# Evaluating the Capacity Coordination in the Urban Multimodal Transport Network
## 2 Methodology
### Impedance Function of Links and Paths
The comfort loss of cars in the roadway network is related to its travel time on road links, which can be expressed as shown in Equation (6). \[D_{a}^{c}=\omega\times S_{a}^{c}\times T_{a}^{c}, \tag{6}\]  
where \(S_{a}^{c}\) is the comfort loss per unit time of link \(a\) in roadway network, \(\omega\) is the comfort loss-time dimension conversion parameter. This study assumed that buses run in exclusive lanes and, thus, will be less affected by other vehicles. The driving time was taken as relatively fixed. In addition, conventional buses adopt fixed pricing, whose fare is charged while boarding and, thus, cannot be allocated to each link. As the transit fare (bus and metro) was added to the previous access connection network, the fare cost in transit networks (bus and metro) was 0. The comfort loss for bus travelers is not only affected by travel time, but also related to the congested degree in bus vehicles. Overall, the components of link impedance in the bus transit network can be expressed as shown in Equation (7). \[T_{a}^{b}=t_{a}^{b0},\ a\in A^{b}, \tag{7}\] \[P_{a}^{b}=0,\] \[D_{a}^{b}=\omega\times S_{a}^{b}\times T_{a}^{b}+\frac{v_{a}}{BC_ {a}^{c}},\]  
where \(T_{a}^{b}\) is the time cost of link \(a\) in the bus transit network, \(t_{a}^{b0}\) is the actual driving time of the bus on link \(a\), \(A^{b}\) is the set of links in the bus transit network, \(S_{a}^{b}\) is the comfort loss per unit time of link \(a\) in the bus transit network, \(B\) is the designed passenger volume of the bus, and \(C_{a}^{b}\) is the bus capacity of link \(a\) in bus transit network. Similarly, the components of the link impedance in the urban rail transit network can be expressed as shown in Equation (8). \[T_{a}^{m}=t_{a}^{m0},\ a\in A^{m}, \tag{8}\] \[P_{a}^{m}=0,\] \[D_{a}^{m}=\omega\times S_{a}^{m}\times T_{a}^{m}+\frac{v_{a}}{BC_ {a}^{c}},\]where \(T_{a}^{m}\) is the time cost of link \(a\) in the urban rail transit network, \(t_{a}^{m0}\) is the actual driving time of the metro on link \(a\), \(A^{m}\) is the set of links in the urban rail transit network, \(S_{a}^{m}\) is the comfort loss per unit time of link \(a\) in the urban rail transit network, \(E\) is the designed passenger volume of the metro, and \(C_{a}^{m}\) is the metro capacity of link \(a\) in urban rail transit network. This study considered two types of transfer: internal transfer between transit lines and transfer between different modes. There were also three items composing the impedance of virtual internal transfer links, as expressed in Equation (9). \[ \begin{array}{l}T_{a}^{i}=t_{a}^{i0}+t_{a}^{i1},\ a\in A^{i},\\ P_{a}=\eta\times\cos t^{i},\\ S_{a}^{i}=penalty_{a}^{i},\end{array}  \tag{9}\]  
where \(T_{a}^{i}\) is the time cost of link \(a\) in the virtual internal transfer network, \(t_{a}^{i0}\) is the walk time of link \(a\) in the virtual internal transfer network, \(t_{a}^{i1}\) is the mean waiting time of link \(a\) in the virtual internal transfer network, \(A^{i}\) is the set of links in the virtual internal transfer network, \(P_{a}^{i}\) is the fare cost of link \(a\) in the virtual internal transfer network, \(\cos t^{i}\) is the transit fare of transferring between bus lines or metro lines, \(S_{a}^{i}\) is the transfer penalty of link \(a\) in the virtual internal transfer network, and \(penalty_{a}^{i}\) is the convert transfer penalty into equivalent in-vehicle time. Bus-to-metro and car-to-metro are the two typical combined modes in an urban multimodal transport network. The corresponding transfer hubs considered in this paper were bus-metro transfer hubs and P + R transfer hubs, and the impedance formulas of the transfer hub virtual links were expressed as shown in Equations (10) and (11).