through Road Topography Preview Information

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Fuel-Efficient Platooning of Heavy Duty Vehicles through Road Topography Preview Information

LUKAS B¨ UHLER

Master’s Degree Project

Stockholm, Sweden August 2013

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through Road Topography Preview Information

LUKAS BÜHLER

Master’s Thesis at the Automatic Control Laboratory of KTH, Stockholm Supervisors: Assad Alam, Jonas Mårtensson

Examiner: Prof. Karl Henrik Johansson Examiner at ETH: Prof. Manfred Morari

XR-EE-RT 2013:020

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I

Abstract

Road freight transport is a growing business and serves as a centerpiece of modern economics.

Traffic congestions due to the increasing amount of vehicles, growing environmental problems due to CO2emissions and rising fuel prizes create a vast demand for solutions to these global issues.

Look ahead cruise control and vehicle platooning can be part of a future transport concept. By using GPS localization and considering road topography data (map data), a Look Ahead Cruise Controller LAC optimizes the velocity profile of a single vehicle when facing steep up- or downhills in order to save fuel. Platooning describes the concept of driving vehicles in a convoy with short intermediate distance to reduce the aerodynamic drag and achieve considerable fuel consumption reductions for all vehicles in the platoon. However, research on platooning with road topography preview information is a novel topic.

In this thesis, look ahead cruise control and platooning are analyzed and results are com- bined in order to develop a platoon look ahead controller (PLAC). The PLAC features low computational complexity due to a simple parametric optimization method and reaches sim- ulated energy consumption reductions of up to 20% for a vehicle in the platoon compared to its single drive on the same road. When velocity limits are considered, the fuel saving potential is even higher. It can be shown that the reached energy consumption with a PLAC lies slightly (1.3%-1.9%) over a minimum energy consumption needed to traverse the road section. Results reveal that in practical situations, a platoon should be maintained whenever possible. Errors in parameters of the vehicle, road parameters and localization errors are analyzed in terms of their influence on the energy consumption. Strategies are presented to reduce the influence of parameter errors on the fuel consumption and especially reduce the chance of a separate of the platoon.

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Preface

With this thesis, I complete my studies for a Master of Science in Electrical Engineering and Information Technology at ETH Zurich. As an exchange student, I wrote this thesis at KTH in Stockholm. However, the thesis will be part of my ETH diploma.

I believe that the reduction of the consumption of fossil fuels and the lowering of CO2 emissions to the environment will become main global issues in the upcoming centuries. In the freight transport sector, there is clearly a large potential for improvement where platooning can be part of a future transport concept. The importance of this issue, the technical chal- lenges and the close relation to reality were my personal motivation. I enjoyed working on this topic and writing my thesis and I hope to have contributed thereby to future transport concepts.

Acknowledgement

I would like to express my deepest gratitude to Prof. Karl Henrik Johansson at KTH, who was my main advisor and examiner. He made it possible for me to write my thesis within his group. I highly valuate his good inputs and his communication on eye-level with students.

With Assad Alam and Jonas Mårtensson, I had the privilege to have two very committed supervisors. I would like to thank both of them for the inspiring meetings we had and the valuable tips and inputs I received. A special thank goes to Prof. Manfred Morari at ETH, who made it possible for me to write my master thesis in Sweden.

Many thanks go to my parents for supporting me during all of my education and studies. Without you, I would not have reached this success in my life. And last but not least, I am very grateful to my beloved girlfriend Caroline, who gave me the support I needed during all my years of studying.

Lukas Bühler Stockholm, August 2013

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Introduction

The road transport sector for goods has grown rapidly over the past centuries. 1723 tonne- kilometers were driven by freight transports on roads of the EU-27 states in 2010 [9] The increasing density of vehicles on roads causes traffic congestions and expensive renewals of the infrastructure are carried out in many places. Beneath the road capacity problem, the increasing traffic accounts for resource shortage and environmental problems.

Global warming is growing to a major environmental problem worldwide, caused by the CO2

exhaust and other greenhouse gasses. Road freight transport, which is carried out mainly through fossil fuel combustion, is responsible for a significant part of the problem. The road transport sector accounts for 71% of all CO2 emissions from transports within Europe [10].

Aside of environmental problems, fossil fuels are a diminishing energy source. Fuel prices have increased during the past years and account for around one third of the life cycle cost for a Heavy Duty Vehicle (HDV) [17].

Both from an economical and environmental point of view, effort is required in order to reduce the usage of fossil fuels for road transports. During the past centuries, significant improvements in the efficiency of HDVs were achieved through technological progress. The focus mainly lied on the increase in efficiency of the motor and drive train, but not on intelligent traffic control.

Modern transport concepts for road transportation as part of Intelligent Transportation Sys- tems (ITS) are being developed. One main objective is the fuel usage reduction of HDVs and the avoidance of traffic congestions. Optimization in terms of avoiding empty drives, better utilization of vehicles and fuel-efficient driving is demanded. New communication standards [1] have been defined in order to implement Vehicle-to-Vehicle V2V and Vehicle- to-Infrastructure V2I communication, which will be important for a future ITS.

The concept of driving vehicles in a convoy (platooning) can be part of a solution to the above mentioned problems of road congestion, CO2 emission and limited fuel resources. By driving in platoons, the road capacity can be increased and the fuel consumption can be reduced [7]. Figure 1.1 shows a vehicle platoon on a highway with short intermediate distance among the vehicles. The aerodynamic drag (airdrag) forces, which are responsible for a major part of the energy losses of a vehicle, can thereby be reduced significantly. Studies on

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Swedish highways show that the fuel consumption of a following vehicle can be reduced by up to 4-7% by using the Adaptive Cruise Controller ACC with a short intermediate distance [3].

This controller measures the distance to the HDV in front and holds a predefined distance.

A framework for a future HDV platooning concept is presented in [2] by using decentralized LQR control.

Figure 1.1. Demonstration of HDV platooning on a highway¹

A typical HDV is not able to hold a constant velocity over steep uphill or downhill slopes on a highway due to the limited engine power. On uphill slopes, the vehicle decelerates while applying full throttle. Whereas on downhill slopes, the vehicle accelerates while applying zero throttle. This fact was addressed in several scientific papers in order to find control strategies to minimize the fuel consumption of a single vehicle by using the knowledge about the road topography in front of the vehicle, hereinafter referred to as look ahead cruise control LAC. Fuel optimal control by keeping an over all constant travel time with road topography preview information for a single HDV has been studied in [11], [12], [13] and [14]. Fuel minimizing velocity profiles for a single HDV on a set of simple road profiles are shown in [12]. The optimal control input for an affine engine model consists of sections of zero fueling, sections of full fueling and sections where the fueling is selected in order to keep the velocity constant [13]. The optimal velocity profile to traverse a downhill section is described by a fuel cut-off before the slope, so that the speed is reduced until the start of the slope. The vehicle accelerates during the downhill and reaches a higher speed than the initial speed. After the slope, it decelerates to its initial velocity where steady fueling is taken up again. Similar for an uphill section, full fueling is applied before the uphill so that the vehicle accelerates until the start of the slope. Even with maximum fueling during the uphill, the vehicle decelerates and the speed is reduced until the end of the slope. After the slope, it gains speed so that it reaches its set velocity after the uphill.

Platooning requires a coordinated control of the vehicles due to their deviating acceleration when it comes to steep hills on a highway.

1Source: Courtesy of Scania CV AB

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CHAPTER 1. INTRODUCTION

1.1 Problem Formulation

The problem addressed in this thesis is fuel-efficient control of a platoon for a given road topography. Therefore, a platoon of N heterogeneous vehicles as shown in Figure 1.2 is con- sidered. The slope of the road, α(s), in front of the platoon is assumed to be known and the vehicles are assumed to be equipped with wireless communication, which makes local centralized control possible.

N-1 2

N ... 1

vN vN-1 v2 v₁

α

Figure 1.2. HDVs driving in a platoon experience a decreased air drag force and can therefore reduce their fuel consumption considerably. Over downhill or uphill sections, the individual vehicles might not be able to maintain a constant velocity, which requires a control strategy for the whole platoon in order to traverse the slope with the objective of fuel consumption minimization.

On the level road in front of a steep road section, the platoon is in steady state with a set velocity and constant spacing between the vehicles. Strategies and control should be investigated in order to minimize the fuel consumption of the whole platoon for the given road topography by maintaining a given average speed (constant travel time) after the slope. Con- trol actions are completed when the platoon is back in steady state on level road.

It is of particular importance that the control solution is of low computational complexity, which makes the implementation on the vehicle feasible. Therefore, the interest lies not only on optimal control in terms of fuel reduction, but also on simple control strategies in order to implement fuel-efficient drive.

In a final step, the influence of erroneous assumptions about parameters (vehicle, map data, localization error) should be investigated and strategies in terms of control should be found.

1.2 Thesis Outline

In chapter 2, the used model for HDVs is presented and simplifications are explained. Look ahead cruise control for a single vehicle is addressed in Chapter 3. The optimal control inputs for a single vehicle are derived and parametric optimization is presented as a solution in order to implement an optimal controller with low computational complexity. Due to the fact that the optimal control problem grows significantly in complexity when considering not only a single vehicle, but a platoon, energy flows and fuel saving potentials in a vehicle platoon is analyzed in Chapter 4. This leads to the design of a PLAC in Chapter 5, which combines the highest energy saving possibilities and achieves energy consumptions, which lie only slightly above the calculated minimum energy consumption. In Chapter 6, the influence of errors is investigated and conclusions are drawn in order to prevent severe effects on the

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fuel consumption. Chapter 7 summarizes and discusses the obtained results and gives an outlook on future work.

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Chapter 2

Model and Definitions

This Chapter presents the model of an HDV used for calculations and simulations. The model assumes a vehicle reduced to its ability of longitudinal movement. This means the vehicle is considered without any sideway steering action. The equation of motion was derived in [2].

This equation of motion provides a general model of a vehicle without any restrictions to a specific type of HDV.

An engine model for the 12 liter Scania DT1211 L02 diesel combustion engine was presented in [16]. This model was used as a reference to derive a general engine model. This engine model should apply to various vehicle engines. Input to the system is therefore not a throttle input, but the torque requested from the engine.

All variables and parameters are described in appendix A.3.

2.1 Equation of motion

The equation of motion describes the acceleration of the vehicle ˙v depending on the forces acting on the vehicle. It was derived in [2].

˙v = 1

m_t(Fengine(ωe, P) − Fbrake− F_airdrag(v, d) − Froll(α) − Fgravity(α)) (2.1) The acting forces are illustrated in Figure 2.1.

2.1.1 Engine Force

The engine force is produced through the torque of the engine Te acting on the power train which drives the wheels of the vehicle. The engine torque mainly depends on the applied throttle position P ∈ [0, 1] and the current engine speed ωe. Over the shifting gear with ratio i_t, efficiency ηt, the gear of the final drive with ratio if and efficiency ηf, the torque is applied to the wheels with radius rw. The wheels then apply a force Fengine acting on the vehicle.

Fengine= i_ti_fη_tη_f rw

Te(ωe, P) (2.2)

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Figure 2.1. Forces acting on the HDV

2.1.2 Brake Force

The brake force is considered as a direct input to the system to slow the vehicle down.

2.1.3 Aerodynamic Drag Force

The aerodynamic drag force is produced by the airflow around the vehicle. It depends on the coefficient for the aerodynamic drag cd(d), the density of the air ρa, the area exposed to the airflow Aa and the velocity of the vehicle v. It should be remarked that the aerodynamic force is proportional to v². For an HDV with a mass m = 40000kg on a flat road α = 0 with a speed of v = 85km/h, the aerodynamic drag force is responsible for around 65% of the repulsive forces acting on the vehicle.

Fairdrag= 1

2cD(d)Aaρav² (2.3)

The aerodynamic drag force of a vehicle is reduced if it drives in a platoon due to the vehicles surrounding it. This reduction of the aerodynamic drag coefficient was found in empirical tests in a wind tunnel and is described in [3].

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CHAPTER 2. MODEL AND DEFINITIONS

Figure 2.2 shows the model for the reduction of the air drag coefficient. It is assumed that the distance to the vehicle ahead (lead vehicle) dl and the distance to the follower vehicle df

is known and the total cD reduction is calculated as a function of the distances as shown in equation 2.4.

c_Dreduction(dl, d_f) = cDreduction-l(dl) + cDreduction-f(df) (2.4) The empirical air drag reduction was approximated with a 3rd order least squares polynomial fit. For all simulations, the approximated polynomial fit was used and the calculation of the c_D reduction was done according to equation 2.4.

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70 80

Distance d l, d

f [m]

cD reduction [%]

Reduction due to following vehicle (d_f) Reduction due to 1 leading vehicle (d_l) Reduction due to 2 leading vehicle (d

l)

Figure 2.2. cDreduction due to a vehicle in the back (follower vehicle) and one or two vehicles in the front (lead vehicles)

2.1.4 Roll Resistance Force

The roll resistance force acts on the HDV due to friction losses in the tires and wheels. It depends on the roll resistance constant cr, the mass of the vehicle m, the gravity constant g and the road angle α.

F_roll(α) = crmgcos(α) (2.5)

2.1.5 Gravity Force

As soon as the road is not flat, α 6= 0, the gravitational force influences the vehicle strongly, especially vehicles with a high mass m.

Fgravity(α) = mg sin(α) (2.6)

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In order to unite rotational movements and longitudinal movements, a total inertial mass mt

is used in the equation of motion. It is calculated as follows:

mt= m + Jw

r²_w +i²_ti²_fηtηfJe

r_w² (2.7)

Distance reference

A given road profile defines road gradients as a function of distance in front of a vehicle. The equation of motion is therefore adapted from time reference to distance reference in order to simplify calculations. The infinitesimal horizontal distance element ds is assumed.

˙v = dv dt = dv

ds ds dt = dv

dsv (2.8)

(2.9)

dv ds = 1

v · mt(Fengine− F_brake− F_airdrag− F_roll(α) − Fgravity(α)) (2.10)

= 1

v · m_t

iti_fηtη_f

r_w T_e(ωe, P) − Fbrake−1

2c_D(d)Aaρ_av²− c_rmgcos(α) − mg sin(α) (2.11)

2.2 Engine Model

The engine torque model derived in [16] was used. The engine torque, Te, depends on the engine speed, ωe, and the fueling, γ:

Te(ωe, γ) =

( aeωe+ beγ+ ce , γ >0

a_dω_e+ bd , γ= 0 (2.12)

The fueling is approximated by γ = P ˆγmax with the throttle input P ∈ [0, 1] and the approx- imation for the upper bound of the fueling:

ˆγmax(ωe) = aγω_e²+ bγωe+ cγ (2.13) The parameters aγ,bγ and cγ are characteristic engine coefficients. The engine speed ωe is coupled with the velocity of the vehicle, v, over the gear ratios and the wheel radius: ωe= ⁱ_r^t_wⁱ^fv

2.2.1 Model simplification

The above shown engine model represents the 12 liter Scania DT1211 L02 diesel combustion engine as presented in [16]. Other engines have different parameters and therefore different engine characteristics. In order to get a general applicable engine model, the above mentioned model was analyzed and characteristics were simplified.

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Figure 2.3 shows the range of the engine torque Te depending on the velocity for the highest gear. The green area shows the engine torque Te which can be reached by variation of the throttle position P ∈ [0, 1] at different vehicle velocities with the highest gear. In order to simplify this model, a lower and upper limit for the engine torque Te was defined which can be reached over the whole relevant speed range with this engine. The new input of the system is the engine torque Te∈[Tmin, Tmax].

This leads to the following motion of equation with the engine torque Te ∈ [Tmin, T_max] as input:

dv ds = 1

v · mt

i_ti_fη_tη_f rw

T_e− F_brake−1

2c_D(d)Aaρ_av²− c_rmgcos(α) − mg sin(α) (2.14) (2.15)

70 75 80 85 90 95 100

−500 0 500 1000 1500 2000 2500

T_{m a x}

T_{m i n}

v [km/h]

Te[Nm]

Figure 2.3. Engine torque range depending on the vehicle velocity.

2.2.2 Engine fueling

The fueling for a given engine torque Teand an engine speed ωe can be calculated for the 12 liter Scania DT1211 L02 diesel combustion engine as shown in (2.16).

γ(ωe) = 1

b_e(Te− a_eω_e− c_e) (2.16) Figure 2.4 shows the fueling of an HDV driving with the highest gear. The fueling depends on the velocity and the applied torque. It can be seen that the dependency on the velocity is negligible compared to the dependency on the torque. This function also changes with the engine. To use a general model for the fueling, the engine torque Te can be integrated over

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70 75

80 85

90 95

100

0 500 1000 1500 0 50 100 150 200 250

v [km/h]

Te[Nm]

Fueling γ [mg/stroke]

Figure 2.4. Fueling function

the distance to get a correct appraisal of the consumed fuel.

The following affine engine fueling model is assumed:

γ = kTT_e+ cT (2.17)

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2.3 Definitions and Assumptions

2.3.1 Definition of road gradients

HDVs are typically not able to hold their velocity constant on various road gradients of highways. On steep uphill segments, a fully loaded HDV is decelerating even if full engine torque is applied. On steep downhill segments, an HDV is accelerating even if minimum engine torque is applied without using the brake.

Therefore, the following definition of road gradients can be stated:

small gradient ⇔ ∃ T_e such that dv

ds = 0

T_max≥ T_e≥ T_min (2.18) steep gradient ⇔ @ Te such that dv

ds = 0

Tmax≥ T_e≥ T_min (2.19) 2.3.2 Definition of expressions

Coasting: Reducing the engine fueling to the minimum, so that the engine applies a braking force on the vehicle.

2.3.3 Definition of road section use cases

A real road can be considered as a sequence of road sections with constant inclinations. The test road profiles were constructed to be easy enough to be used for the derivation of fuel- efficient velocity profiles and still capture the properties of real roads. Use cases 6 and 7 were designed in order to investigate the influence of an asymmetrical road profile, especially when velocity limits are considered.

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Figure 2.5. Road profiles

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2.3.4 Assumptions

Equal Power Trains: All vehicles are assumed to have the same powertrain and therefore the same torque range Te∈[Tmin, T_max].

Velocity Range: HDVs are assumed to drive in a limited velocity range of 70- 100km/h. This range contains the upper speed limits on euro- pean highways (80km/h or 90km/h) and provides a reasonable range for variation. This gives also reason to the engine model simplifications (affine engine model) done in Section 2.2.

Gearshifts: Gearshifts require a more complex engine and fueling model.

Due to the limited velocity range, no gearshifts are considered.

One exception are lower velocity limits. The vehicle is then assumed to shift gear immediately without any temporary loss of engine torque.

Constant Travel Time: A constant travel time or an average velocity respectively is assumed as constraint to be fulfilled after traversing the slope. This assumption is important since otherwise the velocity would drop to zero for the optimization criteria. The average velocity should be maintained as soon as possible after the steep slope. This requirement comes due to the fact that the controller of the platoon acts in a wider context together with other more global acting controllers. Other platoons depend on the schedule of the controlled platoon.

Inter-Vehicle Distance: Investigations about the minimum distance between two vehicles in terms of safety and crash avoidance were done in [4].

If no time delay in the communication between two vehicles is assumed and two vehicles have the same velocity, it is reason- able to define a minimum distance dmin which is independent of their velocity. This minimum distance depends on vehicle parameters like the weight or the tires but remains constant while driving. Therefore, the distance between two vehicles has to fulfill the following constraint: d ≥ dmin

Velocity Limit Constraint: Upper velocity limit constraints are required due to speed limits on roads. If an upper velocity limit is considered, it makes sense to also consider a lower velocity limit. Otherwise the velocity is likely to exceed the velocity range where the engine simplification is reasonable.

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2.3.5 Fuel and energy consumption measurement

One of the main subjects of this thesis is the fuel consumption minimization for an HDV on a given road profile. Therefore, the fuel and energy consumption of a vehicle is often given in simulation results. It will be shown in Section 4.1.6 that the fuel consumption is closely related to the energy consumption of a vehicle and that the minimum of the fuel consumption is equal to the minimum energy consumption. The calculation of the fuel and energy consumption is done according to (2.20) and (2.21).

Fuel: F =^Z ^s^b

sa

i_tγds (2.20)

Energy: E =^Z ^s^b

sa

F_airdrag+ Froll+ Fbrakeds (2.21) The distance sa describes the point where the first control action in front of a steep road section starts, i.e. the point at which the first vehicle starts accelerating or decelerating. sb

describes the point where the last control action is completed in order to get the last vehicle back to a constant velocity.

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Chapter 3

Look Ahead Cruise Control for a single HDV

A typical HDV is unable to maintain a constant velocity on uphill or downhill sections of a normal highway. On steep uphill slopes, the engine can not provide enough power to maintain the velocity and the vehicle decelerates. On steep downhill sections, a heavy vehicle might accelerate even if no fueling is applied to the engine (Section 2.3.1). This behavior of an HDV gives rise to the question of finding an optimized velocity profile which minimizes the fuel consumption.

The problem of fuel consumption minimization for a single vehicle on a given road segment with a given travel time was addressed in several scientific works. Optimal velocity profiles for simple road topographies can be found analytically [12]. Fuel optimal control with road topography preview information for a single HDV has been studied in [11], [12], [13] and [14], hereinafter referred to as Look Ahead Cruise Control (LAC).

In all cases of finding an optimal velocity profile for a given road topography, it is assumed that the travel time Tt will be maintained with every strategy. In other words, the average velocity vav is constant over the investigated section of the road.

3.1 Optimal control

The problem of minimizing the fuel consumption of an HDV by controlling the velocity on a given road topography with a given travel time has been analyzed in [12] and [13]. Below, the optimal control inputs to the system are derived in brief. The method was adopted from [13].

An affine engine fueling model as presented in this thesis (2.17) was used for the derivation of optimal control solutions. The vehicle is assumed to drive in the highest gear and the gear ratio it is therefore assumed to be constant. The optimal control problem can be formulated as fuel consumption minimization over the distance sf which is equivalent to the minimization of the integration of the input Te over the distance by(2.17). A constant travel

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time Tt will be maintained:

min ^Z ^s^f

0

γds ⇔ min ^Z ^s^f

0

Teds (3.1)

s.t. ^Z ^s^f

0

1

vds = Tt (3.2)

With the states x = [v, Tt]^T and the state equations for the dynamics of the velocity and the travel time:

dv ds = 1

mtv

itifηtηf

rw

Te− 1

2cDAaρav²− c_rmgcos(α) − mg sin(α)= fv (3.3) dTt

ds = 1

v = fT (3.4)

Pontryagin’s minimum princeple is used in order to find the optimal input torque T_e^∗(s). The Hamiltonian can be stated as follows:

H= Te+ λvfv+ λTfT (3.5)

The adjoint state variables, λv and λT are calculated according to (3.6).

dλ

ds = −∇xH(x(s), Te(s), λ(s), s) (3.6)

dλ_v

ds = λ_v m_tv²

iti_fηtη_f

r_w T_e− c_rmgcos(α) − mg sin(α)+1

2c_DA_aρ_a+λ_T

v² (3.7) dλ_T

ds = 0 (3.8)

The optimal input Te^∗(s) minimizes the Hamiltonian:

T_e^∗(s) = argmin

Te

{H(x(s), Te(s), λ(s))} (3.9)

It can be seen that the Hamiltonian is linear in Te, which means that the optimal control will consist of sections of minimum engine torque, maximum engine torque and sections where

dH

dTe = 0. The latter sections are called singular arcs.

dH

dTe = i_ti_fη_tη_fλ_v

mtrwv + 1 = 0 (3.10)

During singular arcs λv is given by λv = −_i_t^r_i^w_f_η^m_t_η^t_fv and it must hold that _ds^d(_dT^dH_e) = 0.

d ds

dH dT_e

= d ds

iti_fηtη_f m_tr_w

λv

v

(3.11)

= i_ti_fη_tη_f m_tr_w

1 v

dλ_v ds −λ_v

v²f_v

(3.12)

= iti_fηtη_f m_tr_w

1

m_tc_dA_aρ_aλ_vv+ λTv³

= 0 (3.13)

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CHAPTER 3. LOOK AHEAD CRUISE CONTROL FOR A SINGLE HDV

By solving (3.10) for λv and inserting into (3.13), the following equation must apply for singular arcs:

1 m_t

c_dA_aρ_a+i_ti_fη_tη_f r_w λ_Tv³

= 0 (3.14)

Due to the fact that λT is a constant, the velocity must be constant during singular arcs. This means that the system is stationary. The optimal input during singular arcs can therefore be defined as the torque which keeps the velocity constant.

The optimal control input Te^∗(s) will therefore consist of the following input torques:

T_e^∗(s) = [Tmin, T_stat, T_max] with Tstat= Te s.t. dv

ds = 0 (3.15)

This result simplifies the complexity of finding the optimal control input for a given road profile. As described in [13], the optimal control solution will consist of the following pattern:

• Constant engine torque Te= Tstat for flat road and small gradients

• Maximum engine torque Te= Tmax in and in the neighborhood of steep uphill slopes.

• Minimum engine torque Te = Tmin in and in the neighborhood of steep downhill slopes.

The problem of finding an optimal speed trajectory was therefore reduced to a parametric optimization problem of finding the positions for switching between optimal control inputs [12]. This result is of particular significance since it highly reduces the complexity of the problem.

3.2 Intuitive explanation of optimal control solution

The above stated conclusions about fuel optimal control can also be obtained intuitively. The energy losses of a vehicle consist to a major part of losses due to the air drag force. As it will be shown in Chapter 4, only the energy losses due to the air drag and the brake usage can be controlled. All other energy losses are independent of the driven velocity profile. Due to the fact that the air drag force is proportional to the square of the velocity Fairdrag∝ v², the strategy of maintaining a constant velocity on a road with small gradients is apparent.

In case of a road profile with steep slopes, the argumentation for full and zero fueling can be comprehended by the illustration shown in Figure 3.1. The green to red lines show the range of fuel cutoff to maximum fueling. If maximum fueling is applied during the uphill, the deceleration of the vehicle is much lower compared to the situation where minimum fueling is applied. Therefore, the distance before and after the uphill where maximum fueling has to be applied is longer. In total, the variation of the velocity is higher and therefore also the aerodynamic drag energy. Similar, the downhill velocity profile can be analyzed. When minimum fueling is applied during the downhill, the variation of the velocity is lower and therefore also the aerodynamic drag energy.

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The objective of finding the optimal control solution for an uphill(downhill) as shown in Figure 3.1 is reduced to the question of finding the position in distance where the torque is increased(decreased) to maximum(minimum) engine torque.

Figure 3.1. The effect of fuel minimum fueling (green line) to maximum fueling (red line) during a steep uphill or downhill. The variation of the velocity profile is minimized by applying maximum fueling during the uphill and minimum fueling during the downhill. During the black velocity segments, the fueling is maximized on the uphill and minimized on the downhill.

3.3 Velocity limits

Optimal control of a vehicle on a track with velocity limits was addressed in [13]. The velocity limit constraints cause the optimal control problem to be more complicated. The solution is described in the following way:

The start of the deceleration on a downhill should be selected so, that the upper speed limit is reached exactly at the end of the slope. This will minimize brake usage and therefore minimize the total energy consumption. The same reasoning can be done for uphill slopes.

The upper speed limit is exactly reached at the beginning of an uphill section.

It can be seen in the next section about parametric optimization (3.4), that exactly the described solution will be found by optimizing the parameters. Therefore, no special atten- tion has to be set to speed limitations in terms of finding the solution.

3.4 Parametric optimization for finding energy optimal velocity profile

As shown in the previous sections, the problem of finding the optimal control solution for the engine torque in order to minimize the fuel consumption can be reduced to a parametric optimization. The switching points between the three possible optimal control inputs have to be found.

For interconnected road sections of steep gradients without any small gradients in between, this parameter optimization is reduced to two parameters. Figure 3.2 shows a downhill profile with an upper velocity limit of 90km/h. A start parameter ds defines the distance to the slope in which the engine torque is reduced to the minimum. During the downhill, the only

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possible input is minimum engine torque due to the fact that no input torque can maintain the velocity of the vehicle. After the slope, a final distance parameter df is defined which is selected such that the average speed of the vehicle (constant travel time Tt) is maintained with the selected ds. Therefore, ds is used in order to find the minimum Energy.

0 500 1000 1500

−20

−15

−10

−5 0 5

Distance [m]

Altitude[m]

0 500 1000 1500

70 75 80 85 90

Distance [m]

v[km/h]

0 500 1000 1500

0 500 1000 1500 2000

Distance [m]

EngineTorque[Nm]

d_s d_f

Figure 3.2. Parameter optimization for finding the optimal velocity profile (blue) for a 40t HDV on a downhill section of -3% over 500m. An upper velocity limit of 90km/h was assumed. The parameters dsand df were found, which minimize the energy consumption within the constraint of a constant travel time. The gray lines show non-optimal profiles with shorter and longer start distance ds. For the sake of clarity, no lower speed limit was defined.

The energy consumption for different values of the start distance ds is plotted in Figure 3.3 for the above shown situation. For every possible start distance, df was selected such that the average velocity remains. The start distance ds was found by the golden section search method [8].

It can be seen that the energy consumption of the vehicle which is proportional to the integration of the engine torque over the distance, finds a minimum at a start distance ds

of 270.5m. The HDV runs into the upper speed limit before the end of the slope if ds is selected shorter. The fact that the vehicle has to brake in this case causes a steep increase of the energy consumption for shorter start distances. If the start distance is selected longer, the final distance during which the velocity is held at the upper speed limit increases. This occurs due to the fact that the average velocity has to be maintained. Therefore, the variance of the velocity profile increases and the energy consumption rises.

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240 260 280 300 320 340 4.44

4.46 4.48 4.5 4.52

x 10⁶

Start distance ds [m]

Energy [J]

Figure 3.3. Energy consumption depending on the start distance at which the vehicle starts to decelerate. Test conditions were a 500m/-3% downhill and a 40t HDV. An upper velocity limit of 90km/h was assumed. The high increase in energy consumption for short start distances is caused by brake usage during the downhill.

3.5 Requirements for a look ahead controller for a vehicle in a platoon

A look ahead cruise controller for a platoon acts as one controller in a hierarchy of controllers for global path planing down to the control of a single vehicle. Therefore, additional basic requirements for an LAC used in platoons arise. In contrast to previously done work about look ahead systems, especially [16], a platoon LAC needs to fulfill the requirement of a constant average velocity as soon as possible after a section with steep gradients. This is important since globally acting controllers rely on the fulfillment of a timeline of one platoon.

In addition, a platoon which was split for any reason will be maintained again at the end of a slope if every vehicle fulfills this requirement.

3.6 Cruise Controller as a reference

In order to assess the performance of an LAC and later on the performance of a look ahead controller for platoons, the cruise controller CC for a single HDV is used as a reference. The objective of a CC is to hold the velocity to the set speed. On level road and small gradients, the CC adjusts the engine torque in order to hold a constant velocity. On steep gradients, a CC is not able to hold a constant velocity within the control input and the vehicle starts to accelerate or decelerate.

A real cruise controller of an HDV does not fulfill the requirement of a constant travel time when it comes to steep gradients in the road profile. On a downhill section, the vehicle gains speed so that the average velocity will increase and vice versa on an uphill section.

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In order to compare the performance of a new controller, a non-look-ahead controller, which maintains an average velocity, was desired. Therefore, the cruise controller was extended with a compensation of the average velocity. Hereinafter, whenever a CC is used, it compensates the average speed so that it can be compared to a new controller which holds an average velocity. Figure 3.4 shows the resulting velocity profile of a CC acting on a vehicle traversing a downhill section. At a distance of 500m, the CC is not able to hold the velocity anymore and reduces the engine torque to the minimum. Nevertheless, the velocity increases during the downhill. At a distance of approximately 1500m, the velocity reaches the set speed.

Due to the fact that the average velocity was too high up to this point, the CC starts do decelerate in order to fulfill the requirement of a constant average velocity as soon as possible.

On an uphill road profile, the CC would on contrast to the downhill profile accelerate after the slope in order to fulfill the requirement of a constant average velocity as soon as possible.

0 500 1000 1500 2000 2500 3000

−20

−15

−10

−5 0 5

Distance [m]

Altitude [m]

0 500 1000 1500 2000 2500 3000

75 80 85 90 95

Distance [m]

v [km/h]

0 500 1000 1500 2000 2500 3000

0 500 1000 1500 2000

Distance [m]

Engine Torque [Nm]

Figure 3.4. 40t HDV with cruise controller on a 500m/-3% downhill section. After a distance of 1500m, the controller compensates for the missed average velocity.

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3.7 Simulation results of LAC vs. CC and discussion

Simulations with an LAC and a CC with post compensation for the missed average velocity were performed with all presented road profile use cases (section 2.3.3). The simulations were performed without speed limits and with an upper speed limit of 90km/h. The results are shown in table 3.1 and 3.2.

Use Case: 1 2 3 4 5 6 7

Energy Saving with −0.48% −0.80% −0.54% −0.87% −0.23% −0.46% −0.94%

LAC comp. to CC

Table 3.1. Energy saving with LAC compared to CC for all road profile use cases. All simula- tions were performed with a 40t HDV. Detailed results can be found in Appendix A.4.1.

Use Case: 1 2 3 4 5 6 7

Energy Saving with −15.9% −0.43% −4.03% −5.49% −9.99% −13.8% −11.6%

LAC comp. to CC

Table 3.2. Energy saving with LAC compared to CC for all road profile use cases. All simula- tions were performed with a 40t HDV and an upper velocity limit of 90km/h. Detailed results can be found in Appendix A.4.2.

It can be seen that a look ahead controller saves up to ∼ 1% energy if no velocity limits are considered. As soon as the upper speed limit is introduced, the savings reach up to ∼ 16%.

This result shows that a look ahead system is of particular advantage when speed limits force vehicles to brake.

The above mentioned results should for the moment serve to familiarize with the dimension of the energy saving potential. In Chapter 4, a detailed analysis of the energy saving possibilities will be given.

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Chapter 4

Energy saving comparison and dimension estimation

The driven velocity profile of a single HDV can be optimized in order to minimize the energy consumption as shown in the last Chapter. The resulting velocity profile depends on various parameters of the vehicle. Especially the mass of a vehicle has a major influence on the driven optimal velocity profile.

In a platoon, several vehicles of different masses travel together in order to save fuel due to the reduced aerodynamic losses. On a road with small gradients where every vehicle is able to maintain a constant velocity, the optimal solution in terms of energy consumption is obvious. By driving a constant velocity and minimizing the distance between the vehicles to the minimum allowed distance [4], this optimal solution is implemented. The air drag force acting on every vehicle will be minimal and will stay constant. Driving a constant velocity is the optimal solution for every vehicle as shown in the previous Chapter. However, when steep road gradients are considered, this solution does not hold anymore. The individual optimal velocity profiles for every vehicle diviate from each other, especially due to the mass differences of the vehicles. And if the vehicles drive different velocity profiles, the distance between each other will vary. Therefore, the air drag force will be higher than the minimum air drag force with minimum distance between the vehicles.

Finding an optimal control solution for a platoon driving on a road with steep gradients which minimizes the total energy consumption of all vehicles is of much higher complexity than the previously shown optimal control solution for a single HDV. The results for the optimal control input found for a single HDV (3.15) do not apply anymore to vehicles traveling in a platoon. This makes the finding of an optimal control solution through parametric optimization computationally extremely intensive due to the increased number of parameters.

Using dynamic programming is conceivable, but of high computational complexity.

The aim of this thesis was to find a controller which is able to drive a vehicle platoon over a given road profile in a fuel-efficient way. This Chapter shows a new approach in order to identify cause and effect of energy flows in a vehicle traveling in a platoon. It sets the premise for the energy saving potential.

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4.1 Fuel and energy consumption

The fuel consumption is a linear function of the generated engine torque Te as shown in Sec- tion 2.2.2 and with assumed constant gear it = const also linearly depending on the engine force Fengine.

In order to calculate the consumed fuel, the fueling γ is integrated over the driven distance s_f:

F =^Z ^s^f

0

itγds⁰ (4.1)

=^Z ^s^f

0

i_tk_TT_e+ itc_Tds⁰ (4.2)

= itkT

Z sf

0

Teds⁰+ itcTsf (4.3)

(4.4) The minimization objective of the fuel can be stated as a minimization of the integration of the engine torque. Due to the proportionality between engine torque and engine force with assumed constant gear, it can also be seen as the minimization of the generated mechanical engine energy. Common constraint is a constant travel time Tt.

minTe (F) ⇔ min

Te

Z _s_f

0

T_eds⁰ ⇔ min

Fengine

Z _s_f

0

F_engineds⁰= min

Fengine

(Eengine) (4.5) s.t. ^Z ^s^f

0

1

vds⁰ = Tt (4.6)

For all integrations, the infinitesimal distances as shown in Figure 4.1 will be used. A detailed description of this forces can be found in Section 2.1.

Figure 4.1. Forces and infinitesimal distance elements

In order to compare the energy consumption of HDV’s driving different velocity profiles on the same topographic street profile, the general road profile shown in Figure 4.2 is used. The track has horizontal length ∆s and road profile length ∆s⁰. The road angle α is assumed to

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CHAPTER 4. ENERGY SAVING COMPARISON AND DIMENSION ESTIMATION

be known for the whole road profile. Therefore, the over all height difference between the start point and the endpoint ∆h is also known. In addition, the initial velocity vi and the final velocity vf are assumed to be known.

Figure 4.2. General road profile

The energy consumption of a vehicle driving over the shown road profile can be calculated by the integration of its engine force over the driven distance as shown in (4.7). The integration can be separated into an integration over every acting force. This corresponds then to the aerodynamic drag energy, the roll resistance energy, the gravitational energy (or potential energy), the acceleration energy (or kinetic energy) and the brake energy as shown in (4.10).

E_engine=^Z F_engineds⁰ (4.7)

=^Z F_airdrag+ Froll+ Fgravity+ Facceleration+ Fbrakeds⁰ (4.8)

=^Z F_airdragds⁰+^Z F_rollds⁰+^Z F_gravityds⁰+^Z Faccelerationds⁰+^Z F_brakeds⁰ (4.9)

= Eairdrag+ Eroll+ Egravity+ Eacceleration+ Ebrake (4.10) In order to analyze the influence of the driven velocity profile to the energy consumption of the vehicle, the following properties of the driven velocity will be used, here shown with respect to the time:

Empirical mean value:

The mean value is the measure of the average. With respect to the time, the mean value of the velocity represents the average velocity.

meant(v) = R vds

R dt = 1 T

Z

vdt = vt (4.11)

Empirical variance:

The variance of the velocity represents the average squared distance between the velocity and the average velocity. It provides a measurement of the variation of the velocity profile.

vart(v) = meant

[v − vt]² (4.12)

Empirical skewness:

The skewness is a measure of the lack of symmetry of the velocity profile around the average

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velocity.

skewt(v) = meant(v³) − 3vart(v)vt− v_t³

vart(v)³² (4.13)

The energies mentioned in (4.10) can be calculated knowing the properties of the road profile shown in Figure 4.2 and the empirical properties of the driven velocity profile. This is done in the following sections.

4.1.1 Roll Resistance Energy

E_roll=^Z F_rollds⁰ (4.14)

=^Z Froll 1

cos(α)ds (4.15)

=^Z c_rmgcos(α) 1

cos(α)ds (4.16)

=^Z c_rmgds (4.17)

= crmg∆s (4.18)

The roll resistance energy does not depend on the driven velocity over the track. It is proportional to the horizontal length of the track ∆s and to the mass of the vehicle m.

4.1.2 Potential Energy

E_gravity =^Z F_gravityds⁰ (4.19)

=^Z F_gravity 1

cos(α)ds (4.20)

=^Z mgsin(α) 1

cos(α)ds (4.21)

=^Z mg dh (4.22)

= mg∆h (4.23)

The gravitational energy is independent of the driven velocity profile. It is proportional to the hight difference between the start and the end point ∆h and to the mass of the vehicle m.

through Road Topography Preview Information

Fuel-Efficient Platooning of Heavy Duty Vehicles through Road Topography Preview Information

LUKAS B¨ UHLER

Master’s Degree Project

Stockholm, Sweden August 2013

through Road Topography Preview Information

Abstract

Preface

Contents

Acronyms and Abbreviations

Chapter 1

Introduction

1.1 Problem Formulation

N-1 2

N ... 1

α

1.2 Thesis Outline

Chapter 2

Model and Definitions

2.1 Equation of motion

2.2 Engine Model

2.3 Definitions and Assumptions

Chapter 3

Look Ahead Cruise Control for a single HDV

3.1 Optimal control

3.2 Intuitive explanation of optimal control solution

3.3 Velocity limits

3.4 Parametric optimization for finding energy optimal velocity profile

3.5 Requirements for a look ahead controller for a vehicle in a platoon

3.6 Cruise Controller as a reference

3.7 Simulation results of LAC vs. CC and discussion

Chapter 4

Energy saving comparison and dimension estimation

4.1 Fuel and energy consumption