Model predictive eco-driving control of internal-combustion-engine vehicles equipped with CVTs in car-following scenarios

ABSTRACT Eco-driving control is a key technology for achieving carbon neutrality of vehicles equipped with automated-driving systems and advanced driver-assistance systems. This paper proposes a model-predictive eco-driving controller for car-following by internal-combustion-engine vehicles equipped with continuous variable transmissions. The effectiveness of our controller is demonstrated through numerical simulations. The results indicate that driving operations like the pulse-and-glide strategy are still candidate optimal solutions to the eco-driving control problem in car-following scenarios even in the receding-horizon setting. Moreover, the results indicate that restricting the engine operation range to the optimal brake-specific-fuel-consumption line is not necessarily optimal in the receding-horizon setting.


Introduction
Climate change is getting more serious [1].To resolve this problem, carbon neutrality should be achieved in the near future [2].In particular, emissions from the transportation sector must be reduced by more than 20% to reach carbon neutrality [3].Therefore, reducing the fuel consumption and emissions of vehicles will be crucial for building a sustainable human society.
Eco-driving control [4] is a key technology for fuel saving in vehicles with automated-driving systems and advanced driver-assistance systems.Eco-driving control aims at computing a desirable profile of vehicle speed, propulsion force, or their equivalent, that reduces fuel consumption.From a control-theoretic perspective, optimal control theory is often used to formulate and solve eco-driving control problems [4].
The fuel consumption of an ICE vehicle (ICEV) depends on the powertrain structure of the vehicle and its operation range.Under the quasi-static model assumption [16], the instantaneous fuel consumption (IFC) of the ICE can be determined by its torque and rotational speed.The relation between IFC and engine torque and speed can be depicted as an engine map.In general, the operation range of the engine during travel is restricted by the transmission.Continuous variable transmissions (CVTs) can make seamless changes in the transmission ratio [17,18].This feature makes it possible to exploit the maximum range of engine operation to improve fuel economy.
Car-following is one of the most important application scenarios, and numerous studies have treated it [12][13][14]19,20].Li et al. [5,6] numerically solved an optimal control problem of eco-driving and reported that the PnG strategy is the fuel-optimal driving strategy for ICEVs in car-following scenarios within a particular range of preceding-vehicle speed.However, they only formulated the problem in an offline setting and restricted the operation range of the engine to the optimal brake-specific fuel consumption (BSFC) line, i.e. the curve defined by connecting the most efficient operating point on each constant-power curve in the engine map [5].Hu et al. [21] proposed an online eco-driving controller for ICEVs in car-following scenarios without restricting the engine operation range to the optimal BSFC line, in which periodic motions did not appear in the numerical results.
In this study, we designed an online eco-driving controller based on nonlinear model predictive control (NMPC) [22][23][24][25][26][27] for car-following scenarios of ICEVs equipped with CVTs.We optimized not only the propulsion force but also the CVT ratio, in order not to restrict the engine operation range to the optimal BSFC line.The cost functional was set to be the fuel economy itself rather than the fuel consumption.Our numerical results indicated that PnG-like strategies are still potential solutions to the optimal eco-driving control problem in car-following scenarios even in the recedinghorizon setting.Moreover, they indicate that restricting the engine operation range to the optimal BSFC line is not necessarily optimal in the receding-horizon setting.These facts were not revealed in Ref. [21].Moreover, the latter fact means that the results reported in [5,6], whose engine operation range was restricted to the optimal BSFC line, do not apply to online settings.
The rest of this paper is organized as follows.In Section 2, a vehicle model is developed for deriving the state equation.In Section 3, the NMPC problem of the eco-driving for car-following scenarios is formulated.Section 4 gives numerical results which indicate the above-mentioned facts.Section 5 gives concluding remarks.

General description
The overall state-space model can be described as In the rest of this section, we derive a detailed description of the right-hand side of (1).

CVT ratio
We assume that the response of CVT ratio to its reference signal can be modelled as a first-order lag system, i.e.
where T cvt is the time constant of the CVT ratio.

Engine torque response
We assume that the response of engine torque to its reference signal can be modelled as a first-order lag system, i.e.
where T e is the time constant of the engine torque.

Vehicle propulsion
As commonly done in vehicle propulsion control [28], the propulsion dynamics of the vehicle are modelled as where M [kg] is the vehicle mass, F [N] the propulsion force acting between the wheels and road, and F dr [N] the driving resistance.The driving resistance is modelled as where c a [-] is the drag coefficient, A a [m 2 ] the maximum vehicle cross-section area, ρ a [kg/m 3 ] the air density, μ [-] the rolling resistance coefficient, and g [m/s 2 ] the acceleration due to gravity.Let v p [m/s] be the speed of the preceding vehicle.We will assume that v p is approximately constant.Accordingly, the dynamics of the preceding vehicle can be modelled as ṡp = v p .

Remark 2.1:
The proposed controller can be applied to the case of time-varying v p provided that the trajectory v p (t) over the prediction horizon can be predicted.The prediction of preceding-vehicle trajectory is beyond the scope of this study.

Driveline
The driveline architecture is illustrated in Figure 1.In what follows, the quantities and ω * [rad/s] with an arbitrary subscript ' * ' stand for the moment of inertia, torque, and rotational speed of the corresponding vehicle element indicated by ' * ', respectively.For these quantities, the subscript ' * ' is one of 'e', 'cvt', 'fd', or 'w', which respectively correspond to the engine, CVT, final drive, and wheels.The symbols i fd [-] and r w [m] denote the gear ratio of the final drive and the wheel radius, respectively.
As is commonly done in powertrain controls [28,29], we assume that • the clutch is always engaged, • the moments of inertia of the engine and wheels dominate the other driveline components, and • the friction losses and torsion effects in the driveline are neglected.We follow the standard driveline modelling (e.g.[28]), except that the transmission ratio is continuously variable in time unlike the literature in which discrete transmissions are treated.We deal with the time variation by substituting the dynamic equation ( 2) into driveline dynamics, as described below.
The engine is modelled as Torque and speed propagations are described as The wheels are modelled as By substituting ( 6) and ( 5) into the first term of the right-hand side of the above equation, we obtain Note that icvt does not vanish, unlike in discrete transmissions, because the transmission ratio is continuously variable in time.Rearranging the above equation yields By substituting (3b) into the last term of the right-hand side of the above equation, we obtain Rearranging this equation and using the equality r w ω w = v yields By substituting (2) into the second term of the righthand side of the above equation, we find that

Fuel consumption
In accordance with quasi-static modelling [16], the fuel consumption rate q BSFC [g/kWh] is modelled as a function of ω e and τ e , as illustrated in Figure 2. q BSFC is approximated by the polynomial, where c ij 's are coefficients.The IFC, denoted by q [mL/s], is described by q(ω e , τ e ) = 1 60 2 ρ q BSFC (ω e , τ e )P e (ω e , τ e ), where ρ [g/mL] is the fuel density, and P e [kW] is the engine output calculated by P e (ω e , τ e ) = ω e τ e × 10 −3 .
Recalling the equalities r w ω w = v and (6b), we can describe q as a function of v, i cvt , and τ e , i.e.
Finally, the fuel consumption dynamics are modelled as

Constitution of state equation
From the above, we obtain the state equation as follows:

NMPC formulation
We set the cost functional over the horizon as where T [s] is the horizon length, t 0 [s] is the initial time of the simulation, w 1 is the weight coefficient, and h d [s] is the target value of time headway [12,13].The first term represents the fuel economy, and the second the car-following error.
The initial condition on the horizon is set as where x t is the measured (or estimated) state-vector at the current time.The path constraints are set as ) where each symbol of the form * min/max means a prespecified minimum/maximum value of the quantity of interest.
The nonlinear optimal control problem is described as follows.

NMPC problem
At each sampling instant, solve the following optimal control problem over the horizon: The initial value of the obtained optimal control input over the horizon is applied to the controlled system at every sampling instant.

Settings
The parameters in the state equation and NMPC formulation are listed in Tables 1 and 2. The simulation duration in units of second was set to [t 0 , t f ] := [0, 100] with a sampling time of 0. We used Falcon.m[30] ver.1.27 with MATLAB R2020b to solve an optimal control problem on each horizon.To focus the investigation on vehicle behaviour, we did not consider the real-time computability and left it to future work.

Results and discussion
For comparison, we conducted two simulations on (a) our eco-driving controller and (b) a pure car-following controller designed with the fuel cost removed; i.e. the terminal cost of ( 8) was removed.
We obtained the time histories of variables from these controllers, as shown in Figure 3. From the top, the vertical axes are the inter-vehicle distance, vehicle speed, accumulated fuel consumption, CVT ratio, engine torque, engine speed, reference input of CVT ratio, and reference input of engine torque.The horizontal axes are the time.The engine operating points are plotted in Figure 4. We divided the time histories in Figure 3 into three phases: rush, coasting, and carfollowing plotted in different colours.These phases were also applied to the plot of the engine operating points in Figure 4.
The results of the two controllers show that the vehicle initially accelerated (rush phase) and then it coasted as it got close to the preceding vehicle and dropped back to the target distance (coasting phase).Eventually, it started following the preceding vehicle at a speed around its target value (car-following phase).
In the car-following phase of the eco-driving controller, the vehicle followed the preceding one with a quasi-periodic motion, unlike the case of the pure carfollowing controller, which drove the vehicle at a constant speed.This fact and the engine torque trajectory  show that the eco-driving controller drove the vehicle in a PnG-like way during the car-following phase.It is reasonable that the terminal cost representing fuel economy yielded the PnG-like behaviour known to be a possible fuel-optimal one.
The eco-driving-controlled vehicle had a step-wise fuel-consumption increment because of the PnGlike operation, while the pure car-following-controlled vehicle had a linear one.The fuel economy of the ecodriving controller was 40.1913 [km/L], which is greater than the pure car-following controller, 28.7501 [km/L].This shows the effectiveness of our eco-driving control method.
Before the first pulse phase of the eco-driving controller started at approx.39 s, the engine speed was regulated to around 2500 rpm by manipulating the CVT ratio.Accordingly, the engine was operated only in a high-efficiency region with less transient engine motion than operations in which the engine speed is escalated after the pulse phase starts.Moreover, the CVT ratio dropped during the pulse phase in order to suppress the escalation in engine speed and keep the engine operating point in the high-efficiency region while accelerating the vehicle to follow the preceding vehicle.These operations were also performed in the subsequent pulse phases.
The resulting engine operating points were quite different between the two controllers.When the vehicle was propelled, the engine operating points of the eco-driving controller were concentrated in the highefficiency region to improve fuel economy, while those of the pure car-following controller were distributed in the high-torque mid-speed region and low-torque low-speed region, which are low-efficiency regions.The above-mentioned CVT manipulation enabled the ecodriving controller to operate the engine in such a fuelefficient manner.The engine operating points resulting from the eco-driving controller were not distributed alongside the optimal BSFC line even when the engine was propelling the vehicle.This is due to the fact that the optimal BSFC line is, in general, calculated only for steady engine operation whereas eco-driving is a dynamic behaviour.
Although the reference inputs u cvt and u e of the two controllers oscillated, their effects were absorbed through the first-order lag systems, so that the fluctuations in the physical quantities i cvt and τ e were small.Moreover, the vehicle motion was smooth.Therefore, the oscillations in these controllers do not matter much.One can still incorporate penalty terms into the cost functional to reduce them, if needed.
The above discussion indicates that PnG-like strategies are still possible solutions to the optimal ecodriving control problem in car-following scenarios even in the receding-horizon setting.Moreover, restricting the engine operation range to the optimal BSFC line is not necessarily optimal in the online setting.These facts were not revealed in Ref. [21].Moreover, the latter fact means that the results reported in Refs [5,6], whose engine operation range was restricted to the optimal BSFC line, do not apply to online settings.

Remark 4.1:
The engine speed resulting from the ecodriving controller did not exceed 4000 [rpm], which is recognized as the lower boundary of the 'yellow zone', whereas no penalty was imposed on violations of this boundary.This fact implies that eco-driving controllers tend to make engine operation safer.

Remark 4.2:
Although PnG-like motions might not be comfortable for human passengers or drivers, this strategy is still useful when no one is in the vehicle, for example, in the case of automated driving of following vehicles in platooning [14].The trade-off between fuel optimality and ride comfort of the PnG strategy is studied in Refs [10,11,13].To achieve real-time optimization, this must be decreased below the sampling time (0.1 [s] in this simulation setting).This computational burden was due to the use of an offline optimal control problem solver with high accuracy.To accelerate the computation, one can use tailor-made NMPC softwares, e.g.C/GMRES [24,25] and ParNMPC [26,27].

Conclusion
This paper proposed a model-predictive eco-driving controller for the car-following scenarios of ICEVs equipped with CVTs.The effectiveness of our controller was demonstrated through numerical simulations.The results showed that car-following could be achieved with a PnG-like operation.This indicates that PnGlike strategies are still candidate optimal solutions to the eco-driving control problem in car-following scenarios even in the receding-horizon setting.Moreover, it was shown that the engine operating points resulting from the NMPC were not distributed alongside the optimal BSFC line.This indicates that restricting the engine operation range to the optimal BSFC line is not necessarily optimal in the receding-horizon setting.
The computational burden was not addressed in this paper because it focused on an investigation of vehicle behaviour.With the goal of implementing the proposed controller, real-time computability should be studied in the future.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Figure 3 .
Figure 3.Time histories of variables.The rush, coasting, and car-following phases are plotted in magenta, green, and blue, respectively.The dashed black lines are the minimum/maximum values of the corresponding quantities.The dotted black lines in the top and second plots are h d v p and v p , respectively.(a) Eco-driving (proposed) and (b) Pure car-following.

Figure 4 .
Figure 4. Engine operating points.The colours of the plot markers correspond to those of the time histories (Figure 3).The solid black curve is the optimal BSFC line.(a) Eco-driving (proposed) and (b) Pure car-following.

Remark 4 . 3 :
The average computation time for optimization at each time instant was 20.7923 [s].

Table 1 .
Parameters in the state equation.