# An Optimal Sizing Design Approach of Hybrid Energy Sources for Various Electric Vehicles
## 2 Development of Global Search Algorithm for HEESs
### The Procedures of the GSA  
In this section, we developed the GSA for HEESs of EVs. The procedure was separated into four steps: (1) vehicle specification and performance requirements of energy sources, (2) determination of cost function and constraints, (3) GSA optimization with for-loops, (4) optimal results. The following is the procedure of the proposed GSA:  
(1) Vehicle specification and performance requirements of energy sources; in this step, two phases are designed separately. The first phase is the performance indices of energy sources, while the second phase is the specification of requirements of the target vehicle. The performance indices of energy sources are: (a) specific energy (kWh/kg or kWh/L) (the energy stored in a constant mass or in a constant volume), as the higher the specific  
Figure 1: Configurations of the HEES. energy, the more electric energy is able to be released to drive the traction motor in an EV so that the traveling mileage is larger; (b) specific power (kW/kg or kW/L) (the maximized power that can be released in a constant mass or in a constant volume), as the higher the specific power, the more electric power is able to propel the traction motor so that the values of vehicle acceleration and maximum vehicle speed are higher; and (c) specific price (USD/kWh) (the cost for energy capacity of \(1\) kW-hr), as the lower the specific cost, at a constant stored energy, the lower the overall cost [24]. The second phase is the vehicle requirements and specification. The key indices are (a) traveling mileage; (b) maximal power; (c) climbability; and (d) acceleration. These indices are criteria for target cost function, which will be described later. For traveling mileage, it is correlated to the energy stored in the HEES. For a chosen specific driving cycle, the consumed energy for one cycle as well as the traveling distance can be calculated by the following two equations:  
\[E=\int_{0}^{t}P_{t}dt=\int_{0}^{t}(F(t)\times V(t))dt \tag{1}\]  
\[D\sum_{i=0}^{n}V(t)\times t_{i} \tag{2}\]  
where \(E\), \(P\), \(F\), \(D\), \(V\), and \(t\) represent energy demand, power demand, traction force of the wheel, distance, vehicle velocity, and time step, respectively; the subscript \(i\) represents the driving time(s). Hence, the total distance decides the minimum requirement of energy storage. For maximal power, it is related to the tested initial speed, target speed, and the varied vehicle mass determined by the HEES sizing. The two following equations are listed as [3,4]:  
\[P_{max}\approx 2\times\Delta E\times\Delta t^{-1} \tag{3}\]  
\[\Delta E=\frac{M_{vch}}{2}\Big{(}{V_{init}}^{2}-{V_{goal}}^{2}\Big{)} \tag{4}\]  
where \(P_{max}\), \(\Delta E\), \(\Delta t\), and \(M\) represent maximum power demanded, kinetic energy, demanded time(s), and mass, respectively; the subscripts \(veh\), \(init.\), and \(goal.\) represent vehicle, initial, and goal, respectively. For climbability, it is correlated with the maximum power requirement as well. Modified from Equations (5)-(8), the calculated power is formulated as:  
\[F_{climb}=M_{vch}g{\rm cos}\theta \tag{5}\]  
\[F_{roll}=\mu M_{vch}g{\rm sin}\theta \tag{6}\]  
\[F_{wind}=\frac{M_{vch}C_{d}PA_{f}V^{2}}{2} \tag{7}\]  
\[M_{vch}\frac{dV}{dt}=\frac{\tau_{w}}{\tau_{w}}-F_{climb}-F_{roll}-F_{wind} \tag{8}\]  
where \(F\), \(g\), \(\theta\), \(\mu\), \(C_{d}\), \(\rho\), \(A_{f}\), \(r\), and \(\tau\) represent force, gravity acceleration, road slope, rolling coefficient, aero drag coefficient, air density, frontal area, radius, and traction torque, respectively; the subscripts \(climb\), \(roll\), \(wind\), and \(w\) represent climbing resistance, rolling resistance, wind resistance, and wheel, respectively. For acceleration, it can be determined by the profiles in Figure 2.  
(2) Determination of the cost function and constraints; to optimize the HEES, the variable of the cost function is set to be the total cost of the HEES.