Analytical model of a vehicle platoon

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Analytical Model of a Vehicle Platoon

Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet

av

Simon Eiderbrant LiTH-ISY-EX--13/4710--SE

Linköping 2013

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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Analytical Model of a Vehicle Platoon

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan vid Linköpings universitet

av

Simon Eiderbrant LiTH-ISY-EX--13/4710--SE

Handledare: Niclas Evestedt

isy_{, Linköpings universitet}

Kuo-Yun Liang

Scania CV AB

Assad Alam

Scania CV AB

Examinator: Daniel Axehill

isy, Linköpings universitet

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Avdelning, Institution Division, Department

Division of Automatic Control Department of Electrical Engineering SE-581 83 Linköping Datum Date 2013-06-11 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.ep.liu.se

ISBN — ISRN

LiTH-ISY-EX--13/4710--SE

Serietitel och serienummer Title of series, numbering

ISSN —

Titel Title

Analytisk modell av fordonståg Analytical Model of a Vehicle Platoon

Författare Author

Simon Eiderbrant

Sammanfattning Abstract

Platooning is a way of reducing traffic flow, road congestion and, above all, fuel consumption by maintaining a short intermediate distance between heavy duty vehicles (HDVs). This is accomplished by every vehicle controlling the distance to the vehicle ahead electronically with the help of vehicle to vehicle (V2V) communication.

When a HDV platoon is travelling from a starting point to a destination, there might be a number of different possible routes. To decide which route is more fuel efficient, simulations of velocities and fuel consumption are usually done. Simulating the velocity trajectories of a platoon with many vehicles can be quite computationally intense. This thesis develops a point-mass model for an HDV platoon to reduce the computation time of these simulations. The simulation time for the point-mass model is in the order of O(N) times faster than the original simulations, where N is the number of vehicles in the platoon. When choosing the most fuel efficient route, the best route can be guaranteed in some cases, due to calculated lower and upper bounds on the fuel consumption.

Nyckelord

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Abstract

Platooning is a way of reducing traffic flow, road congestion and, above all, fuel consumption by maintaining a short intermediate distance between heavy duty vehicles (HDVs). This is accomplished by every vehicle controlling the distance to the vehicle ahead electronically with the help of vehicle to vehicle (V2V) com-munication.

When a HDV platoon is travelling from a starting point to a destination, there might be a number of different possible routes. To decide which route is more fuel efficient, simulations of velocities and fuel consumption are usually done. Simulating the velocity trajectories of a platoon with many vehicles can be quite computationally intense. This thesis develops a point-mass model for an HDV platoon to reduce the computation time of these simulations.

The simulation time for the point-mass model is in the order of O(N) times faster than the original simulations, where N is the number of vehicles in the platoon. When choosing the most fuel efficient route, the best route can be guaranteed in some cases, due to calculated lower and upper bounds on the fuel consumption.

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Acknowledgments

This thesis project has been conducted between January and June 2013 in the Pre-development of Intelligent Transportation Systems Department (REPI) at Scania CV AB in Södertälje and was examined at the Automatic Control department at Linköping University.

I want to thank my supervisors at Scania, Kuo-Yun Liang and Assad Alam for their many ideas on how to improve this thesis project. I am also grateful for the report comments and discussions with my LiU supervisor Niclas Evestedt and my examiner Daniel Axehill. Without the company of my friend Fredrik Glans, this thesis project would not have been half the fun. I am deeply grateful to my wife Lisa, for her endless support and patience even when the thesis required many late nights of work.

Lastly, I can never thank God enough for creating me and giving me the abilities I needed to complete this thesis.

Linköping, June 2013 Simon Eiderbrant

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1

Introduction

1.1 Background

Our society relies greatly on fast transportation of goods all over the world. Road transportation by heavy duty vehicles (HDVs) are responsible for much of these transports. Lately, because of the competition from rail transports and more im-portantly because of the threat of global warming, other environmental and fi-nancial issues, the need for more energy efficient and environment friendly trans-portation techniques has become apparent.

1.1.1 Platooning

Platooning is one of the techniques that is thought to have a great potential for making the transportations more fuel efficient. Platooning means that one tries to drive HDVs as close as possible after each other in a, so called, platoon to reduce the air drag forces that the vehicles are subjected to. This air drag reduction saves a lot of fuel because the force that the vehicles’ engines need to overcome is reduced. In fact, [3] shows that the fuel savings can be as high as 4.7-7.7% for the right platoon configuration. This means big savings for the fleet companies since fuel costs amount to as much as 30% of the costs of a fleet company. [1]

1.1.2 Eco-routing

Another way of saving fuel and lowering the greenhouse gas emissions is eco-routing. [12]. Eco-routing is a way to determine what route to choose from a given initial position to a given target position to drive as fuel efficient as possible. At first you determine what routes you can take where the speed limit does not stop you from arriving in time. After that, traffic data is considered for the

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2 1 Introduction

sen routes. If there is heavy traffic congestion on one route or the speed is signifi-cantly reduced for some other reason, that route is excluded from consideration. After that a vehicle model is used, together with elevation data, to calculate target velocities for the vehicle for different parts of the route and then calculate which route consumes the least fuel. This route will then be chosen.

1.2 Problem formulation

Doing eco-routing simulations for platoons with many vehicles can be be rather complex and time consuming. The calculation complexity grows as O(N), where N is the number of vehicles in the platoon.

In this thesis, the possibility of modelling the platoon as one object, to reduce calculation complexity, is investigated. The model that is developed will be used and evaluated in an eco-routing context.

1.3 Thesis outline

The thesis is organized in eight chapters. Chapter 2 gives a background of pla-tooning and describes a survey done on railway modelling. Chapter 3 describes heavy vehicle modelling and some simplifications used in this thesis. Chapter 4 describes a point-mass platoon model that has been developed and compares this model with a multiple vehicle model with every vehicle modelled as in Chapter 3. In Chapter 5, the platoon object model is compared to the multiple vehicle model both analytically and numerically. Chapter 6 is a rather comprehensive sensitivity analysis of the model. Chapter 7 uses the platoon object model in an eco-routing algorithm. Lastly, Chapter 8 gives conclusions and some thoughts about method critique and future work and Appendix A gives an overview of the simulation setup.

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2

Platooning

Platooning is a technique that has a great potential for reducing the traffic flow, fuel consumption and greenhouse gas emissions in the future. This chapter gives an introduction to the concept of platooning. Section 2.2 also describes a survey of the modelling techniques in railway engineering since this might give some insight into the problem.

2.1 Background

In many vehicles these days there is a cruise control software (CC) that lets the driver choose a desired velocity and makes the vehicle maintain this velocity re-gardless of the environmental conditions such as grade, wind, rolling resistance etc. Some vehicles also have an adaptive cruise control (ACC). This means that when there are no vehicles ahead, the ACC acts like a regular CC, but when the vehicle approaches another vehicle from behind, the ACC adapts the speed to the vehicle ahead. If the vehicle ahead is leaving the road or is overtaken, the vehicle returns to its own set speed.

Platooning can be seen as the next step in the evolution. Platooning uses vehicle-to-vehicle (V2V) communication via Wi-Fi to make it possible to drive closer be-hind the vehicle ahead. The V2V communication means that when the vehicle ahead is braking, the following vehicle will immediately be alerted via the Wi-Fi connection so that it can take the appropriate action. This means that the hu-man reaction times will not be the limiting factor as to how close the vehicles can drive.

When driving in a platoon, the leading vehicle is the one that controls the speed in the longitudinal direction; the other vehicles keep the same velocity by

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4 2 Platooning

trolling the distance to the vehicle ahead.

There are a number of advantages with platooning.

• The fuel consumption goes down due to a much smaller air drag resistance. This fuel consumption reduction depends on the intermediate distance be-tween the vehicles.

• The traffic flow will be reduced because the vehicle distances are smaller. • The driver comfort and safety will be better as a result of less dramatic

velocity changes and the fact that panic braking can be optimally handled by the vehicle.

• Slinky effects1_{can be avoided in traffic queues.}

Looking a step further into the future, platooning can be seen as one step closer to fully autonomous vehicles which is probably where the future of transportation lies.

2.2 Railway train modelling survey

Since the notion of a HDV vehicle platoon in many ways is similar to a railway train a survey of railway train modelling was done to see if there was any common ground. If railway engineer actually do model railway trains as one object instead of multiple coupled cars, that would be a good starting point for the platoon model.

2.2.1 Longitudinal dynamics

Since the focus of this thesis is on longitudinal forces and velocities and fuel consumptions, those are the kinds of models that are interesting to examine in the railway train case. There are very complex models of railcars like the one in [5] with 27 equations describing the vehicle motion. This type of model is more useful when analysing in-train forces such as derailing investigations or calculations on in vehicle vibration and for the platoon a much simpler model should be developed.

[6] has a longitudinal model that is much less complex than the model in [5]. The cars actually have only one degree of freedom and that is longitudinal veloc-ity. The model contains resistance forces, grade forces, coupling model (which is treated extensively in this source) and motion equations. The model is for three railcars but can easily be extended to more cars and to have distributed power, i.e., that the train has locomotives not only in the start of the train but also in the middle and/or in the back. This would resemble a HDV platoon more. The idea in [6] is to simulate the train as a group of coupled masses and this might not be

1_{Slinky effects are when velocity fluctuations of a vehicle in a platoon get bigger and bigger as}

they propagate to the following vehicles. This means that if a vehicle brakes suddenly, this can force vehicles behind it to stop altogether causing unnecessary queueing.

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2.2 Railway train modelling survey 5

a good idea for the platoon model since its purpose is to replace these kinds of models.

According to [6], the grade influences not only the grade resistance but also the normal forces of the wheels and thereby the rolling (and maybe even the air) resistance. This effect is however very small and coupled with great uncertainties so it is usually better to ignore it in the modelling.

[11] says that the two most important resistances are air and rolling resistances. Inelastic wheel deformation can be disregarded due to the stiffness of the railcar

wheels. There is an air resistance calculation with parameter cL that is a drag

coefficient for the whole train. This coefficient is a sum of coefficients for every railcar. In the calculation, the terms for the first and the last wagons are calcu-lated separately because there is a bigger contribution from those wagons. The paper also develops a model of trains that have cars with inhomogeneous frontal areas. The model adds a contribution for the extra area of the car that will give rise to extra air drag. This might be something to look into, especially if the sensi-tivity analysis shows that the platoon model is very sensitive in respect to frontal area.

[8] describes all of the different types of resistance that influences the train. Espe-cially curve and grade resistance models are introduced as well as Davis’ formula. There are also equations of motion and solutions to these equations. All of these models are similar to the ones used to model road vehicles so no extra informa-tion can be found here.

[9] means that the three most important factors influencing the air drag are gap length between railcars, position in train and yaw angle of wind. Yaw angle of wind tends to cancel out over long distances so the most important factors are in fact gap length and position in train. That means that it is better to have a larger gap length in the end of the train than in the beginning. The head ve-hicle experiences the most resistance due to headwind impact. Then resistance declines down to around the tenth vehicle from where the resistance is more or less constant for the remaining cars.

2.2.2 Conclusions

The thought behind this survey was to find common ground between railway train modelling and platoon modelling. However, it turned out that in the rail-way engineering field there is little evidence of anyone simulating a train as one object instead of a string of coupled cars. Thus, the survey did not give the ex-pected results.

There are some interesting ideas about air drag modelling but they will not be used in this thesis because the platoon model is supposed to be a simple as possi-ble for calculation complexity reasons.

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3

Heavy duty vehicle model

3.1 Velocity model

Heavy duty vehicle modelling is a well covered area of research. The models are derived from Newtons second law which states that:

ma =XF (3.1)

where m is the vehicle mass [kg], a is the vehicle acceleration [m/s2] andP F is

the sum of all the forces acting on the vehicle [Nm]. Since we are only interested in the longitudinal motion of the vehicle we will only consider forces in the longi-tudinal direction. This means that the acceleration and the forces are all scalars defined as in Figure 3.1.

The most important forces acting on the vehicle are the air drag resistance, the rolling resistance, the grade resistance and the wheel force that drives the vehicle forward. [7] This is summed up in (3.2).

ma =XF = Fw−Fa−Fr−Fg (3.2)

3.1.1 Wheel forces

The force driving the wheel forward comes from the combustion of diesel fuel in the engine cylinders. The force produced by the engine reaches the wheels via the driveline. This means that the force output is a function of the engine properties, the transmission, the driver’s gas pedal input, losses in the engine and driveline

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8 3 Heavy duty vehicle model

F

r

F

g

F

_a

F

α

w

Figure 3.1:The forces acting on a vehicle driving in a grade. Fais the air drag

resistance, Fg is grade resistance, Fr is rolling resistance, Fw is the engine

force acting on the wheels and α is the road grade.

etc. The driver’s gas pedal input can also be seen, more or less, as a function of the road properties, traffic conditions, speed limits and driving style. It is a pretty complex system and thus there are great opportunities to reduce complexity to make the simulations faster.

Driveline

The model that will be used assumes no driveline at all, i.e. the force output from the engine is the force that acts on the wheels. The internal losses, driveline torsion and the transmission is not included in the model. That means that the assumptions are made that there are no losses, no torsion and that the vehicle is always in the right gear.

Engine

In Figure 3.2 the maximum power output from the engine can be seen for differ-ent engine speeds. A simple model is to say that the maximum power can always be attained. This will almost always overestimate the vehicle’s power and thus the vehicle speed, but since only relative results will be of interest in this thesis, this might not matter much.

Braking

The assumption is made that the braking power of a vehicle can be no larger than 60 kNm. This model comes from Scania.

Conclusion

The driving force Fw will be an input signal to the vehicle model so the model

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3.1 Velocity model 9 800 1000 1200 1400 1600 1800 2000 300 350 400 450 500 550

Engine power as a function of engine speed

Engine speed [r/min]

Engine power [hp]

Figure 3.2: The maximum power output from the engine as a function of

engine speed. The figure is adapted from [2].

The vehicle model will have a maximum power value, Pmax, and a maximum

brake force, Fb_max, that pose limits on the power output. They act as saturations

on the force output to the wheels. The upper limit of the saturation, Pmax, will

vary with the vehicle speed since P = Fwv so that the limit on Fwwill be Pmax/v,

v , 0.

3.1.2 Air drag resistance

When the vehicle travels on a road, with a given speed, the air in front of the vehicle is compressed. This high pressure gives a force on the vehicle opposing its motion called the air drag resistance force (or air drag). The higher the velocity, the higher the pressure and the higher the force trying to slow down the vehicle. The force is actually proportional to the square of the velocity.

The air drag also depends on the shape of the vehicle, the size of the vehicle and on the air density. The formula for air drag resistance force, (3.3), is taken directly from fluid dynamic theory.

Fa=

1

2ρaircwAfv

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where Fais the air drag [Nm], ρair is the mass density of air [kg/m3], cwis the

air drag coefficient, Afis the frontal area of the vehicle [m2] and v is the vehicle

velocity [m/s].

When driving in a platoon, the air drag is reduced because of the slip stream effect. The slipstream is a low pressure zone behind the moving vehicle that counteracts the higher pressure caused by the following vehicle. The impact of

platooning on the air drag force, also known as the air drag reduction φi(d), can

be seen in Figure 3.3, adapted from [4].

Figure 3.3:Measured air drag reduction for the first, second and third

vehi-cles in a platoon. Adapted from [4].

The notation φi(d)stands for the air drag reduction of the i:th vehicle with

inter-mediate distance d [m]. This gives an air drag equation as the one in (3.4).

Fa=

1

2ρaircwAfv

2_{(1 −}φi(d)

100 ) (3.4)

where i is the vehicle’s position in the platoon. For the first vehicle in the platoon the intermediate distance d is actually the distance to its following vehicle but for the rest of them d is the distance to the vehicle in front.

A first order least-squares approximation is used to fit the data to a function inside a suitable operating range in [10]. In this thesis, the air drag reduction

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3.2 Fuel consumption model 11

function will instead be linearly interpolated from the data in the graph. This will give a more precise air drag reduction function. For vehicles behind the third one in the platoon the same air drag reduction as for the third vehicle is assumed.

3.1.3 Rolling resistance

The rolling resistance force is the resistance between the road and the truck tires that opposes the vehicle movement. The force is proportional to the normal force of the road acting on the vehicle, i.e. m g cos α. There is also a factor that depends on the types of tires and road surface. This is modelled with a constant coefficient,

cr, called the rolling resistance coefficient. For trucks on asphalt the value of cris

typically around 0.01 according to [7].

Fr = crmg cos α (3.5)

3.1.4 Grade resistance

When the vehicle encounters a upgrade or downgrade on the road, the grade resistance force either opposes or aids the vehicle movement. That means that the grade resistance force is the only one of the resistant forces that can have negative values, giving the vehicle a positive acceleration contribution in the longitudinal direction.

The grade resistance is the effect of gravity in the longitudinal direction and is therefore just the gravitational force of the vehicle projected on the negative di-rection of movement.

Fg = mg sin α (3.6)

3.2 Fuel consumption model

In [12] there is a quite thorough survey done on fuel consumption modelling. In this thesis a model that resembles one of the models in [12] is used. The model used here is a modified version used at Scania. The difference is only how the

efficiencies and the fuel energy density are modelled and the fact that fuel cut1

is used.

3.2.1 Driving force

The force driving the vehicle forward is modelled almost exactly as in the velocity

model. See (3.7). The only difference is the equivalent mass mj of the rotating

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parts2that is not considered in the velocity models.

Fw(a, v, α) = (m + mj)a + 1 2cwAaρav 2_{+ mgc} rcos α + mg sin α (3.7)

3.2.2 Power output

The power output to the wheels from the engine that drives the vehicle is force times velocity.

P (a, v, α) = Fw(a, v, α)v (3.8)

where P is power [W], Fwis driving force [N] and v is vehicle velocity [m/s].

3.2.3 Idle fuel consumption

When the vehicle is idling there is still a small amount of fuel consumed. Because the vehicles are assumed to use fuel cut when coasting, no idle fuel consumption is considered.

3.2.4 Coasting

When the vehicle is coasting, i.e. neither powering nor braking, fuel cut is em-ployed and no fuel is consumed at all. This is modelled with a parameter δ that is 0 when coasting and 1 when not coasting. Since the model does not take braking into account, the parameter can be represented as:

δ =

(

1 if Fw > 0,

0 if Fw ≤0. (3.9)

where Fw is the driving force from before. This gives a model that has a zero

consumption when the engine is not driving the vehicle forward.

3.2.5 Auxiliary Power

In a vehicle there are a lot of auxiliary units also powered by the engine, e.g. the cooling fan, air compressor and power steering pump. All of these units are modelled as a constant load on the engine.

3.2.6 Efficiencies

The transmission efficiency, ηt, and the brake thermal efficiency β are the two

efficiencies in the model from [12].

2_{The rotational inertia of the rotating parts of the vehicle (wheels, drive shafts, flywheels etc.)}

makes up an inertia opposing the vehicle acceleration. Since this inertia opposes acceleration in the same manner as the vehicle mass, it is often modelled as a mass equivalent.

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3.2 Fuel consumption model 13 The transmission efficiency is a measure of losses in the power train, i.e. the differences between the power output at the engine and the power output at the wheels.

The brake thermal efficiency is a measure of how good the engine is at using the energy stored in the fuel.

At Scania, these two efficiencies are usually combined into a mean combustion

efficiency, η_eng. This is the model that will be used from now on.

3.2.7 Full fuel consumption model

The original model from [12] looks like in (3.10).

f c = δ βρd

(P (a, v, α)

ηt

+ Paux) (3.10)

where fc is the fuel consumption [cm3/s], δ is the coasting parameter, β is the

brake thermal efficiency, ρdis the energy density of diesel [J/cm3], P(a, v, α) is

the power output to the wheels [W], ηtis the transmission efficiency and Pauxis

the auxiliary power output.

The model with fuel cut and the Scania efficiency parameter instead looks like in (3.11).

f c = δ

η_engρd

(P (a, v, α) + Paux) (3.11)

where η_engis the mean combustion efficiency of the engine.

These models calculate the fuel consumption of one vehicle. In Chapter 4 these velocity and fuel models will be the starting point for developing a model of a platoon as one object.

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4

Platoon object model

4.1 Mathematical modelling of platoon velocities

When we try to model the platoon movement we need to sum up the forces acting on the different vehicles in the platoon as in (4.1).

X

i

miai =

X

i

(Fw,i−Fa,i−Fr,i−Fg,i) =

X i Fw,i− X i Fa,i− X i Fr,i− X i Fg,i (4.1)

This can also be written as in (4.2).

X

i

miai = Fw_tot−Fa_tot−Fr_tot−Fg_tot (4.2)

4.1.1 Point-mass model

We consider a point-mass model for the platoon. The platoon is then character-ized by one mass and has only one position, velocity and acceleration at any given time. This is a significant simplification of the system but one that makes sense to start with. We want to make the platoon model as simple as possible to be able to simulate its behaviour with as little computational effort as possible. Try simple things first, is a good rule of thumb when it comes to modelling and if a point-mass model is good enough why bother trying out a more complex one? Our point-mass assumption gives that we can rewrite (4.2) as:

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16 4 Platoon object model

Map = Fw_tot−Fa_tot−Fr_tot−Fg_tot (4.3)

where M is platoon mass [kg] and apis platoon acceleration.

The point-mass approximation will also affect the force terms as will be seen below.

4.1.2 Wheel force

If the Pmax values were to be summed up, the value of the platoon maximum

power would be much too high. That is because the platoon cannot travel faster than the slowest vehicle, i.e. the platoon has not reached its destination until all of the vehicles have. Therefore, the platoon model finds the smallest of the

ratios of Pmaxdivided by mass for all the vehicles and gives that ratio to the whole

platoon as seen in (4.4). This assumes that the ratio of Pmaxdivided by mass is a

good measure of the strength of the vehicle.

Pmax_platoon= min i

Pmax,i

mi

M (4.4)

where i are the vehicles in the platoon.

The maximum brake force will be modelled as in (4.5)

Fb_max_platoon= N Fb_max (4.5)

where N is the number of vehicles in the platoon.

4.1.3 Air drag resistance force

If we sum up the air drag forces for all of the N vehicles in the platoon, and assume a point-mass model, we get (4.6).

N X i=1 Fair,i= N X i=1 1 2ρaircw,iAf ,iv 2 i(1− φi(d) 100 ) = 1 2ρairv 2 p N X i=1 cw,iAf ,i(1− φi(di) 100 ) (4.6)

where vp is the platoon velocity [m/s]. Assuming that all of the vehicles have

the same form and therefore the same cw value, that the air drag reduction is

independent of frontal area and air drag coefficient and that the intermediate distances of the platoon vehicles are the same, the equation can be written as (4.7).

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4.1 Mathematical modelling of platoon velocities 17 Fa_tot= 1 2ρairv 2 pcwAftot N X i=1 (1 −φi(d) 100 ). (4.7)

(4.7) has only platoon specific constants, platoon velocity and the air drag re-duction function that depends on the intermediate distance and the number of vehicles in the platoon.

4.1.4 Rolling resistance force

Summing up the rolling resistance forces is quite a lot easier. A small angle approximation is used which says that cos α ≈ 1 for small angles α. This is a good model since the road grade is usually small, often between -5 and 5 % grade. Another assumption that is used is that all the vehicles have the same rolling resistance coefficient. This is probably not a big approximation since the vehicles have wheels of more or less the same type and they drive on the same road surface. With these simplifications the summation is easy.

Fr_tot= N X i=1 Froll,i= N X i=1 crmig = crMg (4.8)

4.1.5 Grade resistance force

When summing up the vehicles’ grade resistance forces a small angle approxima-tion is used, as for the rolling resistance, i.e. sin α ≈ α. Then the equaapproxima-tion looks like (4.9). Fg_tot= N X i=1 Fgrade,i= N X i=1 migαi = g N X i=1 miαi (4.9)

The point-mass model suggests that the platoon has only one position and

there-fore it only experiences one grade. If this grade is called αpthe equation becomes

as in (4.10). Fg_tot= g N X i=1 miαp = gαp N X i=1 mi = Mgαp (4.10)

It is not intuitively certain that this is a good model. When the platoon is climbing a small hill, sooner or later the first vehicle might find itself in a downhill while some of the other vehicles are still in the uphill. What grade will the platoon experience? This is however a start and the simulations will reveal if the model is not accurate enough.

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18 4 Platoon object model

4.2 Mathematical derivation of platoon fuel model

What happens if we sum up a number of models such as the one in (3.11)? We end up with something like (4.11).

X

f c =X δ

η_engρd

(P (a, v, α) + Paux) (4.11)

If we assume that Fw> 0 then we can calculate the maximum platoon fuel

con-sumption (“maximum” because of the ascon-sumption) for the platoon. This means

that we get rid of the δ in (4.11). Or rather that we consider δ a saturation on Fw

instead. This will not affect the model but it will be much easier to work with and simplify.

The equation now looks like in (4.12).

f ctot = 1 ρd XP (a, v, α) + P_aux η_eng (4.12) Pplatoon(ap, vp, αp) = Fw_tot(ap, vp, αp)vp=

Mapvp+ Fa_totvp+ Fr_totvp+ Fg_totvp (4.13)

The air drag, rolling resistance and grade resistance forces in (4.13) are handled exactly the same way as in the velocity model. This is shown in (4.7), (4.8) and (4.10).

Then the only parameter that is not taken care of in the fuel model is η_eng. This is

approximated with the mean of the efficiencies of all the vehicles. The resulting equation is shown in (4.14).

f ctot =

Pplatoon(ap, vp, αp) + Paux

ηmean_eng ρd

(4.14) The platoon object fuel consumption model only contains parameters that can be seen as platoon specific, i.e. we do not need data about every single vehicle but can simulate the behaviour of the whole platoon as one object.

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5

Model comparison

5.1 Mathematical analysis of the simplifications

When the model is used in the eco-routing algorithm, it would be of interest to know the maximum size that the model error can have. This would mean that the most fuel efficient road could be guaranteed given that the maximum bound on the fuel consumption for the best road is lower than the minimum bound of all of the other roads. Such bounds will be found in this section and then used later in Chapter 7.

The fuel consumption is given in (4.14). Since the auxiliary power term is con-stant for all vehicles in the node model, this term does not need to be over-/underestimated. Focus is on the term concerning consumed power. For the maximum bound, the mean combustion efficiency is estimated with the small-est of the vehicles’ efficiency values, giving a larger fuel consumption value (see (5.1)). In the same fashion, the lower bound on the fuel consumption uses the largest of the vehicles’ efficiency values.

δ η_engmaxρd P (a, v, α) ≤ δ η_engmeanρd P (a, v, α) ≤ δ η_engminρd P (a, v, α) (5.1)

where η_engmin= miniηiengand ηengmax= maxiηiengand i is the vehicles in the platoon.

The consumed power consists of four different terms that can be seen in (4.13). They will be handled separately here below.

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20 5 Model comparison

5.1.1 Acceleration

The acceleration term is M apvp. The total fuel consumption contribution of this

term is seen in (5.2).

Wacc=

Z

Fw>0

Map(t)vp(t)dt (5.2)

This is the work done by the engine trying to accelerate the vehicle from one velocity in the beginning, to another velocity in the end of the trip. All other accelerations and decelerations along the way would cancel out over the whole journey. However, since the platoon employs a fuel cut strategy, this energy is lost in the parts where fuel is cut. That means that the integral should be evaluated only over the parts where fuel cut is not used.

Fuel cut is used only in steep downgrades when the platoon keeps the set speed already. This means that the set speed will always be kept, more or less, during fuel cut. The assumption is made that the platoon maintains the same set speed in the beginning and in the end of the trip. This gives that the total work done by the acceleration force is close to zero as can be seen in (5.3).

Wacc= Z Fw>0 Ma(t)v(t)dt ≈ tf Z 0 Ma(t)v(t)dt = Mv 2 f −Mv 2 0 2 ≈0 (5.3)

This is of course a simplification but the truth is that this term contributes very little to the total of the fuel consumption and the over-/underestimations of the other terms affect the bounds much more. This makes it plausible to approximate this term with zero.

5.1.2 Air drag

The air drag term in (4.13) looks as in (5.4).

Fa_tot = 1 2ρairv 3 pcwAf _tot N X i=1 (1 −φi(d) 100 ). (5.4)

There are three parameters that can be over-/underestimated: the frontal area sum, the air drag reduction and the velocity.

The frontal area is the easiest one. If the area is overestimated this becomes an upper bound on the fuel consumption contribution. All of the vehicles are as-sumed to have a frontal area as big as the the largest of all the vehicles’ frontal

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5.1 Mathematical analysis of the simplifications 21

areas for the upper bound and the smallest of the vehicles’ frontal areas for the lower bound.

To get a lower bound on the fuel consumption, the highest possible air drag re-duction should be used. The highest theoretical fuel rere-duction possible occurs when the vehicles have an intermediate distance of zero meters. This will be very conservative since the PI-controller controlling the distance penalizes negative distance errors very hard. In reality, the intermediate distance between two vehi-cle will never be even close to half of the desired distance so this can be used as a lower bound. The theoretical upper bound is one, i.e., the case when there is no air drag reduction. This will be used as an upper bound on the fuel consumption. The velocity is a bit more difficult to put an upper or lower bound on. One way to do this is to use the over-/underestimations of parameters that will give the lowest possible platoon velocity and simulate this platoon configuration over the whole road profile. The velocity profile from this simulation will then be used in the calculation of the fuel consumption bounds.

The PI-controller will force the velocity to be close to the set speed when possible. This is true for the lower and upper bound simulations too. Because it is not a perfect controller, the velocities will vary and it can not be guaranteed that the velocity is always lower/higher than the original platoon velocity when maintain-ing the set speed. However, in upgrades, where the platoon can not maintain the set speed, the speed differences are much bigger so given that the road has at least one upgrade, this will give a lower/higher bound on the fuel consumption. (5.5) shows the bounds on the air drag contribution to the total fuel consumption.

1 2ρairv 3 mincwAminf N ≤ 1 2ρairv 3 pcwAf _tot N X i=1 (1 −φi(d) 100 ) ≤ 1 2ρairv 3 maxcwAmaxf N N X i=1 (1 −φi(d/2) 100 ) (5.5)

5.1.3 Rolling resistance

The rolling resistance term in (4.13) is M g cr. The work done by this force is seen

in (5.6). The same minimum/maximum velocity is used as in the air drag case. This leads to (5.7). Wroll= Z Fw>0 X i migcrvi(t)dt = gcr Z Fw>0 X i mivi(t)dt (5.6)

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22 5 Model comparison W_rolllower/upper = gcr Z F_w>0 vmin/max(t) X i midt = Mgcr Z F_w>0 vmin/max(t)dt (5.7)

This equation shows that it is sufficient to calculate these lower/higher velocities to get lower and upper bounds on the rolling resistance fuel consumption contri-bution.

5.1.4 Grade resistance

The grade resistance term in (4.10) is M g αp. The contribution of this term to the

total work done by the platoon is seen in (5.8).

Wgrade= Z F_w>0 X i migαi(t)dt (5.8)

The surrounding traffic is not considered in the eco-routing algorithm in Chapter 7. The assumption of no surrounding traffic leads to the conclusion that the platoon always chooses its own set speed. That means that the only time the platoon uses fuel cut, i.e. when the platoon is coasting, is when there is a quite steep negative slope. This means that a lower bound on this term would be to integrate over the whole trip as seen in (5.9).

tf Z 0 Fg_tot(t)dt ≤ Z Fw>0 Fg_tot(t)dt (5.9)

However, when the road has a number of long, steep downgrades, this lower bound will be very conservative. Therefore, a less conservative bound would be preferred.

Putting a lower or upper bound on the slope is more difficult than simulating lower and upper velocities. In theory, if the road is very hilly, one vehicle could be in a steep upgrade while another one is in a steep downgrade for the whole trip. That would give a very low lower bound and a very high upper bound resulting in very different fuel consumption values. Since the real distances between the vehicles is not known, some kind of assumption must be made to be able to find less conservative bounds.

To decide what grade should be used in the bounds, an assumption will be made. In the platoon object model, the assumption is made that the vehicles keep the platoon formation in the upgrades even if some of the vehicles in the beginning of the platoon would be able to drive faster. If this assumption is used, the con-trollers are the only things that cause variations in the distances between the

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5.2 Comparison with node model 23

vehicles. These variations will be assumed to be smaller than double the desired distance.

Then the grade values for the bounds are chosen as the highest/lowest slope value

in the road segment from 2 (N - 1)₂ dpmeters in front of the platoon mass centre

to 2 (N - 1)₂ dp meters behind it. This is the road segment on which the platoon

vehicles theoretically can be found using the distance assumption from above. The new bounds can be seen in (5.10).

Mgαs≤Mgα ≤ Mgαl (5.10)

5.2 Comparison with node model

In this thesis, two different platoon models are used. There is the platoon node model which models the platoon as N different point-mass objects, each one using the model described in Chapter 3. This is a more complex model that takes time to simulate.

There is also a platoon object model where the platoon is modelled as a single point-mass. This model is much less complex and the complexity, and simulation times, should be reduced by O(N). The model is described more in Chapter 4. In Chapter 7 the platoon object model is used in an eco-routing algorithm. One tries to find the most fuel efficient route out of a set of possible ones. The simula-tion time is approximately

NvNrt1s

where Nvis the number of vehicles in the platoon, Nris the number of routes and

t1

s is the simulation time for one vehicle and one route.

As can be seen in Section 5.2.4, the simulation time with five vehicles and one

route is approximately 230 seconds. That means that t1_s should be around 230/5

seconds. If the node model is used on a platoon of, e.g., 20 vehicles, and five dif-ferent routes, the simulation time will be around 20*5*(230 / 5) = 4600 seconds > 75 minutes. This is a long simulation time. With the platoon object model this time would be reduced to around 1*5*(230 / 5) = 230 seconds ≈ 4 minutes. To verify the accuracy of the platoon object model the outputs from the models, with the same inputs, will be compared and evaluated. The node model will be considered a reference that the platoon object is validated against.

5.2.1 Whole model

The comparison starts with a study of the fuel consumption differences when the different models are simulated over the same road segment. The road segments

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24 5 Model comparison

Road Node model fc Platoon object fc

[litres] [litres]

Flat road 29.67 29.71

Uphill 56.23 56.38

Real road 31.13 31.15

Table 5.1: Comparison between total fuel consumption of the two models.

Absolute differences.

[%] [%]

Flat road 0 +0.13

Uphill 0 +0.25

Real road 0 +0.053

Table 5.2: Comparison between total fuel consumption of the two models.

Relative differences.

that are used are the same as for the sensitivity analysis, i.e., a flat road, a constant upgrade of 2% and a real road profile which can be seen in Figure 6.2. All profiles are 12 km long. The results are summed up in Tables 5.1 and 5.2.

In all of the simulations, the fuel consumption of the platoon object is very close to the node model value. This hints that maybe the point-mass approximation is a good approximation.

5.2.2 Velocities

To compare the velocity calculations the same road segments are simulated with the two models and the velocities are compared. The same simulations as in Section 5.2.1 are used. The results are seen in Table 5.3.

Model Road Mean Min Max

[km/h] [km/h] [km/h]

Node model Flat road 89.997 89.573 90.033

Platoon object Flat road 89.997 89.829 90.000

Node model Uphill 73.881 72.417 90.000

Platoon object Uphill 73.873 72.434 90.000

Node model Real road 88.119 78.061 90.423

Platoon object Real road 88.117 77.843 90.225

Table 5.3: Velocitiy comparison of the two models on the three different

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5.2 Comparison with node model 25

The table shows that the velocities are very accurate. The models give very similar velocities for all of the road segments.

5.2.3 Fuel consumption

The comparison will continue with the fuel consumption models. To be able to compare these models fairly the same velocity trajectory (calculated as a mean of the node model velocities) and the same grades will be input to the fuel consump-tion models. This will give as fair comparisons as possible.

The total fuel consumption is summed up in Table 5.4. The platoon object model show results that are very similar to the results in Table 5.1.

[litres] [litres]

Flat road 29.67 29.71

Uphill 56.23 56.37

Real road 31.13 31.17

Table 5.4: Fuel consumption comparison of the two models with the same

velocity input.

5.2.4 Performance

The reason for approximating the node model with a simpler model is that the calculation complexity will be reduced. The complexity is in fact reduced. Table 5.5 shows the simulation times for the different simulations done earlier in this chapter.

Road Node model simulation time Platoon object

simulation time Time reduction [s] [s] [times] Flat road 252 58.6 4.30 Uphill 281 63.9 4.40 Downhill 224 52.6 4.26 Real road 230 53.9 4.35

Table 5.5: Simulation time comparison between different models and road

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6

Sensitivity analysis

To evaluate the platoon model and how sensitive it is to uncertainties in the in-puts and parameters a sensitivity analysis has been performed. The main ap-proach is to vary the value of the studied parameter or input and evaluate how this affects the outcome of the simulation. The variations in the parameters will be quite big. This is to make the variations in fuel consumption clear in the graphs and tables. The real uncertainties may not be this big.

6.1 Grade

The first input that was examined was the grade. Since the grade force is a big part of the total force acting on the vehicle it can be assumed that the grade will have quite a big impact on the velocities as well as on the fuel consumption. This is however an assumption and after this analysis it will be clear if this is the case. Two different road profiles will be used to analyse the sensitivity of the parame-ters. Both of them have 1000 samples and a sample interval of 12 meters; that is 12 km of road. The first one is a completely flat road. The second one is a constant upgrade of 2%. Downgrades will not be considered since fuel cut is assumed for steep downhill.

The grade will be varied with 30%, 50% and 70% up and down and the effect on the outputs (velocity and fuel consumption) will be analysed. Only the upgrade roads will be used for analysis of grade sensitivity because it does not make any sense to vary the flat road grade.

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28 6 Sensitivity analysis

6.1.1 Uphill

The first road segment that will be used is a steep uphill climb. The segment is, again, 12 km long and has a constant slope of 2%. The effects on fuel consump-tion and velocities that the variaconsump-tions give are shown in Figure 6.1.

50 100 150 200 250 300 350 55 60 65 70 75 80 85 90 Time [s] Speed [km/h] 50 100 150 200 250 300 350 1.8 1.85 1.9 1.95 2 x 10−3 Time [s]

Fuel consumption [litre/s]

Simulation with increased road grade

Original grade 10% increased 20% increased 30% increased 50 100 150 200 250 75 80 85 90 Time [s] Speed [km/h] 50 100 150 200 250 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 x 10−3 Time [s]

Simulation with reduced road grade

Original grade 10% reduced 20% reduced 30% reduced

Figure 6.1:The uphill simulation grades with grade variations.

The platoon’s ability to maintain speed is greatly affected with the grade varia-tions. Apparently the platoon velocity is quite sensitive to grade variations in up grades. However, when the grade increases, the instantaneous fuel consumption stays the same. This is because the platoon is already using the maximum power. The maximum instantaneous fuel consumption for the platoon is determined by the maximum power limit and both of these limits are reached simultaneously. The total fuel consumption is higher for bigger grades because the platoon needs to travel for a longer time with the same instantaneous fuel consumption. When the grade is reduced with 50% and 70%, the platoon does not reach the maximum power limit and thus it does not drop its velocity and the instanta-neous fuel consumption is lower.

Table 6.1 shows that the total fuel consumption goes up when the grade is in-creased and down when it is reduced even though the instantaneous fuel con-sumption is the same (at least for grade increases and small grade decreases). This is because of the velocity dropping/increasing, meaning that the total trip time for the platoon is changed. A longer time at the same fuel consumption means a higher total.

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6.1 Grade 29

Fuel consumption differences [%] Grade

var.

−_70% −_50% −_30% ±_0% _+30% _+50% _+70%

Uphill -28.7 -17.1 -10.6 ±₀ _+12.2 _+21.1 _+30.5

Real road -3.66 -3.19 -2.98 ±₀ _+4.05 _+7.08 _+10.3

Table 6.1: The influence of grade variations on the total platoon fuel

con-sumption for two road profiles.

6.1.2 Real road profile

It is interesting to see how the platoon reacts to variations in grade for a real road profile. A part of the E4 highway between Södertälje and Norrköping will be used. The road grade profile is seen in Figure 6.2. And the simulation results are shown in Figure 6.3. 0 2000 4000 6000 8000 10000 12000 −3 −2 −1 0 1 2 3 4 Road profile Distance [m] Road grade [%]

Figure 6.2:The real road grade profile.

Figure 6.3 shows that the big differences in instantaneous fuel consumption is for small upgrades or downgrades were the platoon has not yet hit the power limit or the limit where gravity does all the work. These differences however lead to different velocities so that the total fuel consumption for these parts are more

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30 6 Sensitivity analysis 50 100 150 200 250 300 350 400 450 500 65 70 75 80 85 90 Time [s]

Speed [km/h] Original grade

10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 x 10−3 Time [s]

50 100 150 200 250 300 350 400 450 75 80 85 90 Time [s]

Speed [km/h] Original grade

10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 0 0.5 1 1.5 2 x 10−3 Time [s]

Figure 6.3:The real road profile simulation with variations in grade.

or less the same. The big differences in total fuel consumption are in the steep uphill slopes where the maximum power limits the velocity and keeps the fuel consumption at a maximum.

6.1.3 Conclusions

In upgrades the velocities and the fuel consumption is highly sensitive to big changes in grade. However, the grade uncertainties are typically smaller than the ones used in the simulations. [13] gives an estimate root mean square error of grade estimation around 0.16% grade. For a steep uphill this is 10% error or less so the model is not overly sensitive to errors in grade estimation. Table 6.1 shows that the fuel consumption error is more or less linearly dependant on the grade. This would mean that a grade error of less than 10% would give a fuel consumption error of less than 4%.

For real road profiles with up- and downgrades, the model is much more sensi-tive to an overestimation of the grade. This is because an underestimation of a negative slope gives almost no difference in total fuel consumption but an overes-timation gives a big difference.

6.2 Air drag coefficient and frontal area

Air drag coefficient and frontal area are two other parameters that are of interest to understand the vehicle behaviour. As seen in 4.6 the two parameters enter the equations in exactly the same way. This means that varying one of them with say 10% is equivalent with varying the other with 10%. Therefore, when evaluating sensitivity, we treat the two parameters as one, i.e., the case where the product of

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6.2 Air drag coefficient and frontal area 31

the two parameters is varied will be considered.

When evaluating the air drag sensitivity, three different road profiles are used. In addition to the two profiles used for the grade sensitivity analysis, a flat road will also be used.

6.2.1 Flat road

To start off, a flat road simulation will be done. The road is 1000 samples long and completely flat. The simulation results can be seen in Figure 6.4.

50 100 150 200 250 300 350 400 450 89 89.5 90 90.5 91 Time [s]

Speed [km/h] Original coefficient

10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 1.2 1.3 1.4 1.5 1.6 1.7x 10 −3 Time [s]

Simulation on flat road with increased air drag coefficient

50 100 150 200 250 300 350 400 450 89 89.5 90 90.5 91 Time [s]

10% reduced 20% reduced 30% reduced 50 100 150 200 250 300 350 400 450 0.9 1 1.1 1.2 1.3x 10 −3 Time [s]

Simulation on flat road with reduced air drag coefficient

Figure 6.4:The flat road simulation with air drag coefficient variations.

The platoon maintains the same speed throughout the trip but the fuel consump-tion differs quite a bit. This can also be seen in the total fuel consumpconsump-tion in Table 6.2.

6.2.2 Uphill

The results of varying the air drag coefficient 30, 50 and 70%, up and down in an uphill slope of 2%, are seen in Figure 6.5.

It can be seen in the simulations that although a 70% increase of the air drag co-efficient is not enough to make the platoon lose speed when driving on relatively flat road, in the upgrades where the platoon cannot maintain the set speed the speed is reduced even more with a larger air drag coefficient.

6.2.3 Real road profile

A simulation over a real road profile will be done here too. The same road profile as for the grade case is used. The road grades can be seen in Figure 6.2.

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32 6 Sensitivity analysis 50 100 150 200 250 300 350 400 450 500 550 70 75 80 85 90 Time [s] Speed [km/h] 50 100 150 200 250 300 350 400 450 500 550 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation in upgrade with increased air drag coefficient

Original coefficient 10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 500 550 74 76 78 80 82 84 86 88 90 Time [s] Speed [km/h] 50 100 150 200 250 300 350 400 450 500 550 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation in upgrade with reduced air drag coefficient

Original coefficient 10% reduced 20% reduced 30% reduced

Figure 6.5:The speeds of the platoon with different air drag coefficients.

50 100 150 200 250 300 350 400 450 500 75 80 85 90 Time [s]

10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 500 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation with increased air drag coefficient

50 100 150 200 250 300 350 400 450 78 80 82 84 86 88 90 Time [s]

10% reduced 20% reduced 30% reduced 50 100 150 200 250 300 350 400 450 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation with reduced air drag coefficient

Figure 6.6:The platoon speeds and instantaneous fuel consumption for the

real road profile.

Figure 6.6 shows the simulation results. We see that the fuel consumption is af-fected most when the platoon is driving on flat roads or in relatively small grades. When it is driving in uphill, the velocity is reduced for a higher coefficient and this compensates for the higher fuel consumption and when it is driving in down-hill, the fuel consumption is not affected much, because the platoon can easily maintain the set speed without having to produce much torque from the engine.

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6.3 Maximum power output to the wheels 33

Fuel consumption differences [%] Air drag coeff. var. −_70% −_50% −_30% ±_0% _+30% _+50% _+70% Flat road -30.6 -21.9 -13.1 ±₀ _+13.1 _+21.9 _+30.6 Uphill -9.05 -6.18 -3.56 ±₀ _+3.21 _+5.20 _+7.09 Real road -23.8 -17.1 -10.2 ±₀ _+9.60 _+15.9 _+22.1

Table 6.2: The influence of air drag coefficient variations on the total platoon

fuel consumption.

6.2.4 Conclusions

The air drag coefficient uncertainties matter the most when driving on flat road. Then the error in fuel consumption is more or less linearly dependant on the parameter error. However, in upgrades the fuel consumption error is not as big and not linearly dependant.

In upgrades, the model is more sensitive to underestimation of the parameter and this is true also for real road profiles.

6.3 Maximum power output to the wheels

The maximum power output is an upper limit on the input force Fwin the model.

This limit is modelled as a very simple constant model but in reality the power that can be produced from the engine is dependent on many things. If the sen-sitivity analysis reveals a very large sensen-sitivity then this might point to the fact that there is a need for a more complex model of the engine/wheel forces.

6.3.1 Flat road

For the simulations the same road profiles have been used. Figure 6.7 shows the results of a simulation on a flat road with the maximum power limit varied up and down 10, 20 and 30%.

The power limit is not important when driving on a flat road. The vehicle must be very weak not to be able to maintain 90 km/h on a flat road.

6.3.2 Uphill

Figure 6.8 shows the results of a simulation on a flat road with the maximum power limit varied up and down 10, 20 and 30%.

As can be expected, the differences between the maximum power limits are ap-parent in the upgrades. The velocities that can be maintained are higher for a higher power limit and the instantaneous fuel consumption is also higher. As we

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34 6 Sensitivity analysis 50 100 150 200 250 300 350 400 450 89 89.5 90 90.5 91 Time [s] Speed [km/h] 50 100 150 200 250 300 350 400 450 1.2 1.22 1.24 1.26 1.28 1.3x 10 −3 Time [s]

Simulation on flat road with increased maximum power limit

50 100 150 200 250 300 350 400 450 89 89.5 90 90.5 91 Time [s]

Speed [km/h] Original limit

10% reduced 20% reduced 30% reduced 50 100 150 200 250 300 350 400 450 1.2 1.22 1.24 1.26 1.28 1.3x 10 −3 Time [s]

Simulation on flat road with reduced maximum power limit

Original limit 10% increased 20% increased 30% increased

Figure 6.7:Speeds of the platoon on a flat road with varied maximum power

limit. 50 100 150 200 250 300 350 400 450 500 550 70 75 80 85 90 Time [s] Speed [km/h] 50 100 150 200 250 300 350 400 450 500 550 0.02 0.04 0.06 0.08 0.1 0.12 Time [s]

Simulation in upgrade with increased maximum power limit

Original limit 10% increased 20% increased 30% increased 100 200 300 400 500 600 700 55 60 65 70 75 80 85 90 Time [s] Speed [km/h] 100 200 300 400 500 600 700 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation in upgrade with reduced maximum power limit

Original limit 10% reduced 20% reduced 30% reduced

Figure 6.8:Speeds of the platoon in an upgrade with varied maximum power

limit.

see in Table 6.3, this means that the total fuel consumption will be higher for a higher power limit and this is because the platoon can maintain a higher speed.

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6.3 Maximum power output to the wheels 35

Fuel consumption differences [%] Power limit var. −_30% −_20% −_10% ±_0% _+10% _+20% _+30% Flat road ±₀ ±₀ ±₀ ±₀ ±₀ ±₀ ±₀ Uphill -7.93 -5.42 -2.76 ±₀ _+2.83 _+5.69 _+8.59 Real road -4.80 -2.65 -1.14 ±₀ _+0.59 _+0.98 _+1.23

Table 6.3: The influence of maximum power limit variations on the total

platoon fuel consumption.

6.3.3 Real road profile

To verify the sensitivity, a simulation is made over the same road profile used in the grade case. See Figure 6.2 for the grade profile. The simulation results are shown in Figure 6.9. 50 100 150 200 250 300 350 400 450 78 80 82 84 86 88 90 Time [s]

10% increased 20% increased 30% increased 50 100 150 200 250 300 350 400 450 0.02 0.04 0.06 0.08 0.1 0.12 Time [s]

Simulation with increased maximum power limit

50 100 150 200 250 300 350 400 450 500 70 75 80 85 90 Time [s]

10% reduced 20% reduced 30% reduced 50 100 150 200 250 300 350 400 450 500 0.02 0.04 0.06 0.08 0.1 Time [s]

Simulation with reduced maximum power limit

Figure 6.9:Platoon speed and fuel consumption on a real road profile with

varied maximum power limit.

The real road simulation shows that the differences occur in the upgrades where the power limit comes into play. On flat roads and downhill, the platoon does not reach the power limit and therefore nothing changes.

6.3.4 Conclusions

On flat road, the power limit uncertainty does not affect the total fuel consump-tion at all. This is because the power limit is not reached on flat roads. In up-grades, the power limit uncertainty matters the most. The fuel error is almost

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36 6 Sensitivity analysis

linearly dependant on the parameter error. On real roads, the total fuel con-sumption is much more sensitive to an underestimation of the maximum power limit.

The differences in the velocities the platoon can maintain in an upgrade is pretty big even for a small decrease/increase of the maximum power limit. This is a little problematic because with the simplification of a constant power limit the difference between the platoon object maximum power and the actual maximum power can easily be off by 10% or more. The velocities are however not the most interesting outputs from the simulation. The big question is how the fuel con-sumption is affected by the variations in maximum power. In Table 6.3 the dif-ference in total fuel consumption can be seen. The fuel consumption goes up for stronger engines due to the fact that they can maintain a higher speed in the upgrades. The differences in total fuel consumption are, however, not as big as for the velocity simulations and this shows that the simple constant model for maximum power might be good enough.

6.4 Air drag reduction from platooning

When driving in a platoon, the air drag is reduced due to the slipstream effect from the leading vehicles. In this section it will be evaluated how the air drag reduction affects the velocities and the fuel consumption of the platoon.

The air drag reduction depends on the intermediate distance of the vehicles in the platoon. The sensitivity will be evaluated in regard to this distance. Because small variations give very small output variation the distance have been subjected to bigger variations. The original distances between the vehicles are time gaps of 0.4 seconds which is 10 meters at a speed of 90 km/h. The variations used are 0.1, 0.13, 0.2, 0.8, 1.2, 1.6 seconds.

6.4.1 Flat road

The results of a simulation of the platoon with varying distances over a flat road is seen in Figure 6.10. The distances do not seem to matter much at all when it comes to the velocities that the platoon can maintain throughout the road seg-ment. The fuel consumption is however affected by the distance.

6.4.2 Uphill

When driving uphill, the velocities that the platoon can maintain are higher ve-locity for a shorter distance. Because the maximum power is reached, the in-stantaneous fuel consumption is the same for all of the cases but with a longer intermediate distance, the platoon will drive for a longer period of time making the total fuel consumption higher. This is seen in Table 6.4.

Analytical model of a vehicle platoon

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Analytical Model of a Vehicle Platoon

Analytical Model of a Vehicle Platoon

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan vid Linköpings universitet

av

Abstract

Acknowledgments

Contents

1

Introduction

1.1

Background

1.1.1

Platooning

1.1.2

Eco-routing

1.2

Problem formulation

1.3

Thesis outline

2

Platooning

2.1

Background

2.2

Railway train modelling survey

2.2.1

Longitudinal dynamics

2.2.2

Conclusions

3

Heavy duty vehicle model

3.1

Velocity model

3.1.1

Wheel forces

F

r

F

g

F

a

F

α

w

3.1.2

Air drag resistance

3.1.3

Rolling resistance

3.1.4

Grade resistance

3.2

Fuel consumption model

3.2.1

Driving force

3.2.2

Power output

3.2.3

Idle fuel consumption

3.2.4

Coasting

3.2.5

Auxiliary Power

3.2.6

Efficiencies

3.2.7

Full fuel consumption model

4

Platoon object model

4.1

Mathematical modelling of platoon velocities

4.1.1

Point-mass model

4.1.2

Wheel force

4.1.3

_a