced by the vehicle velocity and contact wire tension. The appropriate length of EVA increases with increasing vehicle velocity and decreases with increasing contact wire tension. Based on the calculation results, the appropriate length of EVA should be no less than 160 m under different situations. This value can be used as a reference for the real-time catenary model and other similar methods based on the EVA. In the present work, a specific experiment to validate and investigate the length of EVA in the pantograph-catenary system under different conditions is not considered due to the limitation of time and funds. In the next work, a new experiment will be conducted to investigate the length of EVA in the pantograph catenary system under different conditions, especially the influence of wind and ice. Methodology, L.C. ; Software, L.P. and Y.X. ; Validation, L.C., L.P. and C.H. ; Formal analysis, Y.X. ; Data curation, C.H. ; Writing--original draft, L.C. ; Writing--review & editing, L.P. and Y.X. All authors have read and agreed to the published version of the manuscript. This work is supported by the Research Project of China Academy of Railway Sciences Group Co., Ltd. (Grant No. 2023Y[271]), China Railway Group (Grant No. N2023G065) and the National Natural Science Foundation of China (grant number 12302048). Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. Authors [PERSON] and [PERSON] were employed by the company China Academy of Railway Sciences Co., Ltd.