Fuel-Efficient Platooning Using Road Grade Preview Information

(1)

UPTEC F 15 070

Examensarbete 30 hp December 2015

Fuel-Efficient Platooning Using Road Grade Preview

Sami Freiwat

Lukas Öhlund

(2)

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Fuel-Efficient Platooning Using Road Grade Preview

Sami Freiwat, Lukas Öhlund

Platooning is an interesting area which involve the possibility of decreasing the fuel consumption of heavy-duty vehicles. By reducing the inter-vehicle spacing in the platoon we can reduce air drag, which in turn reduces fuel consumption. Two fuel-efficient model predictive controllers for HDVs in a platoon has been formulated in this master thesis, both utilizing road grade preview information. The first

controller is based on linear programming (LP) algorithms and the second on quadratic programming (QP). These two platooning controllers are compared with each other and with generic controllers from Scania. The LP controller proved to be more fuel-efficient than the QP controller, the Scania controllers are however more fuel-efficient than the LP controller.

(3)

Popul¨arvetenskaplig sammanfattning

Dagens v¨agtransporter idag best˚ar till stor del av tunga fordon.

Majoriteten av dessa fordon kör l˚anga sträckor och förbrukar mycket bränsle som i sin tur bidrar till bl.a. ökande halter av växthusgaser och trafikstockningar. Dessutom är bränsle˚atg˚angen en viktig ekonomisk aspekt för ˚akerier.

Fordonst˚ag eller platooning som det även kallas, är en idé där man till˚ater flera tunga fordon färdas i led med korta relativa avst˚and.

M˚alet med detta är att minimera luftmotst˚andet för att i sin tur minimera bränsleförbrukningen. I examensarbetet har tv˚a bränsleeffek- tiva regulatorer tagits fram och blivit utvärderade. Dessa regulatorer bygger p˚a regulatorstrategin model predictive control som försöker op- timera en kostfunktion över en viss horisont, givet en model för systemet som ska styras. Detta innebär att en model för systemet m˚aste utformas, samt att en matematisk kostfunktion som regulatorn ska minimera m˚aste bestämmas. Regulatorerna använder dessutom information om kommande väglutning i optimeringen för att göra den smartare. De tv˚a regulatorerna som har utformats bygger p˚a tv˚a typer av algoritmer för att beräkna den optimala körprofilen: linjär programmering (LP) och kvadratisk programmering (QP). Skillnaden mellan dessa är formatet p˚a kostfunktionen, linjär programmering vill ha kostfunktionen p˚a linjär form och kvadratisk programmering vill ha den p˚a kvadratisk form. Kostfunktionens format och innebörd är det som avgör hur regulator kommer resonera. I detta examensarbete har den linjära kostfunktionen definierats som den mängden bränsle som kommer förbrukas under en viss tid framöver, och den kvadratiska kostfunktionen har definierats som ett m˚att p˚a avvikelsen fr˚an en önskad snitthastighet under en viss tid framöver. Dessa tv˚a kostfunktioner har visat sig ge ganska olika beteenden, trots att man tidigare trott att de skulle bete sig snarlikt.

Det kan verka självklart att den kostfunktion som beskriver förbrukat bränsle borde vara mest bränsleeffektiv, men de modeller som regulatorerna bygger p˚a är endast approximationer av verkligheten. S˚a vad regulatorn önskar och vad den f˚ar i form av systembeteende överenstämmer inte alltid.

De tv˚a kostfunktioner har jämförs mot varandra, samt mot regulatorer som Scania har utformat sedan tidigare, med hjälp av simuleringar. Simuleringarna har genomförts med en analytisk fordonsmodell och en mer avancerad fordonsmodell framtagen av Scania, den sistnämnda inkluderar en mängd olinjäriteter i systemet som der ana- lytiska modellen inte tar hänsyn till. Regulatorerna utvärderas i dessa simuleringar genom att mäta ˚atg˚angen bränsle över en viss sträcka med olika fordont˚agskonfigurationer. Resultaten fr˚an simuleringarna visar att LP regulatorn är mer bränsleeffektiv än QP regulatorn, men att Scania regulatorerna är mer bränsleeffektiva än LP regulatorn. Resul- taten visar även att de tv˚a regulatorerna som utformats är känsliga för modellfel vilket resulterar i att regulatorerna blir mindre bränsle- effektiva.

(4)

1 Introduction

Platooning is the concept of driving multiple vehicles with short inter-vehicle distances. It has gained interest in the last couple of years. Volvo has presented a concept where cars can enter a platoon led by an heavy duty vehicle (HDV) [13]. DAF and TNO (Netherlands Organisation for Applied Scien- tific Research) has presented a solution where two HDVs form a platoon, called EcoTwin [2], which they state will yield fuel savings of 10 %. Other HDV-manufacturers such as Mercedes have stated that they will have platooning capability on their HDVs within 10 years [1].

One of the reasons for this interest in platooning is the possibility of a significant reduction in fuel consumption. A study has shown that the air resistance is reduced by up to 50 percent for HDVs traveling in a platoon [15] and fuel savings of up to 10 % have been observed [14], which is an enormous saving.

Although fuel consumption is the main motivation for realizing platooning, platooning also enables better utilization of the road infrastructure, allowing for an increase of the capacity of the current highways. Furthermore, platooning combined with an active steering could enable the driver to do other tasks while driving [13], which could be a first step towards autonomous vehicles.

Despite this sudden interest in platooning, it is not a new idea. Already in 1939 General Motors sponsored an exhibit that depicted self-driving vehicles traveling on highways at high velocities with short inter-vehicle distances [8]. One of the difficulties with platooning is to be able to keep a short and stable distance between vehicles. This problem has been addressed by the introduction of the adaptive cruise control (ACC) which most vehicle manufacturers offers as of today. The ACC usually acts like a traditional cruise controller, which tries to maintain a constant speed, except for when there is a preceding vehicle that drives with an equal or lower speed. Then the ACC will try to keep a constant distance to the preceding vehicle. The purpose of commercial ACC:s are most often to increase comfort by enabling cruise control functionality in dense traffic.

Given that the follower vehicles will experience a reduced air drag, using an ACC with sufficiently short inter-vehicle distance should be able conserve fuel. It has however been observed that the fuel that could be saved through reduced air drag is most often lost due to brake action [4]. The brake action occurs for several reasons, in dense traffic it mostly originates from insufficient information of the traffic ahead. With knowledge of upcoming traffic disturbances a controller could cut off fuel to the engine and decelerate

(7)

essentially to convert kinetic energy to heat through friction, which will increase the required energy to transport a vehicle over a certain distance - given a required travel time.

For HDVs in a platoon however, it can be impossible to maintain a constant inter-vehicle distance due to physical limitations. For example, consider two HDVs traversing the steep uphill shown in Figure 1.1. Their power to weight ratio is such that the vehicles will not be able to maintain a constant speed during the uphill, not even at full throttle. So when the lead vehicle enters the uphill it starts to decelerate and in order to keep a constant inter-vehicle distance the follower vehicle will decelerate simultaneously, note however that the follower vehicle has not yet reached the uphill. The vehicles continue to decelerate until the lead vehicle reaches the end of the uphill. It will then accelerate to its previous set speed. Here a problem occurs, as the lead vehicle accelerates, the follower vehicle is still in the uphill and unable to accelerate. Hence it will inevitably fall behind.

Figure 1.1: A scenario where two HDVs traverse a steep uphill and the follower vehicle tries to maintain a constant distance to the lead vehicle, but fails to do so despite its best efforts. The distance increases as the speed of the follower vehicle falls below that of the lead vehicle.

(8)

Downhills are even more troublesome for HDVs, since their large mass often causes them to accelerate to the degree where they have to brake in order to avoid speeding. If a heavy HDV travels behind a lighter one downhill, the heavier one will increase in speed more, forcing it to brake in order to avoid a collision.

So in order to achieve fuel efficient platooning, a controller which utilizes both information regarding preceding vehicles and road grade preview information has to be developed, and that is the aim of this thesis.

1.1 Problem Formulation

A model predictive controller (MPC) shall be formulated and implemented, the purpose of which is to solve the optimization problem given in (1.1).

minimize Fuel consumption subject to Platoon dynamics

Road grade preview information

Constraints on input and output signals Travel time

Safety

(1.1)

In other words, the controller shall utilize road grade preview information and information regarding vehicles in the platoon, in order to reduce fuel consumption. It shall also maintain a set average speed, take into consideration limitations on input and output signals of the vehicles. Above all it shall ensure safety of all the vehicles in the platoon. Note however that surrounding traffic will not be considered, and that information regarding other vehicles in the platoon shall be acquirable both with and without vehicle-to-vehicle communication (V2V).

In the quest in solving (1.1) numerous decisions will be considered, a selec- tion of which is listed below.

• Shall all vehicles be controlled by a single controller, or shall each vehicle decide its own action?

• How shall V2V be utilized, and how shall the absence of V2V be handled?

• What information is of interest for the vehicles in the platoon?

• How can road grade preview information be utilized by the controller?

(9)

1.2 Reading Advise

In Chapter 2 the physical model upon which the controller will be built is formulated, and Chapter 3 will define the energy expenditure as an HDV traverses a road segment.

Chapter 4 consists of a quick review of the theory of optimization and the theory behind the two cost functions presented in this thesis. Some general theory on MPC and what it means is also presented.

In Chapter 5 and 6 the controller structure is presented. The main contri- butions from this thesis lies in these chapters.

Chapter 7 presents the main results from implementation of the controller and these are discussed in Chapter 8.

First time readers are encouraged to skim through Chapters 2-4 and refer to them when needed.

1.3 Related Work

In [3] [2006] Anders Fr¨oberg et al. formulates an optimization problem for minimizing fuel consumption of a single HDV. Results obtained of optimal speed profiles for a few constructed test roads are found. In [10] [2010]

Erik Hellstr¨om formulates a truck model with a detailed empirical model describing the fuel consumption. This model is then used in a dynamic programming LAC for a single HDV. In [11] [2007] Erik Holma investigates the sensitivity of LAC systems and what requirements on input data to the system should be fulfilled.

In [4] [2014] Assad Alam investigates fuel-efficient platooning control and evaluates the fuel-saving potential in these applications. A dynamic programming algorithm is utilized to formulate a longitudinal controller which uses road grade preview information. Results show a fuel reduction of 3.9- 6.5% on average for a heterogenous platoon of HDVs on a Swedish highway.

In [12] [2012] Josefin Kemppainen formulates an MPC for controlling vehicles in platoons. This controller does not use road grade information, and is deduced into two types of controllers: centralized and decentralized. In [6] [2013] Lukas B¨uhler formulates a platoon LAC with low computational complexity. Analysis on when and when not a platoon should be maintained is made.

In [7] [2014] Henrik Felixson formulates an algorithm to estimate properties of a preceding vehicle, called Target Ahead Property Estimation (TAPE).

(10)

In [14] [2015] Valerio Turri formulates a two layer platoon controller, consist- ing of a platoon coordinator controller and a vehicle controller. The platoon controller is based on dynamic programming and calculates fuel optimal velocity profiles for all the vehicles in the platoon, the cost function used describes the fuel consumption for all the vehicles in the platoon. The vehicle controllers were linear MPCs and their purpose was to steer the vehicle according to the given velocity profile.

(11)

2 Physical Model

In order to describe the physics of a system, such as HDVs in a platoon, a detailed mathematical model is needed. The model is only an approximation of the real system but is sufficient for its use in the intended model based controller, described in later sections. First a non-linear model for a single HDV is introduced which is then linearized and discretized. The model is then extended to describe several HDV’s in a platoon where the reduced air drag between the vehicles is taken into consideration.

2.1 Nonlinear Vehicle Model

The intended controller only requires a vehicle model in the longitudinal direction. The aim of this section is to model the forces acting on an HDV in the longitudinal direction. Any notation about the vehicle index in the platoon is skipped, this section is only for a single HDV. The forces that are considered are illustrated in Figure 2.1

Figure 2.1: Illustration of the forces acting on a HDV in longitudinal direction.

where Feis the resulting force on the propelling wheels from the powertrain, F_b is the brake force from the brake system, F_r is the roll resistance, F_a is the air drag and Fg is the gravitational force.

(12)

2.1.1 Powertrain

The force F_e is modeled by modeling the powertrain and how the engine torque is converted to a propulsion force through the clutch, gearbox, final drive and wheels.

Engine

Internal dynamics such as friction, pumping losses, etc. are not considered when modeling the engine. Through combustion of diesel the engine pro- duces torque which is transferred through the engine flywheel. This torque transfer can be described by Newton’s second law

(2.1) Jeω˙e= Te(ωe, δ) − Tc

where T_e(ω_e, δ) is the produced engine torque, T_c is the external load from the clutch, J_e is the moment of inertia of the engine flywheel and ω_e is the angular speed of the flywheel. By considering negligible losses between the engine and the clutch the angular speed of the clutch is modeled as:

ω_c= ω_e (2.2)

where ωc is the angular speed of the clutch.

Clutch

The clutch is modeled as a frictional clutch with negligible losses at the connection point with the gearbox when the clutch is engaged. We can model the clutch dynamics as

TG = Tc (2.3)

ωG = ωc (2.4)

where TG is the output torque from the gearbox, Tc is the output torque from the clutch and ω_G is the angular speed of the gearbox.

Gearbox

The gearbox consists of several gears, each with their own conversion ratio γG which transforms the input torque from the clutch. We assume that the gearshifts are instantaneous. There are always frictional losses when changing gears and this is modeled as an efficiency η_G for each gear. We can model the gearbox dynamics as:

T_f = γGηGTG (2.5)

γGωf = ωG (2.6)

where T is the output torque from the gearbox and ω is the angular speed

(13)

Final drive

The final drive is modeled similarly as the gearbox. The frictional losses is assumed to be negligible at the connection between the gearbox and final drive. It has an conversion ratio γ_f and an efficiency η_f. We can model the final drive dynamics as:

T_w = γ_fη_fT_f (2.7)

γ_fω_w = ω_f (2.8)

where Tw is the output torque from the final drive and ωw is the output angular speed.

Wheels

The wheels are the last part of the powertrain that transfers the engine torque as a force on the road. We assume that the wheel speed is the same for all tires and that there is no wheel slip. We can model the wheel dynamics as:

Jwω˙w= Tw− r_wFe

where r_w is the wheel radius, J_w is the moment of inertia of the wheels and Fe is the resulting propulsion force.

Powertrain Model

By combining (2.1)-(2.9), the vehicle speed v as a function of the angular speed ωe is:

v = r_wω_w = rw

γ_Gγ_fω_e (2.9)

and by adding a resulting brake force Fb, the summarized model for the powertrain and brakes is

F_e+ F_b = γ_Gγ_fη_Gη_f rw

T_e(ω_e, δ) −Jw+ γ_G²γ_f²ηGηfJe

r²_w ˙v

2.1.2 Air Drag

Since the HDVs has a very large front area and therefore disadvantageous aerodynamic properties, the air drag has a strong impact on the total forces acting on the HDVs in longitudinal direction. The air drag for a HDV can be decreased significantly by driving behind another HDV or even several of

(14)

these. Empirical results by [15] have shown that for two van-shaped vehicles the air drag coefficient decreases with increasing order in the platoon. To model this decrease in air resistance empirical results have been derived through measurements in a wind tunnel with commercial buses. This is illustrated in Figure 2.2.

0 10 20 30 40 50 60 70

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Φ(d)

Relative distance [m]

HDV1

HDV2

HDV3

Figure 2.2: Empirical results for the air drag ratio Φ as a function of the relative distance between HDV’s in a platoon.

The figure shows that the air drag ratio Φ decreases with decreasing relative distances. This decrease is greater for vehicles further back in a platoon.

By modeling the air drag for a single HDV and combining this with the empirical results for Φ we obtain:

F_a= 1

2c_DA_aρ_aΦ(d)v²

where cD is the air drag coefficient, Aa is the front area of the vehicle and ρ_a is the air density.

Since the intended controller requires a linear model we chose to approximate the data in Figure (2.2)

Φ(d) = a_lind + b_lin and d≥0

where alin and blin are the coefficients for the linear function.

2.1.3 Rolling Resistance

The roll resistance consists primarily of the non-elastic deformation of the

(15)

surface. This resistance can be modeled in many different way. However, to keep it simple the model is as follows:

Fr= cRmgcos(α)

where c_R is the roll resistance coefficient, m is the total mass vehicle mass, g is the gravitational constant and α is the road grade.

2.1.4 Gravitational Force

HDVs usually weigh many tons, the gravitational forces have a big impact on the longitudinal forces when driving on an uphill or a downhill segment.

By simple mechanics this is modeled as:

Fg = mgsin(α)

2.1.5 Vehicle Model

By combining all of the internal and external forces for a HDV in the longitudinal direction a non-linear model for the vehicle is derived:

m_tdv

dt = F_e− F_b− F_a(v, d) − F_r(α(s)) − F_g(α(s))

= γGγfηGηf

r_w T (ωe, δ) − F_b− 1

2c_DAaρaΦ(d)v²− c_Rmgcos(α) − mgsin(α)

= k_eT (ω_e, δ) − F_b− k_dΦ(d)v²− k_rcos(α(s)) − k_gsin(α(s)) (2.10) where

m_t= J_w

r²_w + m + γ_G²γ_f²ηGηfJe

r²_w .

2.2 Platoon Model

To model a system with several HDVs moving together in a platoon two things are needed: a vehicle model for each of the HDV and a model describing the relative distances between the vehicles in the platoon. A vehicle model is already defined from the last section so the only thing left is a model for the relative distance. This is modeled as

dd_i−1,i

dt = v_i−1− v_i (2.11)

where di−1,i is the relative distance between HDV i − 1 and i. The notation is explained in Figure 2.3.

(16)

Figure 2.3: Explanation of the notation used where vehicle i − 1 is in front of vehicle i

2.2.1 Time Versus Space

The road grade data, which is denoted as α, plays a big role in determin- ing the external forces acting on a HDV. The road grade is received as a function of position but the vehicle model is time-dependent. Thus the road grade data and the physical vehicle model are not compatible. To solve this either the road grade data must be made time dependent or the expressions for relative distance and the vehicle model must be made position dependent. Both approaches have been examined and only the latter alternative have been proven to be possible. Hence the following section will cover the procedure of translating the vehicle model and the model for relative distance.

Transforming to Position Dependence

By using the chain rule (2.11) is transformed to be position dependent ddi−1,i

dt_i = ddi−1,i

ds_i dsi

dt_i = ddi−1,i

ds_i vi = vi−1− v_i and thus we have

dd_i−1,i dsi

= vi−1− v_i vi

= G(vi−1,vi) and vi≥0 Similarly the following is transformed

dv_i dti

= dv_i dsi

ds_i dti

= dv_i dsi

v_i (2.12)

Inserting (2.12) in to (2.10) yields m_tdv_i

ds_i = k_e

v_iT_i(ω_eδ) − k_dΦ(d_i−1, i)v_i− k_r

v_icos(α_i) −k_g

v_isin(α_i) −F_b v_i or by simplifying it even further

(2.13) dvi

ds = ce

v Ti(ωeδ) − cdΦ(di)vi−cr

vcos(αi) −cg

v sin(αi) − cb

v Fb,i

(17)

The constants are given by the equations below ce = ke

m_t = r_wγ_Gγ_fη_Gη_f J_w+ mr²_w+ γ_G²γ_f²η_Gη_fJ_e c_d= k_d

m_t =

1

2r²_wc_DA_aρ_a Jw+ mr_w² + γ_G²γ_f²ηGηfJe

cr = kr

m_t = cRr²_wmg

Jw+ mr²_w+ γ²_Gγ_f²ηGηfJe

cg = kg

m_t = r_w²mg

J_w+ mr_w² + γ_G²γ_f²η_Gη_fJ_e cb = kb

mt

= r_w²

J_w+ r²_wm + γ_G²γ_f²n_Gn_fJ_e

2.2.2 Linear Platoon Model

The intended controller is based on a linear model. Since the model is non- linear it needs to be linearized first. By choosing suitable equilibrium points for equation (2.13) the model can be linearized by using a first order Taylor series. The equilibrium points are:

v₀= v_ref α₀= α_ref d0= d_ref

T0= cdΦ(d0)v₀²+ crcos(α0) + cgsin(α0) c_e

Fb,0= Fb,ref

Observe that T₀ is, by definition, the required engine torque to maintain constant speed in the equilibrium point. With (2.13) describing the non- linear vehicle dynamics, and the equilibrium points, a first order Taylor approximation is applied:

(2.14) dvi

dsi

= F0+ ∂F

∂Ti

(Ti− T₀) + ∂F

∂vi

(vi− v₀) + ∂F

∂αi

(αi− α₀) + ∂F

∂d_i(d_i− d₀) + ∂F

∂F_b,i(F_b,i− F_b,0)

where the different parts of (2.14) is described by F₀=ce

v₀T₀− c_dΦ(d₀)v₀− cr

v₀cos(α₀(s)) − cg

v₀sin(α₀(s))

(18)

=ce

v0

c_dΦ(d₀)v²₀+ c_rcos(α₀) + c_gsin(α₀) ce

− c_d(d0)v0

−c_r v0

cos(α0(s)) − c_g v0

sin(α0(s))

=c_dΦ(d₀)v₀+ c_r v0

cos(α₀(s)) + c_g v0

sin(α₀(s)) − c_d(d₀)v₀

−c_r

v₀cos(α₀(s)) − c_g

v₀sin(α₀(s)) = 0

∂F

∂Ti

= c_e v0

+ O(T_i²)

∂F

∂vi

= −c_eT₀

v²₀ − c_dΦ(d₀) + c_r

v²₀cos(α₀) + c_g

v₀²sin(α₀) + O(v²_i)

∂F

∂α_i = cr

v₀sin(α₀(s)) − cg

v₀cos(α₀(s)) + O(α²_i)

∂F

∂d_i = −v₀c_da_lin+ O(d²_i)

∂F

∂F_b,i = −c_b

v₀² + O(F_b,i² )

Note that F0 is canceled out because of the way T0 is defined by the equilibrium points. With all of the partial derivatives the linearized model can now be formulated as

(2.15) dvi

ds_i = ∂F

∂T_i(Ti− T₀) +∂F

∂v_i(vi− v₀) +∂F

∂d_i(di− d₀) + ∂F

∂αi

(α_i− α₀) + ∂F

∂Fb,i

(F_b,i− F_b,0)

Analogous with the linearized vehicle model the relative distance is linearized in the same way. The equilibrium points are defined as

v_i−1,0 = v_i−1,ref v_i,0 = v_i,ref

and by applying first order Taylor expansion on G(v_i−1, v_i) we get ddi−1,i

dsi

= G0+ ∂G

∂vi−1

(vi−1− v_i−1,0) +∂G

∂vi

(vi− v_i,0) (2.16) where the different parts of equation (2.16) is described by

G0= vi−1,0− v_i,0 v_i,0

∂G

∂v_i−1 = 1 v_i,0

∂G vi−1

(19)

2.2.3 Discrete Platoon Model

Since the intended controller requires a discrete model the non-linear vehicle model must be discretized. This is done by applying the Euler Forward Method on (2.15)

vi(k + 1) − vi(k)

T_s =∂F

∂T_i(Ti(k) − T0) +∂F

∂v_i(vi(k) − v0) +∂F

∂d_i(di(k) − d0) + ∂F

∂α_i(αi(k) − α0) + ∂F

∂F_b,i(F_b,i(k) − F_b,0) v_i(k + 1) =T_s(∂F

∂Ti

(T_i(k) − T₀) + ∂F

∂vi

(v_i(k) − v₀) + ∂F

∂di

(d_i(k) − d₀) + ∂F

∂αi

(αi(k) − α0) + ∂F

∂F_b,i(Fb,i(k) − Fb,0)) + vi(k) (2.17) and on (2.16)

di−1,i(k + 1) − di−1,i(k)

T_s = G₀+ ∂G

∂v_i−1(v_i−1(k) − v_i−1,0) +∂G

∂v_i(v_i(k) − v_i,0)

di−1,i(k+1) = Ts(G0+ ∂G

∂vi−1

(vi−1(k)−vi−1,0)+∂G

∂vi

(vi(k)−vi,0))+di−1,i(k) (2.18) where Ts is the sampling distance and k is the discrete step.

2.2.4 State Space Platoon Model

With (2.17) and (2.18), all of the required building blocks to form a state space model of a platoon system of any size are available. The equations are both linear and discrete versions of the non-linear continuous models. The complete platoon equations are:

v_i(k + 1) = T_s((−ceT0

v²₀ − c_dΦ(d₀) + cr

v²₀cos(α₀) + cg

v₀²sin(α₀))v_i,δ

− v₀c_da_lind_i,δ + c_e V0

T_i,δ+ (c_r v0

sin(α₀) −c_g v0

cos(α₀))α_i,δ

−c_d

v₀F_b,i) + v_i(k)

d_i−1,i(k + 1) = T_s vi,0

(v_i−1,0− v_i,0+ v_i−1,δ(k) − v_i−1,0 vi,0

v_i,δ(k)) + d_i−1,i(k)

(20)

where v_i,δ denotes v_i(k)−v₀and so forth for all other δ notations. To simplify (2.19) the assumption that v_i,0 = v_i−1,0 is made for all vehicles

d_i−1,i(k + 1) = T_s v0

(v_i−1,δ(k) − v_i,δ(k)) + d(k) (2.19) The state space model of the platoon, with a specific choice of state vector, becomes for three vehicles







v₁(k + 1) d1,2(k + 1)

v2(k + 1) d_2,3(k + 1)

v3(k + 1)







=







A_1,1 0 0 0 0

A2,1 A2,2 A2,3 0 0 0 A3,2 A3,3 0 0 0 0 A_4,3 A_4,4 A_4,5 0 0 0 A5,4 A5,5











 v₁(k) d1,2(k)

v2(k) d_2,3(k)

v3(k)







+







B_1,1 B_1,2 0 0 0 0

0 0 0 0 0 0

0 0 B3,3 B3,4 0 0

0 0 0 0 0 0

0 0 0 0 B_5,5 B_5,6











 T1(k) F_b,1(k)

T2(k) F_b,2(k)

T₃(k) F_b,3(k)







+







Bα1,1 0 0 0 0

0 0 0 0 0

0 0 B_α3,3 0 0

0 0 0 0 0

0 0 0 0 B_α5,5











 α1(k)

0 α₂(k)

0 α₃(k)







where the matrix elements are determined by the platoon model equations.

Additional trucks in the platoon simply adds two new states to the state vector, two new states to the input vector and one new state to the α vector.

(21)

3 Fuel Consumption and Energy Losses

The aim of this work is to reduce the fuel consumption of HDVs in platoons.

By modeling the fuel consumption in the engine and describing how it is related to forces acting in longitudinal direction, we can formalize a quantity we want to minimize.

3.1 Fuel Flow Equations

The fuel flow model used in this chapter is derived in [11] which represents the 12 liter Scania DT1211 L02 diesel combustion engine. The fuel flow mass, m_diesel, per time unit can be stated as

dm_diesel dt = f δ

where f is the frequency of diesel combustion in the engine cylinders and δ is the amount of fuel that is consumed in each combustion, which can be formulated as

f = f_crankn_cyl nr

, and fcrank = ω_e

2π (3.1)

where fcrank is the frequency of the crankshaft, ncyl is the number of cylinders and ω_e is the angular speed of the engine. The division by n_r is explained by taking into account how many rotations the crankshaft has to make to complete a full cycle, which in a four-stroke engine is equals two.

The amount of fuel δ that is consumed in each explosion is

δ = δmaxP, where δmax = aδω²_e+ bδωe+ cδ and P ∈ [0, 1] (3.2) P can be seen as the applied accelerator signal. There is also a model for the torque Te

T_e= a_eγ_Gγ_f v rw

+ δ_maxb_eP + c_e (3.3)

The expression for the fuel flow, with (3.1) and (3.2) , is now formulated as

dm_diesel

dt = vγGγfncyl(aδ(γGγf v

rw)²+ bδ(γGγf v

rw) + c_δ)

2πn_rr_w P (3.4)

Equation (3.4) is transformed to be position dependent with the chain rule

dm_diesel

dt =dm_diesel ds

ds

dt = dm_diesel ds v

(22)

which yields dmdiesel

ds =

aδ

γ³_Gγ_f³

r²_w v²+ bδ γ_G²γ_f²

rw v + cδγGγf

n_cyl 2πn_rr_wP

Since the vehicle model has the engine torque as an input signal it is more convenient for our purposes to describe the fuel flow as a function of torque instead of P . If we rewrite (3.3)

(3.5) P = T_e− a_eω_e− c_e

beδmax

and given the expression for fuel consumption we get

(3.6) dm_diesel

ds = 1 vf δ

= γGγfncyl

2πn_rb_er_w(T_e−aeγGγfv r_w − c_e)

3.2 Energy Consumers

Besides looking at the fuel consumption one could also investigate energy loss, in order to better understand where the energy is spent. The energy can be converted to a volume of diesel fuel, but this is a tedious task since the efficiency of the powertrain has to be known to convert the energy into diesel, thus we will not investigate this conversion. Instead we want an expression for the total energy loss which can then be used as a performance measurement for a given speed profile. Consider a HDV traversing a distance s, the engine must produce a propulsion force big enough to overcome forces acting in the opposite direction, forces described in section 2. By integrating the produced engine force over the traversed distance s an expression for the total energy loss is given

Eengine= Z

Feds

= Z

(Fa+ F_b+ Fr+ Fg+ Facc)ds

= E_a+ E_b+ E_r+ E_g+ E_acc

(3.7)

where ds is the infinitesimal traversed distance. Since the different forces can be separated in the integration the expressions for the energy losses can be treated alone. This is done in following sections.

(23)

3.2.1 Kinetic Energy

With Newton’s second law we formulate the kinetic energy as Eacc=

Z

Faccds

= Z

mdv dtvdt

= mv_end² − v_start² 2

(3.8)

where vstart and v_end is the initial and final speed. The kinetic energy is proportional to the vehicle mass and is the required energy for accelerating or decelerating a vehicle over the distance ∆s.

3.2.2 Potential Energy

With (3.7) the potential energy is formulated as E_g =

Z F_gds

= Z

mgsin(α)ds

= Z

mgsin(α) 1 cos(α)ds⁰

= Z

mgdh

= mg(hend− h_start)

(3.9)

where ds⁰ is the infinitesimal horizontally traversed distance. h_start and hend is the initial and final altitude. The potential energy is proportional to the vehicle mass and is the required energy for increasing or decreasing the altitude.

(24)

3.2.3 Roll Resistance Energy

With (3.7) the air roll resistance is formulated as Er=

Z Frds

= Z

crmgcos(α)ds

= Z

c_rmgcos(α) 1 cos(α)ds⁰

= Z

crmgds⁰

= c_rmg∆s⁰

(3.10)

where ∆s⁰ is the horizontally traversed distance. The roll resistance energy is proportional to the distance ∆s⁰.

3.2.4 Air Drag Energy

With (3.7) the air drag energy is formulated as Ea=

Z Fads

= 1

2c_DA_aρ_a Z

v²ds

(3.11)

As seen from the equation above the air drag energy is proportional to the square of the speed.

3.2.5 Brake Energy

Since the brake force is an input to the system it is very simple to calculate the energy losses by integration

E_b = Z

F_bds (3.12)

(25)

3.3 Total Energy Loss

By combining (3.8)-(3.12) the total energy loss E_engine is formulated as

Eengine=Ea+ E_b+ Eg+ Er+ Eacc

Eengine=mv_end² − v_start²

2 + mg(h_end− h_start) + crmg∆s⁰ +1

2c_DA_aρ_a Z

v²ds + Z

F_brakeds

(3.13)

From the results we observe that Er, Egand Eaccis independent of the speed profile of the control strategy between the start and the final point. The other energy losses, Eairdrag and Ebrake, are however losses that depends on the control strategy and are therefore subject to be minimized.

(26)

4 Optimization

Optimization aims to make things better in the sense of increasing profit by using resources more efficient, decreasing the needed amount of material for a structure through clever design or it could be decreasing air resistance by altering vehicle geometry. These are all examples of some measure being improved by changing something.

4.1 Optimization Theory

Expressed in mathematical terms their respective cost function J (u) (also known as objective-, loss- or goal function) has been improved by altering their design variables. The general definition of an optimization problem is

minimize

u J(u)

subject to g(u) = 0, h(u) ≤ 0

which means find a set of design variables u that minimizes the cost function J(u) while satisfying the constraints g(u) = 0 and h(u) ≤ 0.

Problems where optimization is usually applied are where there are conflict- ing interests, such as ensuring strength of a bridge while minimizing material cost, maximizing expected profit of a stock portfolio while minimizing risk or minimizing fuel usage while ensuring safety and arrival time of vehicles in a platoon. The cost function does not necessarily have to be a quadratic form as in equation (4.1), it can be linear or nonlinear. It does not even have to be known - the cost function could be a measurement of an unknown system.

But in this thesis we will only look at quadratic and linear cost functions with linear constraints g(u) = 0 and h(u), known as quadratic programming (QP) and linear programming (LP) problems. Both QP and LP are well researched areas and there are many fast solvers (optimization algorithms) available. We have used the Matlab function quadprog and the c-based li- brary GNU Linear Programming Kit (GLPK, which is freely available at www.gnu.org/software/glpk/ ).

A very common cost function is the sum of a quadratic function

(27)

where the weighted norm ||y||²_Q means the quadratic function

y^TQy = Q1y₁²+ . . . + Q_Ny_N² (4.2) where Q is a symmetric positive semidefinite matrix

Q =







Q₁ 0 . . . 0 0 Q₂ . . . ... ... ... . .. 0 0 . . . 0 Q_N







. (4.3)

The weighting matrix R is defined in the same way as Q. The weighting matrices Q and R in equation (4.1) makes it possible to value the two measures against each other. Higher values for Q indicate that y is more expensive in relation to u. Hence the relative size of Q and R will affect the optimal solution.

4.1.1 Optimization Algorithms

I this section we will, in a simplified manner, attempt in explaining how an optimization algorithm works.

Given an initial guess at the solution u0 the optimization algorithm will determine in which direction the cost function decreases the fastest, it wants to reach the minimum as fast as possible if a small change in u0 is made.

This is most commonly done by measuring and approximating the gradient of the cost function.

Figure 4.1: Illustration of an iteration step done by an optimization algorithm.

(28)

After the direction of the next step is known the step length is determined by some clever strategy. If the optimization problem has constraints the algorithm must also ensure that these are not violated.

Three categories of optimization problems that occur in control theory are:

• Linear programming (LP). The problem consists of a linear cost function and linear constraints. Most common solver is the Simplex method and its idea is to move between the vertices of the feasible region in the direction of of steepest decent until the minimum is reached. (BILD p˚a ett inneslutet plan!)

• Quadratic programming (QP). A quadratic cost function and linear constraints. There is no universal solver as for LP, instead there are many different strategies such as interior point-, active set- and trust region reflect methods. These methods are all slower than the Simplex method, however they are significantly faster than solvers for the next category.(bild)

• Nonlinear programming (NLP). These problems accepts cost function and constraints on virtually any form, given that a solution can be found. Some of the solvers for these problems are dynamic programming (DP), genetic algorithm and sequential quadratic programming (SQP).(bild)

The main differences between these optimization problems are the time required to solve them and their guarantees regarding the optimal solution.

While LP-solvers are guaranteed to yield the optimal solution, QP- and NLP-solvers may result in suboptimal solutions, i.e no assurance of finding the global optimum is given.

4.1.2 Global and Local Optimum

Consider the cost function illustrated in Figure 4.2. It illustrates a cost function with two minima, x1and x2. As earlier mentioned, an optimization algorithm will try to find a minimum by striving towards lower cost function values. If the optimization is initialized ”to the left” of x₁ in figure 4.2, the algorithm will most likely find the minimum x1, which in this case is the lowest value the cost function takes in the feasible region – the global minimum. However, if the optimization is initialized ”to the right” of x₂ the algorithm will most likely search for lower values until i reaches x2, where it will find a minimum and terminate. So the optimization algorithm will have found a minima, but not the lowest possible value for the cost function – a local optima.

(29)

Figure 4.2: Example of a non-convex cost function.

Since the goal of optimization is to find the optimal solution to a problem, we want to find the global optima. It has been shown that if the optimization problem is convex then if a solution is found, it is guaranteed to be the global optima [5].

Proving that a problem is convex consists of two steps. Both the feasible region for the optimization variables and the cost function must be convex.

4.1.3 Proving Convexity

Convexity is an important concept within optimization theory, and a brief introduction is given.

Let U be the feasible set of optimization variables and let x and y be two points such that x, y ∈ U . Define the line between x and y

c = (1 − α)x + αy, where α ∈ {R|0 ≤ α ≤ 1}. (4.4) Then if ∀α, c ∈ U then the set U is convex. An illustration of the proof is shown in figure 4.3

Proving that the cost function is convex is done in a similar fashion. Define the cost function J : R² → R and two points x, y ∈ U. J is then convex on

(30)

Figure 4.3: Example of a convex and a non-convex set.

U if and only if

J ((1 − α)x + αy) ≤ αJ (a) + (1 − α)J (b), where α ∈ {R|0 ≤ α ≤ 1}.

(4.5)

Figure 4.4: Illustration of the proof for convexity of a cost function. In order for the cost function to be convex, the grey line described by αJ (a) + (1 − α)J (b) should never go below the black curve J ((1 − α)a + αb.)

Note that a concave function f can be made convex through −f . Examples of convex and non-convex function can be seen in figure 4.5.

In the case of LP and QP, where the constraints are linear, the feasible set of optimization variables is convex. So only convexity of the cost function needs to be examined.

The cost function for a LP problem is linear. A linear function is always convex and thus, the LP problem has a convex cost function guarantee- ing a global minimum. In the case of a cost function representing a two- dimensional plane the argument for linear convex function still applies, so

(31)

Figure 4.5: Examples of a concave (left) and a non-convex function (right).

4.1.4 Non-Convex Problems

If the optimization problem is not convex guarantees regarding a global optimal solution can in general not be given, however there are optimization/search algorithms such as genetic algorithms and simulated annealing (with logarithmic cool down) that guarantees the global optima as the number of iterations goes to infinity.

−4

−3

−2

−1 0

1 2

3 4

−4

−3

−2

−1 0 1 2 3 4

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

Figure 4.6: Example of a non-convex function that has been sampled.

For non-convex problems one could argue that it might be possible to sample the cost function and ”see” the global optimal solution, as in figure 4.6.

While this may work in problems in one and two dimensions, higher di-

(32)

mensional problems become impossible to visualize and the computational complexity grows exponentially. If a two dimensional cost function takes 1 second to sample with 10 samples per variable, a cost function in 10 dimensions will take, with the same amount of samples per variable, approximately 3.2 years to sample. The decentralized controller in this thesis can have 200 variables, which would take 3.2 ∗ 10¹⁹⁰ years to compute through sampling.

What an optimization algorithm essentially does is walk through this enormous space towards an optima, and effectively shortening the computation time down to less than a second.

4.2 Controller Choice

With the theory of optimization algorithms briefly explained, we will now look at its use within control theory.

The purpose of a controller is to steer a system towards a desired state by controlling its input signal. The most fundamental controllers simply set the input signal to be a function of some user input (e.g. old electrical radiators) or as a function of the control error, which is defined as the difference between the desired and current output. These controllers are in most cases tuned by hand in order to achieve satisfactory results. A common controller that is used in this way is the PID controller [9], which weighs together a proportional-, integral- and derivative-component of the control error to form the input signal. Although these methods works remarkably well, it gets more difficult to tune the PID-gains as the system grows in complexity. This is where the model dependent controllers enters the scene and offers a different approach. Although there are plenty of them we will only focus on one, the Model Predictive Controller (MPC).

4.2.1 Model Predictive Control

The idea behind MPC is to control the system in such a way that some pre- determined cost function is optimized over a future horizon, given knowledge of the system. The controller algorithm can formally be described as:

1. At time t predict the future system states x(t+k|t), k = 1, 2, . . . , N as a function of the current state x(t) and future input signals u(t+i|t), i = 0, 1, . . . , M .

2. Optimize the cost function J (u) with respect to u(t+i|t), i = 0, 1, . . . , M 3. Use u(t|t) from the optimizing set of input signals as input to the

system.

(33)

A more informal way of describing it would be something like this: We know in what state our system is in right now (e.g. the current vehicle speed, distance to the vehicle in front of us, etc) and we know how the input signals will affect its state (e.g. the throttle will typically increase the vehicle speed and braking will decrease it), we can even account for future events that we know or believe will affect our system. Given this information we have information about how to make our system behave the way we want. We then let an optimization algorithm calculate the optimal input signals for the whole future horizon, but we only feed the first input signal to the system and wait for the next system tick. Then the procedure is repeated. The MPC is basically a repetitive real-time implementation of an optimization algorithm. See figure 4.7 for a graphical explanation.

Figure 4.7: Illustration of a step done by an optimization algorithm.

Note that there is virtually no limits on what behavior we could demand from our system. We could tell it to keep one or several of our states as close to some predefined values as possible, generally known as the servo problem [9]. But we could also tell the system to behave in such a way that some measure (e.g. travel time, wear on the vehicle, travel distance, consumed fuel, etc) is minimized or maximized. This is done through step 2 in the algorithm, by choosing a cost function J that describes what the sought after behavior from the system is.

In many ways the MPC resembles the way a human reasons and compromises when faced with a problem. We usually take in several factors which we weigh against each other and find a compromise that conforms with our

(34)

preferences at the moment and satisfy any constraints we may have.

4.3 Cost Functions

The purpose of the controller was to minimize fuel consumption while ensuring safety of the vehicles in the platoon.

In this thesis two main approaches have been examined and evaluated, and the most important results will be presented in the next few sections.

4.3.1 Minimizing State Deviations - QP

A commonly used cost function for an MPC is a simple quadratic function of the states and input signals:

J (k) = ¯x^TQ¯x + ¯u^TR¯u (4.6)

where ¯x and ¯u are vectors containing the system states and input signals for the whole future horizon, Q and R are the positive semi definite weight matrices. When using this text book MPC-approach, the weight matrices are used to tune the controller into desirable behavior. This cost function will be referred to as the QP-cost function for the remainder of the report.

Lets say the states are vehicle speed x₁ and distance to the preceding vehicle x₂, and the input signals are torque from the engine u₁and brake force u₂. If the horizon length is N the complete state and input vectors become

¯ x =







x₁(1) x2(1) x₁(2) x₂(2)

... x₁(N ) x2(N )







, and ¯u =







u₁(1) u2(1) u₁(2) u₂(2)

... u₁(N ) u2(N )







and the weight matrices from (4.6) become

Fuel-Efficient Platooning Using Road Grade Preview Information

Examensarbete 30 hp December 2015

Fuel-Efficient Platooning Using Road Grade Preview

Sami Freiwat

Lukas Öhlund

Abstract

Fuel-Efficient Platooning Using Road Grade Preview

Contents

1 Introduction

2 Physical Model

3 Fuel Consumption and Energy Losses

4 Optimization