## 3 Equilibrium Conditions
\]  
The first HV commuter parking at the peripheral parking cluster experiences the same generalized travel cost as the last HV commuter parking at the central parking cluster, i.e., \(c_{t_{2},2}^{h}=c_{t_{1},1}^{h}\), which gives  
\[t_{2}=t^{*}-\frac{N-K_{1}}{s}-\frac{\beta K_{1}}{\alpha^{h}s}+\frac{(\eta^{h}- \beta)\Delta w-\Delta p}{\alpha^{h}}. \tag{7}\]  
As the remaining \(N^{h}-K_{1}\) HV commuters choose to depart from home during the time interval \((t_{2},t_{3})\), we have the condition that  
\[t_{3}-t_{2}=\frac{N^{h}-K_{1}}{r^{h}}.\]  
Thus, the end of the departure times for HV commuters is given by  
\[t_{3} = t^{*}-\frac{N^{a}}{s}-\frac{\beta N^{h}}{\alpha^{h}s}+\frac{( \eta^{h}-\beta)\Delta w-\Delta p}{\alpha^{h}}. \tag{8}\]  
where \(t_{3}\) also represents the start of the departure times for AV commuters. Considering that the AV commuters prefer the farther (peripheral) cluster with sufficient parking slots,we have the relation between the end of the departure times for AV commuters \(t_{4}\) and \(t_{3}\), i.e., \(t_{4}-t_{3}=\frac{N^{a}}{r_{4}}\). Therefore, the critical time \(t_{4}\) is  
\[t_{4} = t^{*}-\frac{\beta N^{a}}{\alpha^{a}s}-\frac{\beta N^{h}}{\alpha^{ h}s}+\frac{(\eta^{h}-\beta)\Delta w-\Delta p}{\alpha^{h}}. \tag{9}\]  
At equilibrium, the individual generalized travel costs of AV and HV commuters are derived below. \[c^{a} = \beta\frac{N^{a}}{s}+\frac{\alpha^{a}}{\alpha^{h}}(\frac{\beta N ^{h}}{s}+(\eta^{h}-\beta)\Delta w-\Delta p)+p_{2}+\eta^{a}v_{2},\] \[c^{h} = \beta\frac{N}{s}+p_{1}+(\eta^{h}-\beta)w_{1}.\]  
**Case 3-2**: \(K_{1}\geq N^{h}\)  
In this case, the capacity of the central cluster is larger than the demand of HV commuters. The sequence of departures and parking choices is as follows: (i) HV commuters depart from home first, and all HV commuters park at the closer cluster; (ii) then, AV commuters depart from home, and they all park at the farther cluster. The parking pattern is \((n_{1},n_{2})=(N^{h},N^{a})\). The start and end of the departure times from home for HV commuters are denoted by \(t_{0}\) and \(t_{1}\), respectively. \(t_{2}\) is also the departure time of the first AV commuter. The end of the departure time for AV commuters is denoted by \(t_{3}\). We can derive the equilibrium travel pattern by combining three conditions. The first condition states that all commuters will pass the bottleneck during the period \((t_{0},t^{*})\), so no commuters will be late. The second condition describes that all HV commuters will depart from home during \((t_{0},t_{1})\). The last condition enforces that AV commuters depart from home during \((t_{1},t_{2})\). Thus, we have  
\[ \begin{cases}t^{*}-t_{0}=\frac{N}{s},\\ t_{1}-t_{0}=\frac{N^{h}}{r^{2}},\\ t_{2}-t_{1}=\frac{N^{h}}{r^{2}},\end{cases} \Rightarrow \begin{cases}t_{0}=t^{*}- \frac{N}{s},\\ t_{1}=t^{*}-\frac{N^{h}}{s}-\frac{\beta}{\alpha^{h}}\frac{N^{h}}{s},\\ t_{2}=t^{*}-\frac{B}{\alpha^{h}}\frac{N^{h}}{s}-\frac{\beta}{\alpha^{h}}\frac{ N^{h}}{s}.\end{cases} \]  
With the derived critical times, the equilibrium travel pattern when \(K_{1}\geq N^{h}\) in Case 3-2 is depicted in Figure 3b. Following a similar way, we can obtain the solutions to each case. Based on the critical times, we depict the equilibrium patterns for each case in Appendix A. The equilibrium parking patterns and the individual travel costs at UE are included in Table 3. In this table,  
Figure 3: User equilibrium for Case 3-2. \(x\) denotes the number of HV commuters who are indifferent between two parking clusters and choose to park at the closer cluster. \(x\) does not exceed the travel demand for HVs, i.e., \(x\leq N^{h}\). \(y\) denotes the number of AV commuters who are indifferent between two parking clusters and choose to park at the closer cluster. And \(y\leq N^{a}\).