# A Simple Multi-Parameter Method for Efficient Charging Scheduling of Electric Vehicles
## 2 Materials and Methods
### Aggregation Model of the PL  
In this scenario, an equivalent aggregation model of the EVs of the PL was developed and the obtained results are compared with those of the proposed charging scheduling method. The objective function used by the aggregation model of PL is the total charging cost of the PL, \(\mathrm{C_{PL}}\), over the examined 24 h optimization period (96 quarters of the hour). Hence, the decision variable is the total active power, \(\mathrm{P_{PL}(t)}\), exchanged by the PL and the electric network over the examined 24 h optimization period. In order to ensure a fair comparison between the methods, the initial and final energy stored is the same in both cases, while the same battery lifetime constraint is applied. The solved optimization problem is defined in Equations (23)-(30):  
\[\mathrm{C_{PL}}\mathrm{=min}\left(\sum_{t=1}^{96}\mathrm{P_{PL}(t)}\cdot \mathrm{EP(t)}\cdot\Delta t\right) \tag{23}\]  
Subject to:  
* Battery lifetime constraint \[\mathrm{change\_sign}\left(\mathrm{P_{PL}(t)}\right)\;\leq\;3,\;\;\;\;\;\forall\;t\] (24)
* Stored energy constraints \[\sum_{t=1}^{t}\left(\mathrm{pf}\cdot\mathrm{c(t)}-\frac{\mathrm{d(t)}}{ \mathrm{pf}}\right)\cdot\mathrm{P_{PL}(t)}\cdot\Delta t+\mathrm{E_{PL}(t)} \left|{}_{t=0}\;\geq\;\sum_{i}\mathrm{Emin}(i,t),\;\forall t,\forall\;i\;\in \;\mathrm{EV_{CS}}\] (25) \[\sum_{t=1}^{t}\left(\mathrm{pf}\cdot\mathrm{c(t)}-\frac{\mathrm{d(t)}}{ \mathrm{pf}}\right)\cdot\mathrm{P_{PL}(t)}\cdot\Delta t+\mathrm{E_{PL}(t)} \left|{}_{t=0}\;\leq\;\sum_{i}\mathrm{Emax}(i,t),\;\forall t,\forall\;i\;\in \;\mathrm{EV_{CS}}\] (26)
* Charging power constraints \[-\sum_{i}\mathrm{P_{limit}^{charged}(i,t)}\;\leq\;\mathrm{P_{PL}(t)}\;\leq\; \sum_{i}\mathrm{P_{limit}^{Charger}(i,t)},\;\forall t,\forall\;i\;\in\;\mathrm{ EV_{CS}}\] (27) \[\sum_{i}\mathrm{P_{limit}^{Charger}(i,t)}\;\leq\;\mathrm{P_{TF}},\;\;\forall t,\forall\;i\;\in\;\mathrm{EV_{CS}}\] (28)
* Final stored energy constraint \[\sum_{t=1}^{96}\left(\mathrm{pf}\cdot\mathrm{c(t)}-\frac{\mathrm{d(t)}}{ \mathrm{pf}}\right)\cdot\mathrm{P_{PL}(t)}\cdot\Delta t+\mathrm{E_{PL}(t)} \left|{}_{t=0}\;=\;\sum_{i}\mathrm{E(i,96)},\;\;\forall\;i\;\in\;\mathrm{EV_{CS}}\] (29)
* Initial stored energy constraint \[\mathrm{E_{PL}(t)}\left|{}_{t=0}\;=\;\sum_{i}\mathrm{E(i,0)},\;\;\forall\;i\; \in\;\mathrm{EV_{CS}}\] (30)  
where \(\mathrm{P_{PL}}\) is the total active power exchanged by the PL and the electrical network, \(\mathrm{E_{PL}}\) is the total energy stored in the plug-in electric vehicles, \(\mathrm{E}\) is the energy stored in a plug-in electric vehicle and \(\mathrm{E_{min(max)}}\) is the minimum (maximum) energy stored in a plug-in electric vehicle. The energy absorbed and injected into the grid were 3900 and 1065 kWh when the aggregation PL model was applied, respectively. The obtained charging cost was 120.73 EUR and it is 2.76% lower than that obtained by the proposed charging schedule method. However, this is not the case in real-world applications, as the aggregation models are highly sensitive to electricity price forecast errors. Moreover, the low application requirements of the proposed method make it more suitable for such applications. tariff to the electricity price and integrating this into the developed algorithm. In addition to the features of the proposed charging schedule method described above, it should be noted that it allows the EVs to utilize V2G operation, which is expected to be a significant ancillary service for the future electric power systems. Different charging scenarios were created for the examined PL in order to compare the proposed charging scheduling method with the other charging strategies. The proposed charging scheduling method demonstrated superior performance compared to the conventional charging strategies in all examined scenarios. The proposed charging schedule method resulted in a significant reduction in total daily charging cost with regard to the conventional charging strategies, by more than 12%.