## 3 Methodology
### Distributed Cooperative Control Model for AVs  
Considering the benefits of distributed frameworks, we will construct a distributed multiple-objective optimization model to address the cooperation issue of AVs and MVs at an intersection. In what follows, we will detail the objectives, necessary constraints, and problem formulation of this paper. (1) _Objectives_: We aim at regulating vehicles to cross the intersection without collisions by controlling all AVs in the CZ, while also improving traffic mobility and fuel economy. Thus, the objectives of this paper cover three parts, i.e., conflict elimination, traffic mobility, and fuel economy, which are detailed as follows. First, the primary goal is to eliminate conflicts within the intersection area. For this purpose, we define the indexes of the virtual car-following tracking errors as  
\[\mathcal{J}_{i1}=\sum_{k=0}^{N_{p}-1}\sum_{j\in\mathcal{N}_{i}(t),f\in \mathcal{C}_{h}}||\bar{p}_{ih}(k+1|t)-\bar{p}_{jf}(k+1|t)+\mathcal{D}_{ij}||^{2} \tag{3}\]  
\[\mathcal{J}_{i2}=\sum_{k=0}^{N_{p}-1}\sum_{j\in\mathcal{N}_{i}(t),f\in \mathcal{C}_{h}}||\bar{o}_{ih}(k+1|t)-\bar{o}_{jf}(k+1|t)||^{2} \tag{4}\]  
where \(N_{p}\) is the prediction horizon, \(\bar{p}_{ih}\) and \(\bar{p}_{jf}\) denote the distance to the stop line of vehicle \(i\), \(i\in\Omega_{A}(t)\) on lane \(h\), \(h\in\mathcal{L}\), and its neighboring vehicle \(j\), \(j\in\mathcal{N}_{i}(t)\) on lane \(f\), \(f\in\mathcal{C}_{h}\), respectively. \(\bar{o}_{ih}\) and \(\bar{o}_{jf}\) are the speeds of vehicles \(i\) and \(j\), respectively. \(\mathcal{D}_{ij}\) is the desired virtual position error between vehicles \(i\) and \(j\), defined as  
\[\mathcal{D}_{ij}=\left\{ \begin{aligned} & l_{i}+d_{ij}+c_{ij},\quad|\bar{p}_{ ih}|\leq|\bar{p}_{jf}|\\ &-l_{j}-d_{ij}+c_{ij},\quad|\bar{p}_{ih}|>|\bar{p}_{jf}|\end{aligned} \right. \tag{5}\]  
where \(l_{i}\) and \(l_{j}\) denote the lengths of vehicles \(i\) and \(j\), respectively, \(d_{ij}\) is the desired spacing between vehicles \(i\) and \(j\), and \(c_{ij}\) is the compensation of distance within the intersection area, which is defined as the difference in the distance between vehicles \(i\) and \(j\) from the stop lines to the conflict point. \(\mathcal{J}_{i1}\) indicates the performance of vehicle \(i\) in eliminating collisions with its neighboring vehicle \(j\), \(j\in\mathcal{N}_{i}(t)\) when crossing the intersection. Vehicle \(i\) and its neighboring vehicles are expected to arrive at the intersection with a desired virtual spacing. A small value of the virtual spacing may cause collisions of vehicles at the intersection, while a large value will reduce the traffic throughput. \(\mathcal{J}_{i2}\) indicates that vehicle \(i\) is expected to keep a similar speed with its neighboring vehicle \(j\) to enhance safety when crossing the intersection. Second, we expect smooth traffic flows that push each vehicle to run at a desired speed. It can also improve traffic mobility by regulating each vehicle on the same lane to approach an anticipated speed. Hence, we define the index of the variance of the desired speed as follows:  
\[\mathcal{J}_{i3}=\sum_{k=0}^{N_{p}-1}||\bar{o}_{ih}(k+1|t)-\mathcal{V}_{h}(k+1 |t)||^{2} \tag{6}\]  
where \(\mathcal{V}_{h}(k+1|t)\) is the desired speed defined as  
\[\mathcal{V}_{h}(k+1|t)=\frac{\sum\limits_{s\in\mathbb{S}_{h}(t)}\bar{o}_{sh}( k|t)+\mathcal{V}_{max}}{\mathbb{N}_{h}(t)+1} \tag{7}\]  
where \(\bar{o}_{sh}\) denotes the speed of vehicle \(s\) on lane \(h\), \(\mathbb{S}_{h}(t)\) is the set of vehicles on lane \(h\), \(\mathcal{V}_{max}\) is the maximum speed, and \(\mathbb{N}_{h}(t)\) represents the number of vehicles on lane \(h\). In Equation (7), the desired speed is defined as the average value of the speed of all the vehicles on the same lane and the maximum speed. This setting can propel vehicles on the same lane to run evenly and also increase the mean speed of vehicles in the network.