# Assignment Approach for Electric Vehicle Charging Using Traffic Data Collected by SUMO
## 3 Proposed Solution
### Assignment Algorithm
#### 3.2.1 Problem Formulation  
The notation used in our work is presented in Table 1. \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline \(n\) & Number of electric vehicles (\(\text{EV}_{1}\), \(\text{EV}_{2}\),..., \(\text{EV}_{n}\)) \\ \hline m & Number of charging stations (\(\text{CS}_{1}\), \(\text{CS}_{2}\),..., \(\text{CS}_{m}\)) \\ \hline Alloc (i, j, t) & \(\text{EV}_{i}\) allocation coefficient at charge station \(\text{CS}_{j}\) at time t \\ \hline K (i, j, t) & Vehicle density instant on the route between the \(\text{EV}_{i}\) and station \(\text{CS}_{j}\) at time t [veh/km] \\ \hline V (i, j, t) & Vehicle speed according to traffic conditions at time t [km/h] \\ \hline \(\text{v}_{\text{f}}\) (i, j, t) & Reference speed on the route between the \(\text{EV}_{i}\) position and station \(\text{CS}_{j}\) at time t [km/h]. \\ \hline \(\text{k}_{\text{jam}}\)(i, j, t) & Traffic density, capacity supported by the road leading from the \(\text{EV}_{i}\) position to charging station CS\(j\) at time t [veh/km]. \\ \hline q (i, j, t) & Vehicle flow [veh/h] \\ \hline d (i, j, t) & Distance between the vehicle \(\text{EV}_{i}\) and charging station \(\text{CS}_{j}\) at time t [km] \\ \hline C (i) & Nominal capacity of \(\text{EV}_{i}\)’s traction battery [KWh] \\ \hline A(i) & Autonomy of the traction battery of vehicle \(\text{EV}_{i}\) [km] \\ \hline T (i, j, t) & Time required to complete distance d (i, j, t) [h] \\ \hline SOC\({}_{0}\)(i, t) & Initial state of charge of the \(\text{EV}_{i}\) battery at time t [\%] \\ \hline SOC\({}_{\text{f}}\) (i, j, t + T) & Final state of charge of the \(\text{EV}_{i}\) battery upon arrival at charging station \(\text{CS}_{j}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Notation. As shown in Figure 2, the system starts with the EV trip. If the SOC of the vehicle's battery is sufficient, then the driver can continue the trip. On the other hand, if the battery SOC reaches a certain limit, the driver will receive a low battery warning notification. In this case, a drop-off request is sent to a collaboration platform, which takes care of finding the CS that matches all of the received requests based on the information stored in its database or by communicating directly with the CS. Once the free CS is found, the platform will use other services to allocate the appropriate EV to the free CS, taking into account their characteristics (load state, path, power of each CS, etc.). Among these services, the collaboration platform uses an optimization algorithm to compute the optimal allocation, taking into account the system characteristics as well as the information provided by other services (traffic information services, road conditions, etc.). When the optimal allocation is found, the EV is informed of a CS corresponding to its criteria. After the charging process is completed, the used charging point is released and the DB platform database is updated. Then, the EV can continue its journey. The goal was to assign each electric vehicle to the appropriate station with minimal energy consumption. We proposed that:  
* The electric vehicles are of the same brand. * An electric vehicle can be assigned to only one station. * The charging stations have the same characteristics (power of charge). * The charging station has several charging points. * The road is flat. Figure 2: General framework of the proposed approach. The coefficient _Alloc_ (\(i\), \(j\), \(t\)) may depend on several parameters, such as: \(Soc_{0}\) (\(i\), \(t\)), \(SOC_{f}\) (\(i\), \(j\), \(t\) + \(T\)), \(q\) (\(i\), \(j\), \(t\)), \(K_{jam}\) (\(i\), \(j\), \(t\)), and \(V_{f}\) (\(i\), \(j\), \(t\)). In this study, we were interested in the expression of the equation as a function of the elements characterizing the road traffic in order to determine their impact on the variation of the energy consumed during the trip. To do this, we worked with the following equations [21; 22].