## 2 Model Formulation
### Parking Preferences  
In this part, we aim to examine the parking preferences of commuters in a morning commute under a partially autonomous environment. Each commuter always prefers a parking location that minimizes their generalized travel cost. By comparing the magnitude of individual travel costs defined in Equations (1) and (2), we can determine the parking preferences of commuters. We examine the parking preferences of AV commuters and HV commuters separately. We define \(\Delta v=v_{2}-v_{1}\) to measure the difference in self-driving time for AV users between parking at the peripheral and the central cluster. Similarly, we introduce \(\Delta w=w_{2}-w_{1}\) to indicate the difference in walking time for HV users between parking at the peripheral and the central clusters. Since the peripheral cluster is more distant than the central cluster, \(\Delta v\) and \(\Delta w\) are both positive. The difference in parking fees charged at the two clusters is \(\Delta p=p_{1}-p_{2}\). We assume \(\Delta p>0\) considering that the parking demand at the central cluster is usually higher than that at the peripheral cluster. Given a fixed departure time, we calculate the difference in generalized travel cost between commuters between using the peripheral cluster and the central cluster. Based on Equation (1), the derived cost difference for AV commuters is given by  
\[c_{t,2}^{a}-c_{t,1}^{a}=\eta^{a}\Delta v-\Delta p. \tag{3}\]  
The parking preference of AV commuters depends on \(\Delta p\) and \(\Delta v\). Specifically, the AV commuters prefer the central cluster when \(\Delta p<\eta^{a}\Delta v\), prefer the peripheral cluster when \(\Delta p>\eta^{a}\Delta v\), and are indifferent between the two clusters otherwise. If \(\Delta p<\eta^{a}\Delta v\), the parking fee or the corresponding self-driving time for choosing the central cluster is lower than the peripheral cluster, which makes the closer parking cluster more attractive to AV commuters. In this situation, AV commuters prefer parking at the central cluster in order to decrease individual generalized travel costs. In contrast, if \(\Delta p>\eta^{a}\Delta v\), the parking fee or the corresponding self-driving time for choosing the peripheral cluster is lower than the central cluster. Thus, AV commuters could reduce individual generalized travel costs by parking at the peripheral cluster. Similarly, the difference in individual generalized travel costs for HV commuters can be derived based on Equation (2). \[c_{t,2}^{h}-c_{t,1}^{h}=(\eta^{h}-\beta)\Delta w-\Delta p \tag{4}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(t\) & Departure time \\ \(N^{a}\) & Demand of AV commuters \\ \(N^{h}\) & Demand of HV commuters \\ \(t^{*}\) & Desired arrival time at destination \\ \(a^{a}\) & Value of travel time for AV users \\ \(a^{h}\) & Value of travel time for human-driven vehicle users \\ \(\eta^{a}\) & Value of self-driving time for AV users \\ \(\beta\) & Unit cost of early schedule delay \\ \(\eta^{h}\) & Value of walking time for HV commuters \\ \(v_{j}\) & Self-driving time from workplace to parking cluster \(j\), i.e., access time to \\  & parking cluster \\ \(w_{j}\) & Walking time from parking cluster \(j\) to workplace \\ \(T(t)\) & Travel time (i.e., queuing time) for commuters departing from home at time \(t\) \\ \(Q(t)\) & Queuing length for commuters departing from home at time \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. The parking preference of HV commuters relies on the difference in walking time between choosing the peripheral cluster and the central cluster, in addition to the difference in parking fee \(\Delta p\). Specifically, the HV commuters prefer the central cluster when \(\Delta p<(\eta^{h}-\beta)\Delta w\), prefer the peripheral cluster when \(\Delta p>(\eta^{h}-\beta)\Delta w\), and are indifferent between the two clusters otherwise.