## 2 Model Formulation
### Parking Preferences
#### 3.1.1 Assumption 3  
\(\eta^{h}>\beta\). Furthermore, we can describe the combined parking preferences of AV commuters and HV commuters by jointly considering the effects of \(\Delta p\), \(\Delta w\), and \(\Delta w\). By characterizing the range of \(\Delta p\) in terms of \(\Delta v\) and \(\Delta w\), we provide all the cases of parking preferences for the morning commute under a partially automated environment with two parking clusters in Proposition 1. Moreover, we can divide the parking preferences of commuters into five cases and nine subcases, as summarized in Table 2.  
**Proposition 1** (The parking preferences). : _For a single-route bottleneck with two parking clusters, the parking preferences of AV commuters and HV commuters depend on the relation between \(\Delta p\), \(\Delta v\), and \(\Delta w\), specifically as follows:_  
* _Case 1: When the difference in parking fee_ \(\Delta p\) _is very low, i.e.,_ \(\Delta p<\min\{\eta^{a}\Delta v,(\eta^{h}-\beta)\Delta w\}\)_, both AV commuters and HV commuters prefer the central (closer) parking cluster, i.e., the central cluster._
* _Case 2: When the difference in parking fee_ \(\Delta p\) _is very high, i.e.,_ \(\Delta p>\max\{\eta^{a}\Delta v,(\eta^{h}-\beta)\Delta w\}\)_, both AV commuters and HV commuters prefer the peripheral (farther) parking cluster, i.e., the peripheral cluster._
* _Otherwise, the cases of hybrid parking preferences are obtained. Moreover, we can further decompose the cases of hybrid parking preferences into three cases (Cases 3-5) and seven subcases, as shown in Table 2._  
Proposition 1 indicates that the parking location preferences of commuters under a partially automated environment depend on the joint impacts of \(\Delta p\), \(\Delta w\), and \(\Delta v\). Specifically, Proposition 1 reveals that (1) the closer parking cluster (the central cluster ) is more \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Cases** & \multicolumn{2}{c}{**Sub-Cases**} & \multicolumn{2}{c}{**HV Commuters**} & \multicolumn{2}{c}{**AV Commuters**} \\ \hline
1 & - & & \(\Delta p<\min\{\eta^{a}\Delta v,(\eta^{h}-\beta)\Delta w\}\) & 1 & 1 \\ \hline
2 & - & & \(\Delta p>\max\{\eta^{a}\Delta v,(\eta^{h}-\beta)\Delta w\}\) & 2 & 2 \\ \hline  & 3-1 & \(\Delta p=\eta^{a}\Delta v\) & 1 & Indifferent \\
3 & \(\eta^{a}\Delta v<(\eta^{h}-\beta)\Delta w\) & 3-2 & \(\Delta p\in(\eta^{a}\Delta v,(\eta^{h}-\beta)\Delta w)\) & 1 & 2 \\  & 3-3 & \(\Delta p=(\eta^{h}-\beta)\Delta w\) & Indifferent & 2 \\ \hline  & 4-1 & \(\Delta p=(\eta^{h}-\beta)\Delta w\) & Indifferent & 1 \\
4 & \(\eta^{a}\Delta v>(\eta^{h}-\beta)\Delta w\) & 4-2 & \(\Delta p\in((\eta^{h}-\beta)\Delta w,\eta^{a}\Delta v)\) & 2 & 1 \\  & 4-3 & \(\Delta p=\eta^{a}\Delta v\) & 2 & Indifferent \\ \hline
5 & \(\eta^{a}\Delta v=(\eta^{h}-\beta)\Delta w\) & \(\Delta p=\eta^{a}\Delta v\) & Indifferent & Indifferent \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parking location preferences. attractive to all commuters when the difference in parking fees charged at the two clusters is significantly low; (2) the farther parking cluster (the peripheral cluster) is more attractive to all commuters when the difference in parking fees charged at two clusters is significantly high; (3) when the difference in parking fees charged at two clusters varies in an intermediate range, the parking preferences are determined by the range of \(\Delta p\) and the relation between \(\eta^{\mu}\Delta v\) and \((\eta^{h}-\beta)\Delta w\). As stated in Proposition 1, when \(\Delta p\) takes very low or very high values, the two groups of commuters have the same preference towards parking location, and they prefer only one of the two parking clusters. When \(\Delta p\) takes very low values, i.e., \(\Delta p<\min\{\eta^{\mu}\Delta v,(\eta^{h}-\beta)\Delta w\}\) such that \(c_{t,2}^{a}>c_{t,1}^{a}\) and \(c_{t,2}^{h}>c_{t,1}^{h}\), both AV commuters and HV commuters parking at the central cluster experience lower individual generalized travel costs compared with parking at the peripheral cluster.