# Research on Optimal Voltage Control of Distribution Network with the Participation of EVs and PVs
## 2 Materials and Methods
### Energy Management Schemes of EV Charging
#### 2.3.2 Modelling of the EV Charging/Discharging Scheme  
During the real-time energy management of EVs, their charging and discharging powers (assuming vehicle-to-grid technology is supported) have to satisfy the power flow limits and the constraints of battery state of charge (SoC). For the power flow constraints, the charging and discharging powers of any EV at time instance \(t_{m}\) (represented by \(P_{tm,c}\) and \(P_{tm,d}\)) should not exceed the power rating of the connected EV charger, \(P_{r}\), as in (9) and (10). Accordingly, the operating power of an EV charger at time instance \(t_{m}\) can be represented by (11) and (12), where \(a_{tm,c}\) and \(b_{tm,d}\) are values either 0 or 1 and have to satisfy (13). The varying operation status of an EV charger can be represented by different value  
Figure 7: The charging demands of EVs with different vehicle applications. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**EV Transport** & **Charing Period** & **Initial Battery SoC** & **Battery Capacity (kWh)** & **Types of Charging** & **EV Arrival Time at the Charging Station** \\ \hline Private car & 7:00–17:00 & N (0.69, 0.12) & 35 & Slow charging (7 kW) & N (20.9, 1.72) \\ \hline Bus & 23:00–5:30 & N (0.5, 0.12) & 100 & Slow charging (7 kW) & U * \\ \hline Taxi & 4:00–7:00 & N (0.3, 0.12) & 80 & Fast charging (30 kW) & N * (5.6, 1.22) \\ \hline Exclusively-used car & 17:00–8:00 & N (0.48, 0.22) & 50 & Slow charging (7 kW) & N * (18.3, 3.92) \\ \hline \hline \end{tabular}  
* N and U stand for normal distribution and uniform distribution, respectively.  
\end{table} 
Table 3: Characterization of EV traveling patterns. combinations of \(a_{tm,c}\) and \(b_{tm,d}\), as tabulated in Table 4. The actual charging power fed to the EV battery can be represented by (14), where \(\eta_{c}\) and \(\eta_{d}\) refer to the charging efficiency and the discharging efficiency of an EV charger. \[0\leq P_{tm,c}\leq P_{r} \tag{9}\]  
\[0\leq P_{tm,d}\leq P_{r} \tag{10}\]  
\[P_{tm}=a_{tm,c}\ P_{tm,c}-b_{tm,d}\ P_{tm,d} \tag{11}\]  
\[-P_{r}\leq P_{tm}\leq P_{r} \tag{12}\]  
\[a_{tm,c}+b_{tm,d}\leq 1 \tag{13}\]  
\[P_{tm,batt}=a_{tm,c}\ \eta_{c}P_{tm,c}-b_{tm,d}\ P_{tm,d}/\eta_{d}, \tag{14}\]  
Regarding the SoC constraints, the feasible operation region of an EV battery SoC is illustrated in Figure 8. For an EV connected to an EV charger at time instance \(t_{ini}\), its upper and lower boundaries of EV battery SoC are marked by the red curve and the blue curve in Figure 6, respectively. For the upper boundary, the EV will be first charged to its maximum SoC, \(SoC_{max}\), at maximum charging power, \(P_{r}\), and then kept constant at \(SoC_{max}\), followed by being discharged to the user-expected battery SoC, \(SoC_{exp}\), at a power of \(P_{r}\). Similarly, for the lower boundary, the EV is first discharged to the lowest battery SoC, \(SoC_{min}\), at maximum discharging power, \(P_{r}\), and then kept constant at \(SoC_{min}\), followed by being charged to \(SoC_{exp}\) at a power of \(P_{r}\). The upper and lower boundaries of EV battery SoC at time instance \(t_{m}\) can be represented by Equations (15) and (16). \[SoC_{up,tm}=min\Bigg{\{}SoC_{ini}+\frac{\eta_{c}P_{r}(t_{m}-t_{ini})}{E_{r}}, SoC_{max},SoC_{exp}+\frac{P_{r}\big{(}t_{exp}-t_{m}\big{)}}{\eta_{d}E_{r}}, SoC_{tm-\Delta t}+\frac{\eta_{c}P_{r}\Delta t}{E_{r}},\Bigg{\}} \tag{15}\]  
Table 4.0  
where \(SoC_{up,tm}\) and \(SoC_{lo,tm}\) refer to the upper boundary and lower boundary of battery SoC at time instance \(t_{m}\), \(SoC_{ini}\) and \(SoC_{exp}\) are the initial battery SoC (i.e., the battery SoC when EV is connected to an EV charger) and the user expected battery SoC respectively, \(P_{r}\) and \(E_{r}\) are the rated operating power and the rated battery capacity respectively, \(t_{ini}\) and \(t_{exp}\) are the EV arrival time and departure time, and \(\Delta t\) is the control time interval.