## 2 Formulation of the Pantograph-catenary Interaction System and Length of EVA
### Model of the Catenary System  
The catenary system is modeled based on the stitched catenary system, which mainly consists of messenger wire, contact wire, droppers, catenary suspensions, stitch wires, and registration arms. First, the contact wire and messenger wire are considered Euler--Bernoulli beams [26], with their vertical, lateral, and torsional motions considered. Based on the MSM, the kinematic energy of the contact wire and messenger wire can be expressed as  
\[T_{m}=\frac{1}{2}\int_{0}^{L}\rho_{m}A_{m}\dot{y}_{m}^{2}dx+\frac{1}{2}\int_{0 }^{L}\rho_{m}A_{m}\dot{z}_{m}^{2}dx+\frac{1}{2}\int_{0}^{L}\rho_{m}\big{(}I_{ ym}+I_{zm}\big{)}\dot{\phi}_{m}^{2}dx \tag{1}\]  
Figure 2: Model of the pantograph–catenary interaction system. \[T_{c}=\frac{1}{2}\int_{0}^{L}\rho_{c}A_{c}\dot{y}_{c}^{2}dx+\frac{1}{2}\int_{0}^{L }\rho_{c}A_{c}\dot{z}_{c}^{2}dx+\frac{1}{2}\int_{0}^{L}\rho_{c}\big{(}I_{yc}+I_{zc }\big{)}\dot{\phi}_{c}^{2}dx \tag{2}\]  
where the subscripts \(m\) and \(c\) correspond to the messenger and contact wire, respectively. \(L\) is the length of the catenary model, and \(y\), \(z\), and \(\phi\) are the vertical, lateral, and torsional motions of the wire cross-section, respectively. \(\rho\) is the density of the beam, and \(A\) is the cross-sectional area of the wire. \(I_{y}\) and \(I_{z}\) are the second moments of the wire cross-section. The overdot on \(y\), \(z\), and \(\phi\) means the time derivative of these motions. Similarly, the strain energies of the messenger and contact wire can be expressed as:  
\[ \begin{array}{l}\hskip-14.226378 ptU_{m}=\frac{1}{2}\int_{0}^{L}E_{m}I_{ym} \big{(}z_{m}^{\prime\prime}\big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}E_{m}I_{zm} \big{(}y_{m}^{\prime\prime}\big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}G_{m}I_{pm} \big{(}\phi_{m}^{\prime\prime}\big{)}^{2}dx\\ \hskip 14.226378 pt+\frac{1}{2}\int_{0}^{L}T_{sm}\big{(}y_{m}^{\prime\prime} \big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}T_{sm}\big{(}z_{m}^{\prime\prime}\big{)} ^{2}dx\end{array}  \tag{3}\]  
\[ \begin{array}{l}\hskip-14.226378 ptU_{c}=\frac{1}{2}\int_{0}^{L}E_{c}I_{yc} \big{(}z_{c}^{\prime\prime}\big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}E_{c}I_{zc} \big{(}y_{c}^{\prime\prime}\big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}G_{c}I_{pc} \big{(}\phi_{c}^{\prime\prime}\big{)}^{2}dx\\ \hskip 14.226378 pt+\frac{1}{2}\int_{0}^{L}T_{sc}\big{(}y_{c}^{\prime} \big{)}^{2}dx+\frac{1}{2}\int_{0}^{L}T_{sc}(z_{c}^{\prime})^{2}dx\end{array}  \tag{4}\]  
where \(E\) and \(G\) are the Young's modulus and shear modulus of the wire material, respectively. \(T_{s}\) is the tension of the wire. \((\cdot)^{{}^{\prime}}\) denotes the partial derivative of a variable \((\cdot)\) with respect to coordinate \(x\). Based on the [PERSON] method, the displacement and rotations of the messenger and contact wire in Equations (1)-(4) can be further expressed as:  
\[y_{m}(x,t)=\mathbf{Y}_{m}(x)\mathbf{q}_{my}(t),\,z_{m}(x,t)=\mathbf{Z}_{m}(x) \mathbf{q}_{mz}(t),\,\phi_{m}(x,t)=\mathbf{\Phi}_{m}(x)\mathbf{q}_{\theta mx}(t) \tag{5}\]  
\[y_{c}(x,t)=\mathbf{Y}_{c}(x)\mathbf{q}_{cy}(t),\,z_{c}(x,t)=\mathbf{Z}_{c}(x) \mathbf{q}_{cz}(t),\,\phi_{c}(x,t)=\mathbf{\Phi}_{c}(x)\mathbf{q}_{\theta cx}(t) \tag{6}\]  
where \(\mathbf{Y}\), \(\mathbf{Z}\), and \(\mathbf{\Phi}\) are trial functions, and \(\mathbf{q}_{y}\), \(\mathbf{q}_{z}\), and \(\mathbf{q}_{\theta}\) are the vectors of the corresponding generalized coordinates. In the present model, the boundary conditions of the messenger and contact wire are considered as simply supported, and the expressions of \(\mathbf{Y}\), \(\mathbf{Z}\), and \(\mathbf{\Phi}\) can be seen in [26]. Based on Equations (5) and (6), in Equations (1)-(4), there are:  
\[\dot{y}_{m}=\mathbf{Y}_{m}\dot{\mathbf{q}}_{my},\,\dot{z}_{m}=\mathbf{Z}_{m} \dot{\mathbf{q}}_{mz},\,\dot{\phi}_{m}=\mathbf{\Phi}_{m}\dot{\mathbf{q}}_{my}, \,\dot{y}_{c}=\mathbf{Y}_{c}\dot{\mathbf{q}}_{cy},\,\dot{z}_{c}=\mathbf{Z}_{c} \dot{\mathbf{q}}_{cz},\,\dot{\phi}_{c}=\mathbf{derivation process of \(\mathbf{Q}_{F}\) is the same as that shown in [24].