# Pantograph-Catenary Interaction Prediction Model Based on SCSA-RBF Network
## 3 Pantograph-Catenary Interaction Prediction Model  
### Model Architecture  
In the investigated pantograph-catenary system, the decision vector \(\mathbf{X}=[v,\,m_{3},\,m_{2},\,m_{1},\,k_{3},\,k_{2},\,c_{1}]^{\mathrm{T}}\) composed of the running speed and six pantograph equivalent parameters was variable, while the other three pantograph equivalent parameters, catenary parameters, and contact stiffness were given as constants. According to engineering practice, the value range for each decision variable in \(\mathbf{X}\) was determined. Using the Latin hypercube sampling method [24], three disjoint sets, \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), were composed in a seven-dimensional sample space of \(\mathbf{X}\). Based on the pantograph-catenary FEM model, dynamic calculations were carried out to obtain the FEM calculated values of the time-domain signal of the panhead acceleration \(\tilde{z}_{3}(t)\), the time-domain signal of the contact force \(F_{\mathrm{c}}(t)\), and the SDCF \(\sigma\). The SCSA-RBF network [21] was used to construct a surrogate model of the panhead acceleration at time \(t_{i}\), with set \(\mathbf{A}\) as the training set and set \(\mathbf{B}\) as the validation set. Minimizing \(\sum\limits_{p=1}^{N_{p}}\left(1-R^{2}\right)/N_{p}\) of the validation set is the optimization objective. Among them, \(R^{2}_{p}\) is the determination coefficient between the model-predicted value and the FEM-calculated value of the time-domain signal of the panhead vertical acceleration of the validation set sample \(\mathbf{X}_{p}\). The closer \(R^{2}_{p}\) is to one, smaller the error between the model-predicted value and the FEM-calculated value; \(p=1,\,\ldots,\) and \(N_{p}\), where \(N_{p}\) is the number of samples in the validation set; \(t_{i}=0.1i/v\) (\(l=0,\,1,\,\ldots,\) and \(11,001\)). The SCSA-RBF network is a radial basis function network (RBF) [25] trained by the selective crow search algorithm (SCSA). SCSA introduces a roulette search operator [26] based on a crow search algorithm [27], which effectively improves the solution accuracy and convergence speed compared with the crow search algorithm. Using \(\Delta\)UB as the training set, the final surrogate model \(\widehat{\tilde{z}}_{3}(t_{i})=h_{i}(\mathbf{X})\) was obtained. For any \(\mathbf{X}_{i}\)\(\widehat{\tilde{z}}_{3}(t_{i})\) was obtained through the above surrogate model. The panhead vertical acceleration \(\widehat{\tilde{z}}_{3}\big{(}t_{i}\big{)}\) at time \(t_{j}=0.005(j-1)\) (\(j=1,\,2,\,\ldots\)) was calculated using the cubic spline interpolation method. Based on the Simpson formula and the trapezoidal quadrature formula, combined with the initial conditions of \(\tilde{z}_{3}(0)=0\) and \(z_{3}(0)=0\), the vertical velocity and vertical displacement of the panhead at \(t_{j+1}\) can be obtained. \[\widehat{\tilde{z}}_{3}\big{(}t_{j+1}\big{)}=\widehat{\tilde{z}}_{3}\big{(}t _{j}\big{)}+0.0025\big{[}\widehat{\tilde{z}}_{3}\big{(}t_{j}\big{)}+\widehat{ \tilde{z}}_{3}\big{(}t_{j+1}\big{)}\big{]} \tag{2}\]  
and  
\[\widehat{z}_{3}\big{(}t_{j+1}\big{)}=\widehat{z}_{3}\big{(}t_{j}\big{)}+0.005 \big{[}5\widehat{\tilde{z}}_{3}\big{(}t_{j}\big{)}+\widehat{\tilde{z}}_{3} \big{(}t_{j+1}\big{)}\big{]}/6+0.005^{2}\bigg{[}3\widehat{\tilde{z}}_{3}\big{(} t_{j}\big{)}+\widehat{\tilde{z}}_{3}\big{(}t_{j+1}\big{)}\bigg{]}/12. \tag{3}\]  
Taking \(\widehat{\tilde{z}}_{3}\big{(}t_{i}\big{)}\), \(\widehat{\tilde{z}}_{3}\big{(}t_{j}\big{)}\), and \(\widehat{\tilde{z}}_{3}\big{(}t_{j}\big{)}\) as the boundary conditions, combined with the initial conditions, the pantograph dynamic Equation (1) can be solved to obtain the predicted value of the pantograph-catenary contact force \(\widehat{F}_{\mathrm{c}}\big{(}t_{j}\big{)}\). In the numerical calculation, the calculation parameters  
Figure 1: Pantograph–catenary finite element model. and the dynamics iteration steps are the same as those in Section 2.