The Vehicle dynamics and the e-bus energy requirement calculations
A.1 The vehicle dynamics calculations
The amount of energy required by the e-bus to travel between consecutive EBSs in the Guwahati city ring road circuit is determined using the New York City Drive Cycle (NYCC) [113, 123]. The e-bus travels through the city traffic conditions to cover the distance between EBSs in the Guwahati city. Also, an average speed of 11 km/h is suitable for the e-buses to cover the distance between consecutive EBSs in the Guwahati city ring road circuit.
Therefore, NYCC is used as a reference to calculate the energy requirement of e-bus. The speed with respect to time graph of NYCC is shown in Fig. A.1.
Fig. A.1 New York City Drive Cycle (NYCC)
The various resistive forces that oppose the motion of the e-bus are shown in Fig. A.2.
M.g Supercapacitor Noral force
Rolling resistance Velocity of the e-bus Aerodynamic
resistance
Acceleration resistance
Fig. A.2 The resistive forces acting on the e-bus
The e-bus should overcome all these resistive forces to move on the surface of a road. The resistive forces usually include: the rolling resistance of the tire (FR), the aero dynamic resistance (FAD) and the acceleration resistance (FA). The vehicle acceleration (Newton’s second law) can be expressed as
∑ ∑
∑∑ ∑∑
∑(F )- (F )t ∑ r dV=
dt δM (A.1)
0 5 10 15 20 25 30 35 40 45
0 100 200 300 400 500 600
Speed (km/h)
Time (sec) NYCC
V is the speed of the vehicle, Σ(Ft) is the tractive force required by the vehicle, Σ(Fr) is the total resistive force, M is the vehicle mass, δ is the mass factor (converts the rolling component’s rotational inertias into translational mass) [123]. The details of different resistive forces that act on the e-bus movement are explained in the following manner.
A.1.1 The rolling resistance (FR):
The rolling resistance/ rolling friction/ rolling drag, is the force which resists the motion of a vehicle, when the wheels of a vehicle roll on a surface (the tires of a vehicle rolls in contact with the surface of the road. The friction is produced due to the relative motion of two hard surfaces. Both, the road and tires are not rigid. Therefore, both reflex slightly under the load.
The gradual deformation exists between the tire and the road (more at the bottom and low at the entry and exit points). The slip of the tire on the road surface causes energy loss and which results in a resistance) [123]. The forward movement produced by the ground reaction force is the rolling resistance movement (Tr) and is given as
= ×
Tr P a (A.2) P is the normal load acting on the center of the wheel and ‘a’ is the deformation of the tire.
The balancing force of the rolling resistance movement (FR) is therefore
×
= = = × × = × ×
r
R r
d d d
T P a a
F M g ( ) M g f
r r r (A.3)
rd is the dynamic radius of tire (m), M is the e-bus’s mass (104 kg), fr is the coefficient of rolling resistance (0.01), g is the gravitational acceleration (9.81 m/s2). ‘fr’ is the function of tire (structure, material and inflation pressure), road (material, roughness and presence/absence of liquids on the road) and tread geometry [123].
A.1.2 The aerodynamic resistance (FAD):
The resisting force which acts on motion of a vehicle when it travels in the air is termed as the aerodynamic resistance (FAD) (when an object moves in the dense medium, it collides with the molecules which are present in the medium. Therefore, the molecules absorb some of the energy from the object. The moving objects feel it as a resistance. This resistance is proportional to the medium density and the speed of an object) [123]. Therefore, FAD is expressed as
0.5 2
= × × × ×
AD a D
F ρ C A V (A.4) where, ρa is the mass density of air (1.184 kg/m3), CD is the aerodynamic drag coefficient (0.5), A is the e-bus frontal area (6 m2) and V is the speed of the vehicle (m/s).
A.1.3 The acceleration resistance (FA):
The acceleration resistance is influenced by the inertia forces of a vehicle. The force required to overcome the acceleration resistance is more for the greater acceleration of the vehicle (the faster change in speed). Similarly, the accelerating force also rises with the mass of the accelerated vehicle [123]. Therefore, the force required to overcome the acceleration resistance is expressed as
= × ×
A
F M δ dV
dt (A.5) where, δ is the rotational inertia factor (1.1) and dV/dt is the e-bus acceleration (m/s2). The road angle is assumed to be zero in these calculations so that the gradient resistance is zero.
The tractive force required by the e-bus (FT e-bus) to overcome all these resistive forces is expressed as
( ) ( 0.5 2 )
= R + AD+ A = + D +
T e-bus r a
F F F F Mgf ρ C AV MδdV
dt (A.6) A.2 The e-bus energy requirement calculations
The tractive energy (ET e-bus) required by the e-bus is expressed as
1
( ) ( )
=
=∑n ×
T e-bus t T e-bus t t
E F D (A.7) D is the displacement (m), t is the time instants of NYCC (t= 1, 2, --- n). The regenerative braking energy produced by the kinetic energy of the e-bus (ER e-bus) [146] in deceleration/
breaking period is therefore
= ×γ
R e-bus T e-bus
E E (A.8) γ is the fraction of gravitational potential energy (0< γ <1) and is considered as 0.4. Actual e- bus energy (Ee-bus) is therefore
= T − R
e-bus e-bus e-bus
E E E (A.9) The size of the supercapacitor in e-bus (Ce-bus) is expressed as
2
- =(2× - ) / ( )
e bus e bus
C E V (A.10) V is the super-capacitor’s rated voltage. The energy quantity required by the e-bus to travel between the consecutive EBSs (Ee-bus T) is therefore
= max −
T T R
e-bus e-bus e-bus e-bus
E E E + E (A.11) Ee-bus max is the e-bus maximum energy and is given as
max = 2×
e-bus e-bus
E E (A.12) The factor 2 in (A.12) is considered to avoid the emergency situations (If the e-bus failed to receive energy from ESD then e-bus crosses consecutive EBSs with Ee-bus max).
Fig. A.3 The tractive, regenerative and actual energy of the e-bus
The tractive energy needed by the e-bus, the regenerative braking energy produced by the e-bus and the actual energy needed by the e-bus is calculated using NYCC and are shown in Fig. A.3. The solid line represents the tractive energy needed by the e-bus to complete a 2 km distance between the consecutive EBSs. The medium dotted line represents the amount of energy produced by the e-bus during deceleration/braking period while travelling through 2 km distance. The small dotted line represents the actual energy required by the e-bus to complete the distance between the consecutive EBSs in the Guwahati city ring road circuit.
The energy rating of the e-bus is given in Table 2.1 (ref. chapter-2). The time required by the e-bus to complete a round trip of travel through the ring road circuit (TR) is expressed as
1
)
=
=∑Bp +
i i
R C i
T (Tx T (A.13) where, ‘Tx’ is the time required by the e-bus to travel from ith EBS to (i+1)th EBS (i = 1, 2…
Bp) and TC is the time required to charge the supercapacitor at EBS.
0 0.5 1 1.5 2 2.5
0 100 200 300 400 500 600
Energy (kWh)
Time (h)
Tractive energy Regenerative braking energy Actual energy