# Evaluating the Capacity Coordination in the Urban Multimodal Transport Network
## 2 Methodology
### Impedance Function of Links and Paths
\[I_{a}^{S}=t_{a}^{s0}+t_{a}^{s1}+\eta\times\cos t^{S}+penalty_{a}^{S},\ a\in A^{S}, \tag{10}\]  
\[I_{a}^{d}=t_{a}^{d0}+t_{a}^{d1}+\eta\times\left(\cos t^{d}+park^{d}\right),\ a\in A^{d}, \tag{11}\]  
where \(I_{a}^{S}\) is the impedance function of link \(a\) in the virtual bus-metro transfer network, \(I_{a}^{d}\) is the impedance function of link \(a\) in the virtual P + R transfer network, \(t_{a}^{s0}\) is the walk time of link \(a\) in the virtual bus-metro transfer network, \(t_{a}^{s1}\) is the mean waiting time of link \(a\) in the virtual bus-metro transfer network, \(\cos t^{S}\) is the transit fare of transferring from bus to metro, \(A^{S}\) is the set of links in the virtual bus-metro transfer network, \(t_{a}^{d0}\) is the walk time of link \(a\) in the virtual P + R transfer network, \(t_{a}^{d1}\) is the mean waiting time of link \(a\) in the virtual P + R transfer network, \(\cos t^{d}\) is the transit fare of transferring from car to metro, \(park^{d}\) is the parking fare of the car, and \(A^{d}\) is the set of links in the virtual P + R transfer network. The impedance along a path is the sum of the impedances along the links comprising that path. As for the five transport modes in urban multimodal transport network mentioned before, the expressions of path impedance function for three types of single modes were as follows:  
\[ \begin{array}{l}I_{a}^{car}=t_{a}^{u0}+t_{a}^{u1}&+\eta\\ &\times\cos t^{u}+t_{a}^{d0}+t_{a}^{c0}\bigg{[}1+\alpha\left(\frac{\
u_{e}}{ \lambda^{c}C_{a}}\right)^{\beta}\bigg{]}+\eta\times\rho\times x_{a}+\omega \times S_{a}^{c}\times T_{a}^{c},\ a\\ &\in A^{c}.\end{array}  \tag{12}\]  
\[ \begin{array}{l}I_{a}^{bus}=t_{a}^{u0}+t_{a}^{u1}&+\eta\times\cos t^{u}+t_{ a}^{f0}+t_{a}^{b0}+\omega\times S_{a}^{b}\times T_{a}^{b}+\frac{\
u_{b}}{BC_{a}^{c }}+t_{a}^{i0}+t_{a}^{i1}+\eta\times\cos t^{i}\\ &+penalty_{a}^{i},\ a\in A^{b}.\end{array}  \tag{13}\]  
\[ \begin{array}{l}I_{a}^{metro}=t_{a}^{u0}+t_{a}^{u1}&+\eta\times\cos t^{u}+t_ {a}^{d0}+t_{a}^{m0}+\omega\times S_{a}^{m}\times T_{a}^{m}+\frac{\
u_{b}}{BC_{a }^{c}}+t_{a}^{i0}+t_{a}^{i1}\\ &+\eta\times\cos t^{i}+penalty_{a}^{i},\ a\in A^{m}.\end{array}  \tag{14}\]According to the link impedance function of each sub-transport network and virtual link impedance function of transfer hubs introduced above, the two types of combined modes were as follows:  
\[I_{a}^{bus-metro}=T_{a1}^{u}+P_{a1}^{u}+T_{a2}^{b}+D_{a2}^{b}+I_{a3}^{d}+T_{a4}^ {m}+D_{a4}^{m}+T_{a5}^{d},\ a1,\ldots,a5\in a, \tag{15}\]  
\[I_{a}^{car-metro}=T_{a6}^{u} \begin{array}{ll}+P_{a6}^{u}+T_{a7}^{c}+P_{a7}^{ c}+D_{a7}^{c}+I_{a8}^{S}+T_{a9}^{m}+D_{a9}^{m}+T_{a10}^{d},\ a6,\ldots,a10\\ \in a,\end{array}  \tag{16}\]  
where \(a1\) is the link of the access connection network, \(a2\) is the link of the bus transit network, \(a3\) is the link of the virtual bus-metro transfer network, \(a4\) is the link of the urban rail transit network, \(a5\) is the link of the egress connection network, \(a6\) is the link of the access connection network, \(a7\) is the link of the roadway network, \(a8\) is the link of the virtual P + R transfer network, \(a9\) is the link of the urban rail transit network, and \(a10\) is the link of the egress connection network.