# Electric-Vehicle Routing Planning Based on the Law of Electric Energy Consumption
## 3 Driving-Cycle-Based Energy Consumption Model
### Energy Consumption Models of EVs Based on Different Driving Cycles  
We develop the energy consumption models of an EV in a few steps, as shown in Figure 4. According to the kinetic energy theorem, the kinetic energy obtained by the vehicle should be equal to the work done by the vehicle's combined external force, while the total external force is the difference between the vehicle's traction and resistance. Therefore, during an EV's driving process, both the kinetic energy \(E_{K}\) and the energy consumed to overcome resistance \(E_{R}\) come from the electric energy of the motor \(E_{M}\):  
\[E_{M}=E_{K}+E_{R} \tag{1}\]  
Figure 3: Cycle B of an EV. Figure 2: Cycle A of an EV. \(E_{M}\) is then translated into the electric energy \(E\) provided by the battery, that is:  
\[E=\frac{E_{M}}{\eta} \tag{2}\]  
where \(\eta\) is the efficiency of the electric motor. Similar to combustion vehicles, EVs are influenced by four types of external forces during driving, namely, rolling resistance \(F_{f}\), acceleration resistance \(F_{j}\), slope resistance \(F_{g}\), and air resistance \(F_{w}\). Resistances are mainly determined by factors of vehicle mass, vehicle load, vehicle dimensions, engine properties, acceleration rate, speed, and gradient. To present the equations of resistance, we use the notation shown in Table 1.  
Then, the rolling resistance \(F_{f}\), the acceleration resistance \(F_{j}\), the air resistance \(F_{w}\), and the slope resistance \(F_{g}\) can be respectively determined as:  
\[F_{f}=f\cdot m\cdot g \tag{3}\]  
\[F_{j}=\delta\cdot m\cdot a \tag{4}\]  
\[F_{w}=\frac{1}{2}\cdot\rho_{a}\cdot C_{d}\cdot A_{f}\cdot v^{2} \tag{5}\]  
\[F_{g}=m\cdot g\cdot sin\ \alpha \tag{6}\]  
In the following, the steps to build energy consumption models for driving cycles A and B are described in further detail. Let \(d\) be the distance between any two nodes, and \(s\) the distance an EV drives. We start with driving cycle B as it has fewer phases. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(m\) & Mass of an EV in kilograms \\ \hline \(g\) & Gravitational acceleration constant \\ \hline \(\alpha\) & Gradient angle \\ \hline \(f\) & Rolling friction coefficient \\ \hline \(\delta\) & The conversion factor of vehicle rotation mass \\ \hline \(v\) & Speed \\ \hline \(\rho_{a}\) & Air density \\ \hline \(C_{d}\) & Aerodynamic drag coefficient \\ \hline \(A_{f}\) & Vehicle frontal area \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation to calculate resistance. Figure 4: Flow of consumed battery energy. **Driving cycle B**, as shown in previous Figure 3, consists of three phases: first acceleration phase, second uniform motion phase, and, finally, the deceleration phase. The first acceleration phase is from speed 0 to \(v_{b1}\) during time 0 to \(t_{b1}\), that is:  
\[E_{B1}=\frac{E_{K}+E_{R}}{\eta}=\frac{\frac{1}{2}mv_{b1}^{2}+E_{R}}{\eta} \tag{7}\]  
and:  
\[E_{R}=F_{f}\cdot s_{b1}+F_{j}\cdot s_{b1}+F_{g}\cdot s_{b1}+F_{w}\cdot s_{w,b1} \tag{8}\]  
where \(s_{b1}\) is the distance the EV travels during this phase, while \(s_{w,b1}\) is the distance traveled when facing air resistance.