## 2 Methodology
### Mathematical Model Construction of Control Plane Deployment  
In terms of the number of controllers, it is hoped that the number of controllers is small, and the number of ordinary RSUs connected to each controller is large. However, if fewer controllers are used, the vehicle OBU far away from the controller RSU may experience considerable delay. In order to ensure that a single RSU controller does not overload, the average distribution of workload between RSU controllers is also a key factor to be considered. The number of ordinary RSUs connected to the RSU controller can be used as a measure of its workload. Because of the higher number of ordinary RSUs, the larger coverage area means a greater number of connected vehicles, and a greater workload. The above situation is particularly important in high-density urban road networks. Based on the above analysis, considering the network load, the number of RSUs, the location of RSUs, and the delay, the controller deployment model is given as follows: the definition \(V\) represents the set of all RSU nodes, and \(Card(V)=N\) and \(V_{h}\) represent the set of candidate RSU nodes that can be used as RSU controllers, \(V_{h}\subseteq V\). Among them, all controllers in the SDVN can communicate directly with each other; \(V_{k}\) denotes the number of RSUs that a single RSU controller can control, \(\forall 0\leq k\leq N\). One or more of these RSU nodes can be specified as controllers, and the optimal number and location of these controllers depend on a variety of factors, such as delay, distance, etc. All controllers can be connected to each other not only with ordinary nodes, but also with controllers. Therefore, there will be a synchronization cost between 'controller pairs'. In addition, \(C=(C_{i}\in\mathbb{R}^{d},i\in V)\) represents the location of the node in the wireless network; \(\overline{C}=(\overline{C_{i}}\in\{0,1\},i\in V_{h})\) represents the controller deployment strategy. If the node is selected to play the role of controller, then the converse is true. Similarly, we determine the controller allocation strategy, where if the node \(i\) is allocated to the controller \(j\), then \(p_{ij}=1\), otherwise \(p_{ij}=0\). \(d_{ij}=d(C_{i},C_{j})\) denotes the communication delay between nodes \(i\) and \(j\), assuming that the delay is proportional to the square of the Euclidean distance, i.e., \(d_{ij}\propto\left\|C_{i}-C_{j}\right\|^{2}\)[30]. According to the above analysis, the objective function of the optimal allocation of the controller is expressed as Equation (3), and the constraint conditions are shown as Equations (4)-(7):  
\[\min_{P,\overline{C}}\left\{\alpha\sum_{i\in V}\sum_{j\in V_{h}}p_{ij}d_{ij}+( 1-\alpha)\sum_{i,j\in V_{h}}\overline{C_{i}C_{j}}d_{ij}\sum_{k\in V}p_{kj}\right\} \tag{3}\]  
Subject to  
\[\sum_{j\in V_{h}}p_{ij}=1\;\forall i\in V \tag{4}\]  
\[p_{ij}\leq\overline{C_{j}}\;\forall i,j\in N \tag{5}\]  
\[\overline{C_{i}}\in\{0,1\},\;i\in V_{h} \tag{6}\]  
\[p_{ij}\in\{0,1\},\;i\in V,\;j\in V_{h} \tag{7}\]  
In Equation (3), the first term of the objective function pertains to the communication delay between all nodes and the controller, while the second term denotes the synchronization delay among all controllers. Moreover, the synchronization delay is also contingent upon the number of nodes assigned to the controller. The constraint (4) indicates that each node is assigned to only one controller. Constraint (5) ensures that the node assignment of the controller is only performed on the specified controller node. The parameter \(0\leq\alpha\leq 1\) represents the relative weight of the communication delay between the controller and the node relative to the synchronization delay between the controllers. Constraint (6) and constraint (7) represent variables, which are binary variables, and their values range from 0 to 1. The above problem is a multi-objective, mixed-integer, non-linear problem (MMINLP), and its complexity is NP-hard.