## 3 Equilibrium Conditions
For each case of parking preference, we can further categorize the equilibrium into subcases based on the relation between the parking capacity and parking demand associated with two clusters. For each possible travel pattern, we examine the equilibrium conditions to provide the explicit number of vehicles parking at each cluster and the commuters' individual travel costs. Instead of presenting the derivation procedures for each case, we focus on Case 3-2 with hybrid parking preferences to illustrate the derivation process considering that the procedures for deriving equilibrium solutions are similar for all cases. In Case 3-2, the HV commuters prefer the central parking cluster, and the AV commuters prefer the peripheral parking cluster. All AV commuters can park at the peripheral parking cluster because of its unlimited capacity. Since a fixed and limited parking capacity is assumed for the central cluster, the equilibrium travel patterns rely on the magnitude of the capacity of the central cluster (\(K_{1}\)) and the demand for the central cluster (\(N_{h}\)). All HV commuters can park at the closer cluster if its capacity is not lower than the demand of HV commuters, i.e., \(K_{1}\geq N^{h}\). Otherwise, the central cluster can accommodate \(K_{1}\) HV commuters, while the rest of the HV commuters will choose to park at the peripheral one. In the following, we derive the equilibrium patterns under conditions \(K_{1}<N^{h}\) and \(K_{1}\geq N^{h}\), separately. **Case 3-2** : \(K_{1}<N^{h}\)  
The sequence of departures and parking choices is as follows: (i) HV commuters depart from home before AV commuters, and \(K_{1}\) commuters can park at the central cluster; (ii) then, the remaining \(N^{h}-K_{1}\) HV commuters choose to depart from home, and they will park at the peripheral cluster; (iii) finally, all AV commuters will park at the peripheral cluster. The corresponding parking pattern is \((n_{1}^{h},n_{2}^{h})=(K_{1},N^{h}-K_{1})\) for HV commuters and \((n_{1}^{a},n_{2}^{a})=(0,N^{a})\) for AV commuters. Thus, there are \(K_{1}\) vehicles parking at the central cluster and \(N-K_{1}\) vehicles parking at the peripheral cluster, which corresponds to a parking pattern \((n_{1},n_{2})=(K_{1},N-K_{1})\). We incorporate the critical times to describe commuters' departure time choices and parking location choices. Specifically, let \(t_{0}\) and \(t_{1}\) denote the departure times of the first HV commuter and the last HV commuter who park in the central parking cluster, respectively. Let \(t_{2}\) denote the departure time of the first HV commuter who parks at the peripheral parking cluster. Let \(t_{3}\) denote the departure time of the last HV commuter or the first AV commuter who parks at the peripheral parking cluster. Let \(t_{4}\) denote the departure time of the last AV commuter who parks at the peripheral parking cluster. The equilibrium pattern is illustrated in Figure 3a. Since all commuters have to pass the bottleneck before the desired arrival time and the first \(K_{1}\) HV commuters park at the central cluster, we have the following relations between \(t_{0}\) and \(t_{1}\):  
\[ \begin{cases}t^{*}-t_{0}=\frac{N}{s},\\ t_{1}-t_{0}=\frac{K_{1}}{t_{h}},\end{cases} \quad\Rightarrow \begin{cases}t_{0}=t ^{*}-\frac{N}{s}\\ t_{1}=t^{*}-\frac{N}{s}+\frac{a^{h}-\beta}{a^{h}}\frac{K_{1}}{s}\end{cases}