## 4 Passenger Flow Assignment Model
### SUE Assignment Model Formulation  
There are many alternative travel paths between O-D pairs, and the choice of travel path is affected by multiple factors. The discrete choice model based on random utility theory simulates the travel behavior made by travelers in the feasible choice set. The multinomial logit model (MNL), based on the discrete choice theory, is often used to solve the problem of multi-alternatives choice. However, due to the property of independence of irrelevant alternatives (IIA), this will lead to excessive flow being assigned to overlapping paths [40]. In this paper, we introduce a size variable in the utility of a path and propose the path-size logit (PSL) model which considers the path length to eliminate the IIA property of traditional logit models [41]. The choice probability of the \(k\)th path can be calculated by \[p^{k}_{(i,j)}=\frac{\exp(V^{k}_{(i,j)}+\ln S^{k}_{(i,j)})}{\sum\limits_{k^{\prime} \in K(i,j)}\exp(V^{k^{\prime}}_{(i,j)}+\ln S^{k^{\prime}}_{(i,j)})} \tag{16}\]  
where, \(V^{k}_{(i,j)}\) is the utility of path \(k\) between O-D pair \((i,j)\) and can be calculated by Equation (17), \(S^{k}_{(i,j)}\) is the correction term and defined by Equation (18), and \(K(i,j)\) represents the path set between O-D pair. Furthermore, we give  
\[V^{k}_{(i,j)}=-\theta\cdot C^{k}_{(i,j)} \tag{17}\]  
\[S^{k}_{(i,j)}=\sum\limits_{a\in\Gamma^{k}_{(i,j)}}\frac{l_{a}}{\sum\limits_{k ^{\prime}\in K(i,j)}\delta_{ak^{\prime}}\frac{l^{*}_{X(i,j)}}{L_{k^{\prime}}}} \tag{18}\]  
where \(\theta\) is the scale parameter that converts the generalized travel cost to the utility, \(\Gamma^{k}_{(i,j)}\) is the set of links in path \(k\) between O-D pair, \(l_{a}\) and \(L^{k}_{(i,j)}\) are the length of link \(a\) and path \(k\), \(\delta_{ak^{\prime}}\) is the link-path incidence variable, \(\delta_{ak^{\prime}}=1\) if link \(a\) is on path \(k^{\prime}\) and \(\delta_{ak^{\prime}}=0\), otherwise, and \(L^{*}_{K(i,j)}\) is the length of the shortest path in \(K(i,j)\).  
Based on the classical logit SUE assignment model proposed by [PERSON] [42], the improved PSL-SUE model of the schedule-based network considering the train capacity is established:  
\[\min Z(f)=\frac{1}{\theta}\sum\limits_{(i,j)\in(O,D)}\sum\limits_{k\in K(i,j )}f^{k}_{(i,j)}(\ln f^{k}_{(i,j)}-\ln s^{k}_{(i,j)})+\sum\limits_{a\in E}\int \
olimits_{0}^{x_{a}}c_{a}(w)dw \tag{19}\]  
\[\text{s.t.}\sum\limits_{k\in K(i,j)}f^{k}_{(i,j)}=q_{(i,j)},\forall(i,j)\in(O,D) \tag{20}\]  
\[f^{k}_{(i,j)}\geq 0,\forall(i,j)\in(O,D),k\in K(i,j) \tag{21}\]  
\[x_{a}=\sum\limits_{(i,j)\in(O,D)}\sum\limits_{k\in K(i,j)}f^{k}_{(i,j)}\delta^ {a}_{(i,j)k^{\prime}}\forall a\in E. \tag{22}\]  
Here \(f^{k}_{(i,j)}\) denotes the passenger flow on the \(k\)th path between O-D pair \((i,j)\), \((O,D)\) is the set of all spatiotemporal O-D pairs, \(x_{a}\) denotes the flow on arc \(a\), and \(q_{(i,j)}\) represents the passenger demand between O-D pair \((i,j)\).  
Equation (20) represents the conservation of path flow and passenger volume of the O-D pair. Formula (21) is a non-negativity restriction of path flow. The relationship between arc flow and path flow is restricted in Equation (22).  
The equivalence and uniqueness of the model can be proved by constructing the Lagrangian function for PSL-SUE problem, and the Kuhn-Tucker conditions for the PSL-SUE problem can be formulated as follows:  
\[\frac{1}{\theta}(\ln f^{k}_{(i,j)}-\ln S^{k}_{(i,j)}+1)+\sum\limits_{a}c_{a} (x_{a})\delta^{a}_{(i,j),k}-\varphi_{(i,j)}-\mu^{k}_{(i,j)}=0 \tag{23}\]  
\[-\mu^{k}_{(i,j)}f^{k}_{(i,j)}=0 \tag{24}\]  
where \(\varphi_{(i,j)}\) and \(\mu^{k}_{(i,j)}\) are the corresponding Lagrangian multipliers to Equations (20) and (21). Equation (23) can be easily transformed into the following PSL model:  
\[p^{k}_{(i,j)}=\frac{\exp(-\theta\cdot C^{k}_{(i,j)}+\ln S^{k}_{(i,j)})}{\sum \limits_{k^{\prime}\in K(i,j)}\exp(-\theta\cdot C^{k^{\prime}}_{(i,j)}+\ln S^ {k^{\prime}}_{(i,j)})}. \tag{25}\]