# Evaluating the Capacity Coordination in the Urban Multimodal Transport Network
## 2 Methodology
### Impedance Function of Links and Paths  
Each network link is typically associated with some impedance that affects the flow using it. Impedance can represent time, cost, utility, or any other relevant measure. The units of measurement of this impedance depend on the nature of the network and the link flows. Only links can be associated with impedance. Nodes represent merely the intersection of links and are not associated with any impedance to flow. Travelers choose the transport mode with the least generalized travel cost, which consists of time, cost, and comfort loss in this paper, which is expressed as shown in Equation (1). \[I_{a}=T_{a}+P_{a}+D_{a}, \tag{1}\]  
where \(I_{a}\) is the impedance function of link \(a\), \(T_{a}\) is the time cost of link \(a\), \(P_{a}\) is the fare cost of link \(a\), and \(D_{a}\) is the comfort loss of link \(a\). The connection network in this paper refers to the access segment from the origin to the network and the egress segment from the network to the destination by walking. Thus, the components of connection network impedance can be expressed as shown in Equation (2). \[ \begin{array}{c}T_{a}^{u}=t_{a}^{u0}+t_{a}^{u1},\ a\in A^{u},\\ P_{a}^{u}=\eta\times\cos\mu,\\ D_{a}^{u}=0,\end{array}  \tag{2}\]  
where \(T_{a}^{u}\) is the impedance function of link \(a\) in the access connection network, \(t_{a}^{u0}\) is the walk time cost of link \(a\) in the access connection network, \(t_{a}^{u1}\) is the mean waiting time cost of link \(a\) in the access connection network, \(A^{u}\) is the set of access links, \(P_{a}^{u}\) is the fare cost of link \(a\) in access connection network, if the transport mode chosen is car, \(P_{a}^{u}=0\), \(\eta\) is the fare cost-time dimension conversion parameter, \(\cos t^{u}\) is the fare cost of bus or metro, and \(D_{a}^{u}\) is the comfort loss of link \(a\) in access connection network. Similarly, the impedance of egress segment in connection network only involves walking time, as expressed in Equation (3). \[ \begin{array}{c}T_{a}^{d}=t_{a}^{d0},\ a\in A^{d},\\ P_{a}^{d}=0,\\ D_{a}^{u}=0,\end{array}  \tag{3}\]  
where \(t_{a}^{d0}\) is the walk time cost of link \(a\) on egress connection network, and \(A^{d}\) is the set of egress links. Figure 1: Schematic topology of the urban multimodal transport network. The main factors associated with multimodal transport are variables related to the trip itself, type of origin and destination area, trip type, and trip length. In order to account for the contribution of different transport modes, the travel type in this paper was concentrated on commuting trips. Specially, the characteristics of three unimodal transport modes are the road capacity, road length, fuel charge, bus/metro ticket fee, bus/metro capacity, designed passenger volume of bus/metro, and speed of modes. The travel time of cars in roadway network is limited by the traffic flow and the road capacity. In this paper, regardless of the intersection delay, the travel time of cars in roadway network was given by the BPR function, as shown in Equation (4). \[T_{a}^{c}=t_{a}^{c0}\Bigg{[}1+\alpha\bigg{(}\frac{v_{a}}{\lambda^{c}C_{a}^{c}} \bigg{)}^{\beta}\Bigg{]},\ a\in A^{c}, \tag{4}\]  
where \(T_{a}^{c}\) is the time cost of link \(a\) in roadway network, \(t_{a}^{c0}\) is the free travel time, \(v_{a}\) is the passenger flow on link \(a\), \(\lambda^{c}\) is the conversion parameter which converts vehicle flow into equivalent passenger flow, \(\alpha,\ \beta\) are the undetermined coefficients, \(C_{a}^{c}\) is the road capacity of link \(a\) in roadway network, and \(A^{c}\) is the set of links in roadway network. Fuel charge is the main fare cost of car travel on the roadway network. Therefore, the fare cost of cars on the roadway network was calculated as shown in Equation (5). \[P_{a}^{c}=\eta\times\rho\times x_{a}, \tag{5}\]  
where \(x_{a}\) is the length of link \(a\) in the roadway network, and \(\rho\) is the fuel fee.