## 2 Context of the Experiments
### Control Algorithm
#### 2.7.3 Control Principle  
The planning algorithm is executed at each start of a new charging session but also periodically every 10 min. Let us consider that \(\mathrm{t_{0}}\) is the current time, that the 24 h after the current time are discretized by time step of \(\Delta\)t = 10 min and that those points in time are noted \(\mathrm{t_{n}}\). As, in our set-up, it is not possible to directly measure the SOC of the cars, the planning algorithm first estimates the SOC of all the EVs at current time \(\mathrm{t_{0}}\). These estimations are based on the initial values of the SOC and on the measurements of the energy meters of the charging stations (see details in Section 2.7.5). Then, for all \(\mathrm{t_{n}}\), the control algorithm follows two steps:  
1. The available power values are calculated, as detailed in Section 2.7.4.
2. These values are allocated amongst the cars that are connected and that are waiting to be charged (i.e., their SOC estimate is lower than 100%). The allocating process is based on a basic scheduling mechanism, i.e., \"earliest deadline first\" [20], which was adapted to achieve our objectives. In our context, we consider that the \"deadline\" of a car, also called \"lead-time\", is the difference between the time before departure (\(\mathrm{t_{dep}}\)) and the time needed to recharge the battery without a charging control (\(\mathrm{t_{end}}\)). The computation of this lead-time is detailed in Section 2.7.6. Then, the planning algorithm estimates the lead-time of all the EVs. The available power is first allocated to the EVs that are not under control (i.e., EVs in set \"Uncontrolled\") and to the cars that have a negative lead-time. The maximum charging power is allocated to each of these cars. Then, the remaining power is distributed in the inverse order of the lead-time. The maximum charging power is allocated to the cars that have the smallest lead-time. If there is not enough available power for the cars that have the highest lead-time, they do not recharge during this time step but they will be charged during the next ones. The powers allocated to each EV, for each time step, are the SetPoints that are sent over the 3G network to the charging points.  
The following simple example enlightens the principle of the control algorithm. Let us consider an EV owner, not participating in the experimentations, who connects his car (a Zoe with a battery capacity of 50 kWh) to the EVCI. Since SIGE has no information about the initial SOC of the car battery, it takes \(\mathrm{SOC_{init}}\) = 25% as a default value. Thus, SIGE considers that around 40 kWh are needed to fully charge the battery. In this example, we assume that there is enough available power to supply the car battery at \(\mathrm{P_{Max}}\) (here 22 kW) during the time needed to fully charge the battery, i.e., about 1 h 50. In such a case, the planning algorithm send the associated set points (i.e., 22 kW during 1 h 50) to the station. Then, SIGE estimates the charging power as explained in Section 2.7.5. Two cases arise during the charge:  
1. The car may either stop charging because it is full prematurely (i.e., before 1 h 50 of charge). SIGE has underestimated the real SOC. It may happen when the real initial SOC is greater than 25%. In that case, the set points are null power values and the charging power is allocated to another car.
2. Or it may continue to charge even after 1 h 50 of charge. SIGE has overestimated the real SOC. It may happen when the real initial SOC is less than 25%. In that case, the set points do not change until the charge stops.  
We have simulated the effect of this control algorithm on the historical charging session data presented in Section 2.3. This simulation step allowed confirmation of the choice of the different parameters and proved the effectiveness of the control algorithm.