# An Efficient GPS Algorithm for Maximizing Electric Vehicle Range
## 3 Mathematical Modeling and Proposed Algorithm  
To ensure precise prediction of vehicular energy consumption, a comprehensive examination of vehicular dynamics is imperative [24]. Electric vehicles (EVs) possess various dynamic parameters categorized into longitudinal, lateral, and vertical dynamics [25]. Understanding the influence of distinct driving resistances on EV performance is crucial [26]. This study focuses specifically on the impact of driving resistance, a facet of lateral dynamic parameters, on vehicular performance. Figure 2(a),b illustrate the longitudinal motion and resistances of a vehicle on level ground and on an inclined road, respectively [26; 27]. The driving force, situated at the drive axle, is the aggregate of the vehicle's propelling force and the resistances impeding its motion. Primary resistances include aerodynamic drag and rolling resistance. Aerodynamic drag is contingent upon factors such as air density, frontal area, drag coefficient, vehicle velocity, and wind velocity. Rolling resistance, on the other hand, is influenced by the vehicle's mass, velocity, and tire pressure [26; 27]. Traction power fluctuates with various factors, including road type (as the coefficient of rolling resistance varies), driving style (acceleration pattern), aerodynamic drag, and road gradient [28]. Auxiliary power encompasses components such as power steering, lighting, audio systems, air conditioning, suspension type, tire slippage, and other installed features [29]. The energy consumption model is primarily built on traction forces, auxiliary power needed, power train efficiency, auxiliary power efficiency, and brake requirements [30]. As shown in the figure below, the total power consumed can be divided into the following five main forces:  
* Resistive Force
* Grade Load
* Inertial Load
* Brake Recuperation
* Auxiliary Loads  
The resistive force (\(F_{res}\)) is the force needed to overcome the wind speed according to the aerodynamics of the vehicle, and includes the rolling resistance coefficient and the vicious friction coefficient of the tires [26]. These forces can be expressed as follows:  
\[F_{res}=A+B\;V+C\;V^{2} \tag{1}\]  
where 'A' is the rolling resistance coefficient, 'B' is the viscous friction coefficient, 'C' is the drag coefficient of the vehicle, and 'V' is the relative vehicle speed in m/s [26]. Figure 3: (**a**) Longitudinal motion and resistances of the vehicle on a flat terrain. (**b**) Longitudinal motion and resistances of the vehicle on an inclined road (\(\theta\) is the gradient under consideration). The second force is the grade force which changes according to the road topology, and this can be calculated as follows:  
\[\textbf{{F}}_{\textbf{grade}}=\textbf{{mgssin}}\textbf{{\theta}} \tag{2}\]  
where'm' is the vehicle mass in kg, 'g' is the gravitational force, and '\(\theta\)' is the inclination of the road in radian [31]. As for the third force, which is inertial load, it is derived from [PERSON]'s First Law of Motion, which stipulates that an item in motion will maintain its velocity unless acted upon by an external force [32]. Therefore, every attempt to modify the motion of a body needs overcoming its inertia, which results in the application of an inertial load, and it can be expressed as follows:  
\[\textbf{{F}}_{\textbf{inertia}}=\textbf{{ma}} \tag{3}\]  
where 'a' is the vehicle acceleration in m/s\({}^{2}\)[32]. The next one is brake recuperation which is the only force that affects the total force positively on the vehicle.