Fuel economy regression analyses for hybrid electric vehicle

2.1. Battery

The battery is modeled as resistance and voltage which changes the load and power flow. A sub-block was modeled for the state of charge (SOC) taking voltage and current as input and the output of the battery voltage is just over 200V is connected to the input of the DC to DC converter. The modeling criteria used for the battery model can be found in (Liu, 2013) and shown in Figure 1.

The differential equation for electrical scheme is

$\frac { d V _ { \text {terminal} } } { d t } + \frac { V _ { \text {terminal} } } { R _ { d y } C _ { d y } } = R _ { \text {ohm} } \frac { d I } { d t } + \frac { R _ { d y n } + R _ { \text {ohm} } } { R _ { d y n } C _ { d y n } } I + \frac { V _ { o c } } { R _ { d y n } C _ { d y n } }$     (1)

where I is the terminal current of the battery (A), V_terminal is the battery system terminal voltage (V), T is the battery temperature, and V_oc, R_ohm, R_dyn, and Cdyn are the scheme parameters which depend on T and SOC.

Battery calculation of SOC can be applied from the integration of current:

$\operatorname { SOC } ( t ) = \operatorname { SOC } \left( t _ { i } \right) + \frac { 1 } { \operatorname { Cap } _ { A h r } 3600 } \int _ { t _ { i } } ^ { t } I ( t ) \eta _ { b a t } ( \operatorname { SOC } , T , \operatorname { sign } [ I ( t ) ] ) d t$     (2)

where $\eta _ { b a t }$ is the efficiency battery Coulombic and Cap_Ahr is the ampere-hours capacity.

Figure 1. Battery in simulink model

2.2. DC-DC converter

The DC - DC converter is seen as DC equivalent to a transformer AC with a continuously turning ratio. Like a transformer, it is used to step up or step-down DC voltage source. The criteria used for the modelling of the DC-DC converter can be found in (Minh & Pumwa, 2013) and (Minh & Hashim, 2012). For this study the DC to DC converter is used to boost the input voltage from 200V to the output voltage of 500V to drive the motor. The Simulink model of DC-DC converter is drawn in Figure 2.

Figure 2. DC - DC converter in Simulink model

The mathematical relationship used for modelling the DC - DC converter

Voltage transformation ratio $= \frac { N _ { \text {Secondary } } } { N _ { \text {Primary } } } = \frac { V _ { \text {Secondary } } } { V _ { \text {Primary } } }$      (3)

where V is voltage and N is the number of winding in turns. A transformer to increase the voltage from primary to secondary is named step-up transformer. A transformer produced to reduce voltage from primary to secondary is named step-down transformer. The ratio of transformation for a transformer being equal to the square rooted of inductance (L) primary to secondary ratio.

2.3. Electrical motor and generator model

The servomotor is used to represent the generator and the electrical motor, the required output torque is tracked from the torque demand reference from the control system. Both the servo motor and servo generator are connected to the DC supply. The function of the electrical motor is to supply torque to the transmission required for the propelling of the vehicle, while the generator is used to start the combustion engine and to charge the HEV battery through regenerative braking at certain mode in the control process. Criteria used for modeling the motor can be referred in (Liu, 2013) and generator can be referred in (Naik & Shim, 2015). Figure 3 and Figure 4 show the Simulink model of the electrical motor and generator, the blue lines in both figures indicate the electrical part of the model while the green parts indicate the mechanical system.

Figure 3. Motor in Simulink model

Figure 4. Generator in Simulink model

Three operational modes of motor and generator are as follows:

2.3.1. Operation of propulsion mode

When an electrical motor runs in this mode, it supplies the propulsion torque, and its behavior is depicted from following equations:

$\tau _ { \mathrm { mot } } = \tau _ { \mathrm { demand } } + \tau _ { \mathrm { spin } _ { \mathrm { loss } } } + J _ { \mathrm { mot } } \frac { \mathrm { d } \omega } { \mathrm { dt } } ( \text {output torque} )$  (4)

$\tau _ { m o t } \leq \max \left( \tau _ { m o t } \right) = f ( \omega ) ( \text {contrain } )$   (5)

$\tau _ { \text {spin } _ { \text {loss } } } = \alpha _ { 1 } \delta ( t ) + \alpha _ { 2 } \omega + \alpha _ { 3 } \operatorname { sgn } ( \omega ) ( \text {loss spin torque} )$   (6)

$\mathrm { P } _ { \text {elec } } = \frac { \mathrm { P } _ { \text {mech } } } { \mathrm { n } _ { \text {mot } } } = \frac { \tau _ { \text {mot } . \omega } } { \eta \left( \tau _ { \text {mot } , \omega } \right) }$ (required electrical power)   (7)

$\eta _ { \mathrm { mot } } = \eta \left( \tau _ { \mathrm { mot } } , \omega \right) ( \text {motor efficiency} )$  (8)

$V _ { \text {mot } } = V _ { \text {bus } } ( \text { motor voltage } )$  (9)

$I _ { \operatorname { mot } } = \frac { P _ { \text {elec} } } { V _ { \text {bus} } } ( \text {motor required current} )$  (10)

where J_mot is the inertia of motor, $\tau_{mot}$ is the torque propulsion delivered by the electrical motor (Nm),

 is the torque lost due to the friction (Nm), $\tau_{spin_loss}$ is the torque demanding from the HEV (Nm), $\tau_{demand}$ is the angular velocity of motor (rad/s), $max(\tau_{mot})=f(\omega)$ is the physical maximum torque of motor, α₁, α₂, and α₃ are the static coefficient friction, viscous coefficient friction, and friction Coulomb coefficient, respectively, $\eta_{mot}$ is the efficiency lumped of the electrical motor, controller and  inverter. V_bus is the bus high-voltage.

2.3.2. Operation in regenerative mode

When an electrical motor runs in the regenerative mode, the generator supplies charge for the battery and provides negative (brake) torque for the vehicle. In this mode, the generator behavior can be determined by the following equations.

$\left| \tau _ { \text {regen } } \right| = \left| \tau _ { \text {demand } } \right| - \tau _ { \text {spin } _ { \text {- } } \text { loss } } + J _ { \text {mot } } \frac { \mathrm { d } \omega } { \mathrm { dt } } ( \text { brake torque } )$   (11)

$\left| \tau _ { \text {regen } } \right| \leq \max \left( \tau _ { \text {regen } } \right) = g ( \omega ) ( \text { contrain } )$   (12)

$\tau _ { \text {spin } \text{_} { \text {loss } } } = \alpha _ { 1 } + \alpha _ { 2 } \omega + \alpha _ { 3 } \operatorname { sgn } ( \omega ) ( \text {lost spin torque} )$    (13)

$\mathrm { P } _ { \text {elec } } = \eta _ { \text {regen } } \mathrm { P } _ { \text {mech } } = \eta \left( \tau _ { \text {regen } } , \omega \right) \tau _ { \text {regen } } \omega ( \text { general electrical power } )$  (14)

$\eta _ { \operatorname { mot } } = \eta \left( \tau _ { \text {regen } } , \omega \right) ( \text { motor efficiency } )$    (15)

$\mathrm { V } _ { \mathrm { mot } } = \mathrm { V } _ { \mathrm { bus } } ( \text {motor voltage} )$  (16)

$\mathrm { I } _ { \text {regen } } = \frac { \mathrm { P } \text { elec } } { \mathrm { V } _ { \text {bus } } } ( \text { generated current } )$    (17)

where J_mot is the inertia of motor, $\tau_{regen}$ is the negative brake motor torque (Nm), $\tau_{demand}$ is the torque demanding from the HEV (Nm), ω is the angular velocity of motor (rad/s), $max(\tau_{regen})$ is the maximum motor regenerative torque, and $\eta_{mot}$ is the regenerative efficiency lumped of the electrical motor, controller and inverter.

2.3.3. Operation in spinning mode

When an electrical motor runs in this mode, it is passively spun and supplies a minor negative torque friction for the HEV as:

$\tau _ { m o t } = \tau _ { s p i n _ { - } l o s s } = \alpha _ { 1 } \delta ( t ) + \alpha _ { 2 } \omega + \alpha _ { 3 } \operatorname { sgn } ( \omega )$  (18)

where $\tau_{mot}$ is the output torque, $\tau_{spin_loss}$ is the loss spin torque, α₁, α₂, and α₃ are the static friction viscous friction coefficient, coefficient, and friction Coulomb coefficient on electrical motor and generator.

2.4. Planetary gear (power split)

Planetary gear is used in the series parallel HEV configuration for power split functionality between the mechanical (ICE) and the electrical link (Electrical Motor). The planetary gear connects the gasoline engine, generator and electric motor together allowing the vehicle to operate at different mode (ICE, Motor, ICE and Motor combine). The set of planetary gear (Power Split) includes of the sun gear, carrier, ring gear, and several pinion gears, carrier. The planetary gear scheme is drawn in Figure 5.

Figure 5. Planetary gear in Simulink model

2.5. Vehicle dynamics

The vehicle dynamics is consisted of gear box, differential, left and right tires, and vehicle body. The vehicle body model is drawn in Figure 6.

Figure 6. Simulink model of vehicle body model

The HEV body provides the resistance rolling and aero-drag resistance forces of the HEV. The retarding forces of the HEV when the motionless is set equal to $F _ { \text {fric} } + F _ { \text {grade} }$. The propulsion force required at the given speed is presented by the relationship below.

$\mathrm { F } _ { \mathrm { wh } } = \mathrm { F } _ { \mathrm { rolling } } + \mathrm { F } _ { \mathrm { aero } } + \mathrm { F } _ { \mathrm { grade } } + \mathrm { m } _ { \mathrm { v } } \mathrm { a }$   (19)

where, F_wh is the force traction on the wheel by the powertrain, mv is the mass (kg), a is the HEV acceleration (m/s²), and $F _ { \text {rolling} } , F _ { \text {aero} } , F _ { \text {grade} }$ are resistance rolling, aero drag resistance, and weighing forces, respectively, expressed as:

$\mathrm { F } _ { \text {rolling } } \approx \mathrm { k } _ { \mathrm { rrc } } \mathrm { k } _ { \mathrm { sc } } \mathrm { m } _ { \mathrm { v } } \mathrm { g }$    (20)

$\mathrm { F } _ { \text {aero } } = \mathrm { k } _ { \text {aero } } \mathrm { V } ^ { 2 }$   (21)

$\mathrm { k } _ { \text {aero } } = \frac { 1 } { 2 } \mathrm { C } _ { \mathrm { a } } \mathrm { A } _ { \mathrm { d } } \quad \mathrm { C } _ { \mathrm { d } } \mathrm { F } _ { \mathrm { a } }$  (22)

$F _ { \text {grade} } = m _ { v } g \sin ( \alpha )$  (23)

where k_rrc is the resistance rolling coefficient, k_sc is the coefficient of road surface, k_aero is the factor aero drag, C_a is the density of air correction at the given altitude, A_d is air density of mass (km/m³), Cd is the drag aerodynamic coefficient of HEV (N $\frac{s^2}{kg}$ m), F_a is the area frontal of HEV (m²), and α is the incline angle of road (rad).

2.6. Internal combustion engine

The engine generates driving torque of $T _ { e } \left( \omega _ { e } , x _ { \theta e } \right)$ and the gearbox applies T_g as the load to the engine. The equation of the engine rotational motion can then be formulated as:

$\mathrm { T } _ { \mathrm { e } } \left( \omega _ { \mathrm { e } } , \mathrm { x } _ { \theta \mathrm { e } } \right) - \mathrm { T } _ { \mathrm { g } } = \mathrm { I } _ { \mathrm { e } }\frac{d_{\omega e}}{dt}$   (24)

where $\omega _ { e } , x _ { \theta e }$ and $I _ { e }$ are engine rotational speed, throttle position, and equivalent rotational inertia of engine respectively. While $x _ { \theta e }$ is assume as a variable in the range of [0%–100%]. The vehicle engine scheme is drawn in Figure 7.

Figure 7. Vehicle engine in Simulink model

2.7. Control system

The control system is designed from five separated sub blocks namely: Motor Controller; Mode Logic; Battery Controller; Generator Controller; and Engine Controller. The system control performs decisions required for the control of the HEV and drawn in Figure 8.

Figure 8. Control system in Simulink model

This study successfully builds up a high-level modeling of HEV with battery, converter, electric motor and generator, planetary gear, internal combustion engine, vehicle body, vehicle dynamics and the control system for HEV. Different HEVs are selected with different weights to simulate the fuel consumption on different tire rolling radiuses. Results of fuel consumption are analyzed and processed on several regression models. The best regression model is the third order polynomial equation with more than 99% of the fuel consumption being able exactly estimated. The fuel consumption regression model helps to identify the HEVs optimal weight subject to their tire rolling radius. Results of HEV optimal weight vs tire radius are also re-verified by simulations of optimal HEV and non-optimal HEV. HEV designers can refer to this study to select their HEV parameter to achieve the best fuel economy performance.