Speed Profile Generation for a Dual Motor Equipped Electrified Series Hybrid Tracked Vehicle Through Dynamic Programming Based Energy Optimization Procedure

Abstract

Electrification is a widely explored area in many fields and is becoming a widely researched topic in tracked vehicles. Achieving efficient operation for such vehicles is crucial similar to traditional vehicles. This paper introduces a speed profile generation procedure for a dual track – series hybrid tracked vehicle with an emphasis on fuel economy and trip time optimization. A Dynamic Programming method is proposed in spatial domain, considering the properties of predefined road geometries. The study includes examination of speed and torque estimation of the sprockets, particularly during steering maneuvers. Unlike wheeled vehicles, tracked vehicles have to overcome the turning resistance moment of skid steered tracks. Taking these dynamics into account, the study investigates efficient operation, presenting over a predefined road track including straight road, inclined road and curve sections, discussing the tradeoff between fuel economy and trip time optimality. As the result of problem solution, speed profiles are generated, and operation trends of the powertrain components are discussed.

1 Introduction

Energy efficient operation is the main driving force of improvement of hybrid electrified vehicles. This subject is mainly researched for straight road conditions, utilizing different type of control strategies e.g. rule based control, power following control, ECMS, etc. Dynamic Programming (DP) is generally used in these studies as benchmark of the optimal control. Most studies apply the rules of the DP in time domain, where speed profile is predefined. A good example for a dual track hybrid vehicle is given in [1], where energy management system is discussed for a tracked vehicle on straight road and includes also cornering sections in time domain. [2] discusses the improvement of fuel economy of a wheeled hybrid vehicle for an upcoming uphill and downhill scenario. In [3] a speed profile generation study for a conventional truck is presented for upcoming inclined roads. This study is an example of the spatial domain DP problem applications. [4] provides a sample of spatial domain problem for a wheeled EV vehicle, covering energy efficiency of electrical motors and trip time minimization tradeoff, for a predefined track including curved road sections. Adaptive on-line optimization strategies are covered in literature, however scope of this paper limited with predefined road profile and off-line problem solution.

This study aims to provide optimized target speed profiles for a series hybrid – electrified tracked vehicle which provides efficient operation in terms of fuel economy & trip time. The DP approach is applied to solve the problem in spatial domain for predefined road geometries. Using the characteristics of created road profile sections, e.g. inclination, turning radius of the curves; required torque, speed and power demand of electrical motors are estimated. Trip time, fuel consumption and losses of other powertrain components are the key objectives used in DP to be minimized. The output of this problem is the optimized target speed profile. For a dual-motor, series hybrid tracked vehicle the traction is provided by only the electric motors. Supply of the electrical motors’ power demand is shared between the generator and the battery with respect to applied control strategy. This relation is presented in Fig. 1. In the figure, two sided arrows show the possible bidirectional power flow.

Fig. 1.

Power flow diagram of the dual motor – series hybrid tracked vehicle

Unlike wheeled vehicles, braking of electrical motors does not occur only while slowing down; the inner track brakes during a steering maneuver to overcome the turning resistance moment of the skid steered tracks. In this scenario, outer tracks consume the supplied traction power, whereas inner track regenerates the kinetic energy during braking. In most generic form, required dynamic torque for electrical motors can be estimated as in Eq. (1) for a cornering maneuver [1]. In the equation, first square bracket section includes required torque due to steady state maneuver (Rolling resistance, resistance due to slope and turning resistance with respectively), whereas second square bracket section covers the inertial moments due to longitudinal and yaw motion with respectively. In the equation $r_{spr}$ describes the sprocket radius, $i_0$ is cumulative gear ratio of the driveline, $\eta$ is driveline efficiency, $B$ is the tread of the vehicle and $\dot{\omega }_{EM_{o - i} }$ is the acceleration of outer and inner electrical motors with respectively.

$$ \tau_{EM_{o - i} } = \left[ {\left( {F_{Roll} + F_{Grad} \pm F_{Turn} } \right)\frac{{r_{spr} }}{i_0 \eta }} \right] + \left[ {\frac{{mr_{spr}^2 }}{i_0^2 \eta }\frac{R}{(R \pm B/2)} \pm \frac{{I_z r_{spr}^2 }}{i_0^2 \eta B(R \pm B/2)}} \right]\dot{\omega }_{EM_{o - i} } $$

(1)

$F_{Turn}$ parameter given in Eq. (1) occurs due to turning moment resistance of the tracks and it is expressed in Eq. (2). $\mu_t$ is effective turning resistance coefficient which decreases with respect to increasing turning radius, as expressed in [5]. For the case of R > B/2 electrical motor velocities can be calculated as shown in Eq. (3) for outer and inner motors with respectively. $V_c$ is the velocity of the centerline of the vehicle.

$$ F_{Turn} = \frac{W \cdot \mu_t (R) \cdot L}{{4 \cdot B}} $$

(2)

$$ \omega_{EM_{o - i} } = \left( {V_c \cdot \left( {1 \pm \frac{B/2}{R} } \right)} \right) \cdot \frac{i_0 }{{r_{spr} }} $$

(3)

2 Formulation of the Dynamic Programming Problem

To reduce the burden of dimensionality problem of DP, it is assumed that vehicle perfectly follows perfectly the trajectory at each stage and lateral dynamics are only reflected in kinematic relations for the tracks in Eq. (3) and in Eq. (1) for calculating the electrical motor torques. Main dimensions of the DP problem are hence selected as 1) Distance, 2) Longitudinal velocity, 3) Longitudinal acceleration, 4) SoC of the battery and 5) Alternator torque. Velocity and SoC are the main states; longitudinal acceleration and alternator torque are considered as the inputs for the optimal control problem. Note that alternator in generator set is assumed operating in torque mode, altering the applied brake torque to provide required power output. Contrarily internal combustion engine operates in speed mode, maintaining its speed at rated speed. Therefore, control variable in the generator assembly is considered as only the alternator torque. At each stage of DP, longitudinal velocity and SoC at next stage is calculated with respect to corresponding inputs as shown in Eqs. (4) and (5). $v_{k + 1}$ is the calculated velocity at next stage using long. Acceleration input $a_{i,k}$ and $\Delta t_{j,k}$ which is the duration to complete constant road segment distance $\Delta s$. $SoC_{k + 1}$ is the calculated SoC at next stage using the battery current $I_{Batt}$ and maximum charge capacity $Q_{max}$.

$$ x_{1k + 1} = v_{k + 1} = v_{j,k} + a_{i,k} \cdot \Delta t_{j,k} ,where\;\Delta t_{j,k} = \frac{\Delta s}{{v_{j,k} }} $$

(4)

$$ x_{2k + 1} = SoC_{k + 1} = SoC_k + I_{Batt} \cdot \frac{{\Delta t_{j,k} }}{{Q_{max} }} $$

(5)

In Eq. (5) battery current is calculated according to Eq. (6). ${\text{V}}_{{\text{oc}}}$ is open circuit voltage and ${\text{R}}_{{\text{Batt}}}$ is the internal resistance of the battery. Battery power depends on the power demand of electrical motors and power supply of the generator as defined in Eq. (7). If the generator power supply exceeds the electrical power demand of the motors, it charges the battery, otherwise they both supply the electrical motors. Also, battery is charged through recuperation during the braking of electrical motors.

$$ {\text{I}}_{{\text{Batt chg}} - {\text{dischg}}} = \frac{{ \mp {\text{V}}_{{\text{oc}}} \pm \sqrt {{{\text{V}}_{{\text{oc}}}^2 \pm 4{\text{R}}_{{\text{Batt}}} \cdot {\text{P}}_{{\text{Batt}}} }} }}{{2{\text{R}}_{Batt} }} $$

(6)

$$ {\text{P}}_{{\text{Batt}}} = - \left( {\left( {P_{Elec - EM - o} + P_{Elec - EM - i} } \right) - P_{Elec - Gen} } \right) $$

(7)

The cost function of the DP problem, $g_k \left( {u_k ,x_k } \right)$ consists of three main parts as shown in Eq. (8). Trip time cost and energy related cost have their own weighting parameters: $w_T$ and $w_E$ respectively. Energy cost include the fuel consumption, and power losses of the battery and the electrical motors. Constraints defined by restrictions of components and physical limits are penalized by the last term “$L_{Constraints}$”. While solving the DP problem backwards from “N-1”^th stage back to 1^st stage, minimum cost at each stage is stored in the so called “cost-to-go matrix” (Eq. (9)), and corresponding optimal control values ($u_k$ → Long. Acceleration & Generator torque) are stored in other matrices. After these steps are completed, a forward calculation loop starts to obtain optimal velocity profile, and SoC calculations according to defined initial conditions. Constraints are applied in the problem according to inequalities (10), (11), (12), (13), (14) and (15):

$$ g_k \left( {u_k ,x_k } \right) = w_T \cdot \Delta t_{j,k} + w_E \cdot \left( {Fc + P_{Losses} } \right) \cdot \Delta t_{j,k} + L_{Constraints} $$

(8)

$$ J_k \left( {x_k } \right) = \begin{array}{*{20}c} {\min } \\ {u_k } \\ \end{array} \left( {g_k \left( {u_k ,x_k } \right) + J_{k + 1} \left( {f\left( {u_k ,x_k } \right)} \right)} \right) $$

(9)

$$ \omega_{EM min} \le \omega_{EM o - i} \le \omega_{EM max} $$

(10)

$$ \tau_{EM min} \le \tau_{EMo - i} \le \tau_{EM max} $$

(11)

$$ \left| {I_{Batt} } \right| < I_{max} $$

(12)

$$ a_{x - min} \le a_x \le a_{x - max} $$

(13)

$$ v_x \le v_{max - limit} $$

(14)

$$ v_x \le \sqrt {{a_{y - max} \cdot R}} $$

(15)

As the nature of DP, final stage has a separate solution step, where a special penalty for velocity and for battery SoC, as shown in Eq. (16), enforcing them to reach their final value at N^th stage to desired value $Vel_{Fin}$ and $SoC_{Fin}$. Quadratic formation of these penalties is beneficial in problem solution of the DP method.

$$ J_N \left( {x_N } \right) = g_N \left( {x_N } \right) + \left( {Vel_N - Vel_{Fin} } \right)^2 \cdot 100 + \left( {SoC_N - SoC_{Fin} } \right)^2 \cdot 10^6 $$

(16)

3 Results of Speed Profile Optimization

The DP problem is solved based on given road profile in Fig. 2. To present the concept in a clear way, length of the designed road is kept short. The profile consists of curvature sections as well as gradient sections including ascending and descending 10% slope. Maximum speed limit is shown in Fig. 2-b calculated and used as a constraint for DP, considering longitudinal and lateral acceleration limits. Solutions are obtained for three different weighting as shown in Fig. 3: Solution 1 reaches to higher speeds at road segments compared to others, whereas Solution 3 have overall better fuel economy as numeric results are listed in Table 1.

Fig. 2.

a) Road Profile b) Velocity Limit & Road Data

Fig. 3.

Velocity profile results for 3 different Time – Fuel Cons. Tradeoff weightings

Table 1. Results for 3 different Time – Fuel Cons. Tradeoff weightings

In Fig. 4 operation results for powertrain components are presented. Note that sign of the power values are with respect to the component perspective. For example, negative battery power means, power is distributed from the battery, whereas positive sign corresponds to power is absorbed: charging. According to results, alternator usage is more aggressive in Solution 1 which results with higher fuel consumption. At acceleration phases, alternator provides power supply, whereas, at near steady state velocity regions, flat road sections, alternator doesn’t contribute. Battery is charged starting from about 70 m up to start of the +10% gradient uphill, then battery power is used at downhill and straight segments. It is also observed that the battery is slightly charged before R10 and R15 curve sections. At the end of 200 m road profile, SoC values and velocity get back to ~60% and ~5 km/h reaching to their initial values as enforced in DP problem.

Fig. 4.

Operation of the powertrain components

4 Conclusion

This study covers a 5D Dynamic Programming method for a Dual Motor – Series Hybrid Tracked vehicle. In the study, speed profile generation method is explained on a predefined road which cover curves, and inclination. According to results, tradeoff between fuel economy, and trip time can be presented operation of powertrain components are observed.