Predictive Cruise Control for Heavy Duty Vehicles T.W.T. Ivens CST 2010.057

Master traineeship report Supervisor: dr. ir. T. Hofman (TU/e) Coach: ir. M.F.M. Pesgens (DAF Trucks N.V.) Eindhoven University of Technology Department of Mechanical Engineering Automotive Engineering Science group Eindhoven, August 2010

Samenvatting In de energiehuishouding van voertuigen wordt normaal gezien het werkpunt van de motor veranderd om brandstof te besparen. Dit geldt zowel voor hybride als conventionele voertuigen. Dit kan echter ook gerealiseerd worden door de vermogensvraag op sommige momenten aan te passen. Dit kan gedaan worden door voertuigeigenschappen te veranderen (bijvoorbeeld de aerodynamica) of door de (gewenste) voertuigsnelheid aan te passen. Dit verslag gaat uit van de laatste optie en geeft inzicht in het potentieel van Predictive Cruise Control (PCC) met betrekking tot brandstofbesparing in zware trucks. PCC gebruikt informatie over het hoogteprofiel van de toekomstige weg om het bijbehorende ideale snelheidsprofiel te bepalen. Daarbij worden kleine afwijkingen van de gewenste snelheid van de cruise control toegelaten. Een wiskundige routine, genaamd Dynamisch Programmeren, is gebruikt om het snelheidsprofiel te optimaliseren. Het DP algoritme in dit verslag bleek erg traag te zijn door de complexiteit van interne definities. Er zijn echter wel simulaties uitgevoerd. Die zijn gedaan voor een zware DAF truck in verschillende scenario’s. Op deze manier is inzicht verkregen in hoe en wanneer het voertuig brandstof kan besparen. Op en rondom een korte heuvel is een gemiddelde besparing van 4.76% gerealiseerd, wat overeenkomt met wat in de literatuur gevonden is. De brandstofbesparing vloeit voort uit het effici¨enter gebruik van de motor en het vermijden van remacties.

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Abstract In vehicle energy management, usually the operating point of the engine is influenced in order to save fuel. This holds for both hybrid as conventional vehicles. However, it is also possible to change the power demand at certain points in time. This can be achieved either by changing vehicle properties (e.g. aerodynamics) or by changing the (desired) vehicle speed. This report deals with the latter case and gives insight in the potential of Predictive Cruise Control (PCC) with respect to fuel savings in heavy duty trucks. PCC uses future road profile information to determine the ideal speed profile, allowing small deviations from the cruise control set speed. A mathematical routine called Dynamic Programming is used for the optimization of the speed profile. The DP algorithm used in this report appeared to be very slow due to complexity of definitions inside it, but simulations have been performed. This is done for a heavy duty DAF truck in several test cases, giving insight in how and when the fuel savings are realized. On and around a small hill, an average fuel saving of 4.76% is realized, which corresponds to findings from literature. The fuel saving results from more efficient use of the engine and avoiding braking.

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Preface Preface This traineeship for DAF Trucks N.V. is part of my Master studies at the mechanical engineering department. Within this department, I have focussed on powertrains in the Automotive Engineering Science (AES) group. As this traineeship includes very different subjects, like powertrains, drivetrains and optimization methods, it is very appealing to me. Although this assignment covers a relatively short time, I hope that my work will contribute to the development of more fuel efficient commercial vehicles.

Acknowledgment My supervisor Theo Hofman (TU/e) was always enthousiastic about the subject and my findings. He stimulated me to investigate new things all the time. My coach Michiel Pesgens (DAF Trucks N.V.) was always very patient and really tried to understand my problems. He encouraged me to get into details and really understand what was going on. They both have guided me through this traineeship and their efforts paid off: I have learned a lot during these months and I am very thankful for that. I assume that there is no need for explaining that Yvette, my family, friends, colleagues and all other supporting people are hereby thanked too.

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Table of Contents Samenvatting

i

Abstract

ii

Preface

iii

Table of Contents

iv

1 Introduction 1.1 Background . . . . . 1.2 Literature . . . . . . 1.2.1 Methods . . . 1.2.2 Driveability . 1.2.3 Fuel economy 1.2.4 Functionality

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1 1 1 1 2 2 3

2 Assignment 2.1 Problem Definition . . . . . 2.2 Objectives . . . . . . . . . . 2.3 Contribution of this report 2.4 Case . . . . . . . . . . . . . 2.5 Outline of this report . . . .

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4 Numerical Optimization 4.1 Algorithm functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Control choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 DP grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Approach and Modeling 3.1 Optimization definition . . . . . . . 3.2 Methods and tools . . . . . . . . . . 3.2.1 Model approach . . . . . . . 3.2.2 Model orientation / causality 3.3 Force balance . . . . . . . . . . . . . 3.4 Model structure . . . . . . . . . . . . 3.5 Maximum slope performance . . . . 3.6 Sensitivity analysis . . . . . . . . . .

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TABLE OF CONTENTS 4.4 4.5

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DP function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Results 5.1 Test cases . . . . . . . . . . . . 5.2 Performances . . . . . . . . . . 5.2.1 Computing performance 5.2.2 Fuel savings . . . . . . . 5.2.3 Trends . . . . . . . . . . 5.2.4 Fuel saving principle . .

17 17

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19 19 19 19 21 21 25

6 Conclusions and Recommendations 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 28 28

A Powertrain Modeling A.1 Engine model . . . A.1.1 Model 1 . . A.1.2 Model 2 . . A.1.3 Model 3 . . A.1.4 Model 4 . . A.1.5 Comparison A.2 Transmission . . .

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B Sensitivity Analysis B.1 Vehicle speed sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Slope sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Mass sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C Nomenclature

40

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Chapter 1

Introduction 1.1

Background

Cruise Control (CC) is a well-known system in the automotive industry these days. It holds the vehicle speed in order to provide more comfort to the driver. Especially for professional drivers, who have a high annual mileage, this is a very welcome invention. However, conventional cruise controls do not take into account road profiles. Therefore, they are not as fuel efficient on hilly roads. A truck, for example, that drives on a hill, uses a lot of energy while going up at constant speed. However, when the truck descends shortly later, a lot of energy is available. If this was taken account for, the truck could have gone up more slowly and use downhill slope to gain speed again. These days, with a shortage of fossil fuels ahead (and therefore increasing fuel prices), the demand for fuel saving technology increases. DAF Trucks N.V. now has (among others) a special project focussing on energy management of commercial vehicles, in cooperation with Eindhoven University of Technology. The topic of this traineeship is applicable to both hybrid and conventional vehicles. The focus in this stage, however, is on conventional vehicles. For clarity, two conventions have to be made at this point: two different methods to calculate a reference speed can be found in literature: the first using current vehicle states (Smart Cruise Control or SCC) and the second using information from the road ahead (Predictive cruise control or PCC). Figure 1.1 shows the global structure for using SCC and PCC. The controller block is discussed in section 1.2.4.

1.2 1.2.1

Literature Methods

Literature shows that some research has already been performed on this subject, mainly focussing on PCC. Daimler has written a paper on PCC [6], which shows a simple cost function, but not the method used for optimization. This company also holds a US Patent [7] and has a truck on the market with PCC: the Freightliner Cascadia, which uses (offline) maps from Navteq. Hellstr¨ om’s thesis [3] gives some more insight in methods: there, a model 1

1.2 Literature

2 Plant Controller

fuel consumption

set speed CC, PCC or SCC

throttle, brake actual speed

Figure 1.1: Global structure for CC/SCC/PCC.

predictive control (MPC) scheme is used and the optimization is carried out through dynamic programming (DP). Furthermore, truck manufacturer Scania seems to have an SCC in its new R-series: Ecocruise saves up to 5% on complete routes without using preview technology.

1.2.2

Driveability

As has been stated in Walker’s paper [8], three key market drivers can be distinguished: legislation, the customer and the original equipment manufacturer (OEM). Legislation prescribes (among others) emissions and fuel economy. The customer asks for cleaner and more efficient vehicles, which are easy to use and cheap to maintain. Finally, the OEM tries to meet customer desires and legislative requirements, while meeting cost and quality targets. This implies that new technology does not only have to be safe and efficient, but also has to take into account the drivers: new features will encounter higher acceptance levels if they are easy to use, relatively cheap and offer more comfort. The PCC/SCC systems that are already implemented in trucks meet this desire: from an operating point of view, they do not differ from a conventional cruise control. The user interfaces are the same and the smart part of this CC will simply be turned off if there is not enough (preview or vehicle) information available: the system then acts as a conventional CC. Also, the truck’s velocity should not be changed significantly: the driver will be tempted to intervene with the throttle if the truck slows down too much. As shown in Hellstr¨om’s controller [2], boundaries can be defined to prevent the truck from slowing down or speeding up too much.

1.2.3

Fuel economy

The fuel savings using PCC/SCC differ somewhat, since these depend on the application and on their complexity. As stated before, fuel savings up to 5% seem to be possible with SCC. Hellstr¨ om’s papers [2], [3] show savings around 2,5% with simulations on long, real road profiles. His colleague Fr¨ oberg [1], however, states that fuel savings above 10% can be realized if speed limitations (according to legislation) are taken into account. This number holds for short road sections (test cases up to 5 kilometers). Latteman [6] shows fuel savings around

1.2 Literature

3

4% for real road simulations. Obviously, a heavier truck saves relatively more fuel, since it has a higher inertia. These savings might seem low, but since these heavy duty trucks have a high annual mileage, cost savings can be very high. Unfortunately, no real-life measurements could be found yet.

1.2.4

Functionality

A conventional cruise control regulates the vehicle speed toward a fixed speed setpoint. The cruise control’s reference speed could also be prescribed by an additional algorithm. Using vehicle and/or road information and the set speed (provided by the driver), the optimal vehicle speed profile is calculated. Scania [5] states that its Ecocruise function (which is not PCC, but Smart Cruise Control) ”detects” hills and responds correspondingly: the truck goes up at full engine power and as soon as the slope decreases again (top of the hill), the truck waits with accelerating, since it can be expected that a down-slope will be ahead and cause the vehicle to accelerate naturally. Once down the hill, Ecocruise holds the higher velocity gained downhill for some time, to be ”ready” for a next hill that might be ahead. If the latter is not encountered, the vehicle will slow down to the set speed again. A PCC involves some more functions: as it is clearly described by Neiss [7], the GPS system gives a signal to the PCC. The latter contains a 3D digital road map and can therefore estimate what hills will be encountered on the road ahead. An optimization algorithm then calculates the optimal control actions for a limited piece of road ahead. The lookahead distance is called the prediction horizon. This horizon should be long enough to determine the optimal controls, but should not be too long. This is because the optimization algorithm requires a high level of calculation power. The simpler and shorter a calculation can be made, the less computing power is needed. The PCC then sends out the optimal reference speed to the speed controller (which is in fact a conventional cruise control). The latter then sends control signals to the engine and brakes (inducing a certain vehicle speed change). The vehicle speed is fed back into the controllers to determine a new control action. Figure 1.2 shows the controller block of Figure 1.1. Controller details will be discussed in Chapter 3.

1.2 Literature

4

Controller

GPS signal (PCC)

3D Digital Road Map road profile

set speed

Optimization Algorithm reference speed

actual speed

−

+

Cruise Control

Figure 1.2: PCC functionality overview.

throttle, brake

Chapter 2

Assignment 2.1

Problem Definition

In vehicle energy management, usually the operating point of the engine is influenced in order to save fuel. This holds for both hybrid as conventional vehicles. However, it is also possible to change the power demand at certain points in time. This can be achieved either by changing vehicle properties (e.g. aerodynamics) or by changing the (desired) vehicle speed. A conventional cruise control keeps the vehicle speed constant in situations where there are no big environmental changes to be expected (so in case of low traffic density and a road with only small bends). The goal of this assignment is to investigate what the fuel savings might be when using a different type of cruise control: predictive cruise control (PCC), which calculates an optimal speed control action by using preview (looking at the road ahead) signals. The main problem is defined here as: Determine the potential of predictive cruise control in heavy duty trucks with respect to fuel savings.

2.2

Objectives

The main objective of PCC is to realize fuel savings in commercial applications. Next to this, the system should keep the vehicle within an allowed range of velocities, since the time lost when driving at lower speeds should be insignificant, as time could represent a certain cost as well. Also, speeds may not exceed certain limits since high speeds may be dangerous and/or prohibited.

5

2.3 Contribution of this report

2.3

6

Contribution of this report

This report gives insight in the potential of PCC with respect to fuel savings in heavy duty trucks. It also provides some tools for further research and expansion of the current technology in future applications and can be seen as a first step towards fuel saving cruise controls.

2.4

Case

For this assignment, calculations are performed for a long-haulage, heavy-duty truck (in US: class 8). These trucks have an average annual mileage of 150,000 km and consume in the order of 30 L/100km on average. The corresponding fuel costs are about 30% of the life cycle cost of a heavy truck [4]. Because of the latter, they also seem to have a high potential: due to their weight, heavy trucks can have a large amount of kinetic energy, which can be used while driving over hills. For simplicity only traffic-free, long and straight roads are assumed, so external influences are minimized. All calculations are performed for a DAF XF105 truck. When not mentioned otherwise, it has a total weight of 40 tonnes (40.000 kg). Figure 2.1 shows such a truck.

Figure 2.1: DAF XF105 long-haulage, heavy-duty truck.

2.5

Outline of this report

For the simulations within this traineeship, a model is constructed, which is described in Chapter 3. Some details of the powertrain and a sensitivity study are described in Appendices A and B, respectively. In order to develop a predictive cruise control, the method dealing with the optimization problem is first described in Chapter 4; this is the dynamic programming method. The results are presented in Chapter 5 and finally, conclusions are drawn in Chapter 6. Here recommendations are also done. The symbols used in this report are clarified in Appendix C.

Chapter 3

Approach and Modeling This chapter will go into more detail of the model and design choices will be clarified.

3.1

Optimization definition

The optimization problem can now be formulated. First of all, the objectives are chosen, based on Lattemann’s [6] paper: • The vehicle should consume as little fuel as possible on a given road profile. • The total travel time is fixed to make the system useful for the transport company. • The velocity should not exceed an upper and not undershoot a lower velocity limit. These objectives can be implemented as constraints to the problem. These will be described in the following chapters; for now, only the fuel cost function is taken into account, which is defined as the consumed fuel volume (Vf ) per traveled distance (sr ):

Jf uel

Vf 1 = = sr sr

Z

m ˙f dt, ρf

(3.1)

where ρf is the fuel density. In order to determine the fuel flow rate (m ˙ f ), the road load has to be known. However, the road load is depending on the vehicle’s speed in history (which determines the current position and hence, the slope on the route) and its current speed. Appendix B shows this relation and the fuel consumption sensitivity to some system variables. Because of this dependency, it is hard to solve this cost function algebraically. Therefore, a model is derived to perform a numerical optimization procedure. In the next sections, the method and tools dealing with this problem are described and some model equations are defined.

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3.2 Methods and tools

3.2

8

Methods and tools

The simulations in this assignment are performed using MATLAB (which also has the possibility to implement algorithms). For simplicity, no lateral forces (for example, due to steering) are considered. There, a longitudinal vehicle model is constructed. In literature, it appears that a relatively simple vehicle model is desired, since that takes little calculation time and is therefore more suitable [1]. Guzzella [14] describes different ways to build such a model: it can have different orientations and use different approaches to solve the problem.

3.2.1

Model approach

A vehicle model, like the one in this report, can use different approaches. Guzzella [14] distinguishes three types: The average operating point approach is used for a first preliminary estimation of the fuel consumption. It uses a representative average operating point to calculate the fuel consumption. The quasistatic approach uses engine maps to calculate the fuel consumption. This is more accurate. The dynamic approach is even better: it uses mathematical descriptions of the powertrain system and is therefore able to find a very good approximation of the real fuel consumption. Because of its complexity however, it makes the model slow. The calculation time can be reduced by neglecting relatively fast effects (like exhaust gas recirculation effects) that have an insignificant contribution to the fuel consumption. In this report, a relatively simple dynamic approach is used: the main behavior of the system is described by mathematical equations.

3.2.2

Model orientation / causality

A forward facing model starts in this case at the driver: the truck responds to his control actions and undergoes some external influences, resulting in a vehicle speed. A model like that follows the natural path of occurrences. A backward facing model uses road information to calculate the power demand from the engine. This is a fast and easy way, but it is not able to handle feedback control problems, like the cruise control in this assignment. In this report, a backward facing model is used to get a quick understanding of the fuel saving potential of PCC.

3.3

Force balance

The model uses a force balance, as shown in Figure 3.1. It is based on Newton’s second law: the vehicle’s mass times its acceleration (m · a) is equal to the sum of all longitudinal forces

3.4 Model structure

9

acting on the vehicle. ΣF = −Fext − Fbrk + Feng = m · a,

(3.2)

which are external and engine force. Fext = Fdf + Fsl

(3.3)

Drag and friction force: Fdf = m · gCr cos(α) + 0.5ρair Cw Av 2

(3.4)

The gravitational force due to the slope is calculated using the road profile angle α: Fsl = m · g sin α

(3.5)

F eng

F brk + t

F ex

sr

α

yr

xr

Figure 3.1: Force balance when driving on a slope. The road angle α can be calculated from the road profile; Figure 3.1 shows the definitions of xr , yr , sr and α. dsr = v, dt α = arcsin

3.4

(3.6) 

yr sr



,

(3.7)

Model structure

The road height h and vehicle speed v are known, since they are prescribed in a backward facing model. Therefore, the drive force demanded from the transmission is equal to the sum of the other forces acting on the vehicle. Fdr = m · a + Fsl + Fdf .

(3.8)

3.5 Maximum slope performance

10

The wheel torque and speed can then be determined: Tw = Fdr Rw , v . Rw

ωw =

(3.9) (3.10)

Now, the engine torque and speed can be calculated directly using the transmission and differential efficiency, which is assumed to be constant. This efficiency describes the whole drive, including the differential. Tw , rg rf ηtrm

Te =

ωe = ωt rg rf .

(3.11) (3.12)

Finally, the engine’s fuel consumption can be determined using a Willans approximation [15]. This means that the fuel consumption (m ˙ f ) at each engine speed is approximated as a linear function of the engine torque. This method is further explained in Appendix A.

m ˙f =

ωe V d (pme + pml ), 4πLHV ηi

(3.13)

This result can be used to calculate the total amount of fuel used:

Vf =

3.5

Z

m ˙f dt. ρf

(3.14)

Maximum slope performance

In order to determine what test cases are realistic, the maximum possible slope first is determined. For simplicity, it is assumed that the truck only uses 12th (highest) gear and taking into account the maximum torque, this yields a maximum slope for each speed. Figure 3.2 shows this dependency for different masses. The effect of the maximum torque line can clearly be distinguished. It appears that for the case in this report (40 tonnes), the maximum slope is about 1.85 % at 80 km/h and 1.93 % at 72 km/h. These numbers seem very small, but the highest gear is mainly for cruising at flat roads. It corresponds with Fr¨oberg’s [1] findings, who also uses a fixed gear ratio (no shifting).

3.6 Sensitivity analysis

11

4.5 4 m = 20 tonnes m = 30 tonnes m = 40 tonnes

slope [%]

3.5 3 2.5 2 1.5 1 0.5 40

50

60

70 velocity [km/h

80

90

100

Figure 3.2: Maximum slope for different masses in 12th gear.

3.6

Sensitivity analysis

In order to gain some understanding of the system, a sensitivity analysis of the fuel consumption is performed. First, equations 3.8 to 3.9 are substituted in the equation for the engine torque (3.11):

Te = =

Tw Fdr Rw = rg rf ηtrm rg rf ηtrm (m·g sin(α)+m·gCr cos(α)+0.5ρair Cw Av2 )Rw rg rf ηtrm

.

(3.15)

As can be seen from this relation, (3.13) and (3.1), it is hard to solve this algebraically. Because of the Willans definition, the vehicle speed is present in all equations and an optimal speed profile cannot easily be derived, as the relations are non-linear. Therefore, an algorithm is needed to determine the optimal speed profile. This is discussed in the next section. From the equations above, it becomes clear that the vehicle speed, road angle and vehicle mass are important variables. These can vary over time or when the operating conditions change. In the figures below, only one of these three variables is varied. The others are assumed to be constant and equal their ”baseline” values: v = 80 [km/h], α = 0 [rad] and m = 40 [tonnes]. Also, a fixed gear is assumed: 12th gear (highest). The equations for these results are evaluated in Appendix B.

3.6 Sensitivity analysis

12

8 7

fuel flow rate [g/s]

6 5 4 3 2 1 0

0

10

20

30

40 50 velocity [km/h]

60

70

80

90

Figure 3.3: Vehicle speed sensitivity of fuel consumption.

14

12

fuel flow rate [g/s]

10

8

6

4

2

0 −2

−1.5

−1

−0.5

0 slope [%]

0.5

1

1.5

2

Figure 3.4: Slope sensitivity of fuel consumption.

It appears that the vehicle speed has a third order influence on the fuel consumption, whereas the road angle and vehicle mass have a (nearly) linear influence.

3.6 Sensitivity analysis

13

6

fuel flow rate [g/s]

5.5

5

4.5

4

3.5 10

15

20

25 mass [tonnes]

30

35

40

Figure 3.5: Mass sensitivity of fuel consumption.

Chapter 4

Numerical Optimization This chapter describes the algorithm functionality, what control method is chosen and how it is applied to the problem in this report.

4.1

Algorithm functionality

As described in the previous chapter, both SCC and PCC algorithms have the same function: minimize the fuel cost function (3.1). This does not mean that fuel consumption at any instant is minimized; fuel consumption might even increase at some parts of the route, as long as the fuel consumption for the whole trip is minimal. Since cruise control is used most of the time during long distance driving at constant speed (for instance at a speed of 80 km/h), a fixed gear ratio is assumed. It is assumed that all hills can be traveled using only 12th (highest) gear. The torque needed from the engine can be influenced by changing the desired vehicle speed. The algorithm should determine what reference speed profile leads to the lowest overall fuel consumption while still driving a reasonable average speed. As mentioned before, the travel time can be seen as a cost, too: if the average speed is low, the vehicle will be less profitable for the transport company. In the backward facing model, this output is handled as a prescribed speed profile. In a forward facing model, it would be offered as a reference speed to the cruise controller. The inputs for the algorithm are the set speed (driver input), vehicle speed and vehicle acceleration (time derivative of speed). These signals are present in the truck, so no extra sensors have to be implemented. In the case of a PCC, also a GPS signal is used as input.

4.2

Control choice

The equation in section 3.1 can be seen as a cost function. The method which minimizes this function is called optimal control. For the problem in this report, three optimal control types can be used: heuristic control, MPC and DP. 14

4.3 DP grid

15

Heuristic control is based on simple rules: the logic is prescribed by the programmer and a quick estimation of a solution is possible. However, it might be challenging to set up these rules. MPC, which stands for Model Predictive Control is a different type of optimal control. The controller uses real-time data and internal models to determine which control action has to be applied. It handles dynamics in the system well, but performance largely depends on the quality of the internal models. DP, or Dynamic Programming, also depends on its models, but really determines the optimal solution. No other controller type performs better, but DP is not the fastest: the number of calculations grows exponentially with the size of the problem. Bellman [11] calls this the curse of dimensionality. The algorithm calculates a cost-to-go matrix for every step: this matrix contains information of the cost to go from that system state to any other possible state. If the cost to go from one state to another is seen as having a length, the optimal solution then is the shortest path from the initial state to the final state. This is called a shortest path problem. It appears that there’s a tradeoff between calculation time and quality of the solution. This work investigates the potential of PCC and therefore, the quality of the solution is of greater importance than the needed calculation time. The solution will be determined offline and is not to be implemented in a truck directly. This makes DP the best control method for this problem: it indicates the maximum achievable fuel savings. The result can later on be used as a benchmark for suboptimal controllers.

4.3

DP grid

Berstekas [12] gives many examples on how to use DP in practice, but Sundstr¨om [10] provides two very complete examples: not only equations are given, but also the complete DP algorithm is described in a MATLAB environment. Therefore, these examples are used as a basis for this report. For the DP algorithm, several definitions have to be made concerning the grid and the constraints. One input and two states are defined here. A backwards facing model is used and the input is the vehicle speed. The cost function is defined as (3.1). The two states are the traveled distance and the previous velocity. This choice is explained in the next section. The DP grid structure is displayed in Table 4.1. Given the constraints, the possible inputs (velocities) can be interpreted as a search space, which is displayed in Figure 4.1. The green field contains inputs, whereas the gray areas indicate the infeasible points due to constraints on initial and final values. At every time step, the cost to go from one state to any other possible state is determined. Therefore, the number of operations is of order

O(N ) = N n2

(4.1)

4.3 DP grid

16 Table 4.1: Grid structure for DP algorithm.

Input: velocity Nu{1} number of grid points in input grid Un{1}.hi upper bound velocity Un{1}.lo lower bound velocity Problem Ts N

time step number of time steps in problem

State 1: traveled distance Nx{1} number of grid points in state grid Xn{1}.hi upper bound traveled distance Xn{1}.lo lower bound traveled distance X0{1} initial traveled distance XN{1}.hi upper bound final traveled distance XN{1}.lo lower bound final traveled distance State 2: previous velocity Nx{2} number of grid points in state grid Xn{2}.hi upper bound velocity Xn{2}.lo lower bound velocity X0{2} initial velocity XN{2}.hi upper bound penultimate velocity XN{2}.lo lower bound penultimate velocity

v.hi velocity

v.N

v.lo Time step

Figure 4.1: Search space for average velocity.

Hereby, N is the problem length and n is equal to the number of possible states.

4.4 DP function

4.4

17

DP function

In order to satisfy the constraints, the average velocity has te be calculated within the optimization. Therefore, the traveled distance is chosen as state: together with the problem length (in seconds), this determines the average velocity (according to the midpoint method). Also, the acceleration of the vehicle has to be determined for every step. Therefore, a second state is introduced, being the previous velocity. Together with the input (”new” velocity), the mean velocity, traveled distance and acceleration can be calculated: vold + v 2

(4.2)

∆x = v · ∆t

(4.3)

v=

a=

2(∆x − vold · ∆t) (∆t)2

(4.4)

The DP function describing the system returns new state variables and cost-to-go. The states and input are loaded at the start of this function. Also, the road profile is loaded. The current position is calculated as the product of the average velocity and the time traveled. Then, the road height is determined by interpolation in the loaded table. DP assumes that during one time step, the states are constant. Since the time step ∆t is 1 second, the calculations in the DP function are simple: the traveled distance in one step ∆x is equal to the speed value and the consumed fuel is equal to the fuel flow rate value. The DP function returns the new states, a cost-to-go matrix and a matrix I containing infeasible points. Solutions which not yield an allowed final state are automatically seen as infeasible. However, constraints can result in more infeasible points, which are stored in this matrix I. In this problem, I contains points where the maximum engine torque is exceeded. Figure 4.2 shows the structure of the cost matrix. The layers (gray) each represent the results for one time step: it contains the cost for every state and input. The marked positions (red) represent how some values are eliminated from the solution: these are the infeasible points.

4.5

Simulation setup

The DP algorithm described above can now be used in a simulation environment. First, a test case is defined, prescribing vehicle data, road profile and allowed velocities. The main file then exchanges information with the reference scenario: a vehicle driving at constant speed over the same road (using cruise control). The main file also prepares the DP problem grid and some settings. The DP routine and the reference both return fuel consumption and vehicle states. The main file outputs this and the calculated fuel savings. The workflow for this setup is displayed in Figure 4.3.

4.5 Simulation setup

18

input 1 to Nu

state 1 to Nx

Time

mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf

State

Input

Figure 4.2: Cost matrix.

DP

- average velocity - velocity band

- optimized fuel consumption

- total mass

- vehicle states

- gear - road - -DP DPgrid settings test case

main

- average velocity

savings and vehicle states

- velocity band - total mass

- average velocity

- gear

- total mass

- reference fuel consumption

- road

- gear

- vehicle states

- road reference

Figure 4.3: Workflow for simulation setup.

Chapter 5

Results 5.1

Test cases

In order to find how a fuel saving speed profile would look like, several test cases have been set up. Simulations are done for hills, valleys and simple steps up and down. Also, several different setups are investigated: average velocity, speed difference (between average and min/max value), gear and mass. The test cases are described in Table 5.1 and the results are discussed later on in this chapter. In simulations, the road angle is prescribed and used to determine the road load. The road height is determined by integration of the road angle over distance. For all cases (either up or down), a slope of 0.018 radians is used. This equals a slope of 1.8%. As mentioned in Section 3.5, the maximum slope for a fully loaded truck in 12th gear is 1.85%. A slope of 1% appeared to yield no significant savings, so a steeper profile has been chosen. A drawback is then that the downhill section is too steep for engine braking only: the truck will have to use a retarder and/or the normal brakes.

5.2

Performances

After simulation of all test cases, several results are interesting. This section discusses the computing performance, fuel savings and the trends that can be found in the speed profiles.

5.2.1

Computing performance

A well-known problem using DP algorithms is the explosion of the problem size when a detailed simulation is desired. A dual core processor (two times 2.50 GHz) in combination with 4 GB memory appeared to be insufficient to solve this problem. Some research learned that a 64-bit system is able to process larger amounts of data. A server was used for the simulations in this assignment. The specifications are displayed in Table 5.2. With a step size of 0.5[s] and a speed discretization of 0.125[m], a problem took about 75 minutes to be solved. An explanation for this large computing time lies in the fact that the DP algorithm 19

5.2 Performances

20 Table 5.1: Definition of the test cases.

case 1 2 3 4 5 6 7 8

v[km/h] 80 80 85 75 80 80 80 80

dv[km/h] 5 2.5 5 5 5 5 5 5

gear 12 12 12 12 12 12 11 10

mass [ton] 40 40 40 40 25 15 40 40

profile hill hill hill hill hill hill hill hill

9 10 11 12 13 14 15 16

80 80 85 75 80 80 80 80

5 2.5 5 5 5 5 5 5

12 12 12 12 12 12 11 10

40 40 40 40 30 20 40 40

valley valley valley valley valley valley valley valley

17 18

80 80

5 5

12 12

40 40

two hills two valleys

19 20

80 80

5 5

12 12

40 40

step up step down

investigates every step defined in the distance grid at every time instant. It does not take into account that only a few steps near the current step will yield a valid solution. Table 5.2: Server specifications. number of CPU’s CPU speed total memory swap total operating system MATLAB version

4 2660 MHz 32959644 KB 14635172 KB Linux 2.6 R2009b (64-bit)

5.2 Performances

5.2.2

21

Fuel savings

After simulation of all test cases, an overview of the fuel savings can be made. Figure 5.1 shows the result. Several things can be noticed here. First of all, the high peak at case 20 (single step down). In combination with the fact that case 19 (single step up) appears to save nothing at all, one can draw the conclusion that fuel saving is realized at downhill parts. The savings profile of case 1-8 looks the same as the one of case 9-16. The latter covers the same cases, but on the valley profile. Cases 6 and 14 save nothing at all: these represent empty trucks. Apparently, the truck has to be heavy in order to save fuel. If one does not take into account the single step cases (19, 20), an average fuel saving of 4.76% is realized for these short test cases. 18 16 14

fuel savings [%]

12 10 8 6 4 2 0 −2

0

2

4

6

8

10 12 case number [−]

14

16

18

20

Figure 5.1: Fuel savings per case.

5.2.3

Trends

By studying the results of the simulations, one can find several trends in the speed profiles. Figure 5.2 shows the results for different speeds on a hill. The profile looks the same: deceleration uphill and acceleration downhill. The speed is raised before the hill in order to be able to decelerate all the way up. Velocity increases again at the downhill section, making sure the average velocity is met. The difference in road height can be explained by investigating the definition: the traveled distance is used to look up the road angle by interpolating in a prescribed vector. This experienced road angle is then used to determine the road height, causing a small deviation in some cases. For different masses, a similar trend can be found: deceleration uphill and acceleration downhill, as can be seen in Figure 5.3. Because of the lower inertia, a lighter vehicle accelerates faster and can reach a higher velocity before the hill. Therefore, the amplitude of this profile

5.2 Performances

22

90

80 km/h 85 km/h 75 km/h

velocity [km/h

85 80 75 70

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

2

height [m]

1.5 1 0.5 0 −0.5

Figure 5.2: Results for different speeds on a hill.

is somewhat larger. The same holds for different gears (Figure 5.4): lower gears enable faster acceleration and therefore a larger amplitude. 86

40 tons 25 tons 15 tons

velocity [km/h

84 82 80 78 76

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

2

height [m]

1.5 1 0.5 0

Figure 5.3: Results for different masses on a hill.

5.2 Performances

23

84

gear 12 gear 11 gear 10

velocity [km/h

83 82 81 80 79 78 77

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

2

height [m]

1.5 1 0.5 0

Figure 5.4: Results for different gears on a hill.

5.2 Performances

24

Simulations of the valley profile show the same trends again: acceleration downhill, deceleration uphill and larger amplitudes for lighter vehicles. Figure 5.5 shows the results for different speeds and Figure 5.6 for different masses. Something remarkable occurs for the 20 tons case: the profile is totally different. This is due to problems in the optimization. For empty trucks, the fuel saving is approximately zero and apparently several solutions are available for zero savings.

velocity [km/h

90

80 km/h 85 km/h 75 km/h

85 80 75 70

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

0.5

height [m]

0 −0.5 −1 −1.5 −2

Figure 5.5: Results for different speeds in a valley.

83

40 tons 30 tons 20 tons

velocity [km/h

82 81 80 79 78 77

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

0

height [m]

−0.5 −1 −1.5 −2

Figure 5.6: Results for different masses in a valley.

5.2 Performances

25

Figure 5.7, finally, shows the results for different gears. The same trends can be found once more: acceleration downhill, deceleration uphill and larger amplitudes for lower gears. 83

gear 12 gear 11 gear 10

velocity [km/h

82 81 80 79 78 77

0

50

100

150

200

0

50

100

150

200

250

300

350

400

450

500

250 300 distance [m]

350

400

450

500

0

height [m]

−0.5 −1 −1.5 −2

Figure 5.7: Results for different gears in a valley.

5.2.4

Fuel saving principle

In order to explain the fuel saving principle, case 1 is taken as an example. Other cases show similar results. Figure 5.8 shows detailed results of this case. It is clear that the engine torque is very low for the deceleration phase. The fuel saving principle is already indicated in Section 5.2.2, as it appeared that mainly heavy loaded trucks on downhill sections are able to save fuel. Figure 5.9 shows the engine map for case 1, in which it becomes clear that for constant speed, the desired torque is lower than what is possible by engine braking. Therefore, the service brakes have to be activated. This type of energy conversion is not encountered by using the PCC algorithm in this report. In order to determine whether this is the fuel saving principle, the energy flows are depicted in Figures 5.10 and 5.11. Here, it becomes clear that using PCC no energy is lost by braking. Moreover, the engine appears to operate more efficiently, as less engine losses occur using predictive cruise control. PCC thus aims at more efficient engine use and braking prevention by changing the speed profile: a low speed at the start of a downhill section, combined with a higher allowed speed, result in a profile without use of brakes.

5.2 Performances

26 case 1

v [km/h]

90 80 70

0

50

100

150

200

250

300

350

400

450

500

−2 0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250 300 distance [m]

350

400

450

500

0

50

100

150

200

250 300 distance [m]

350

400

450

500

Te [Nm]]

h [m]

2 0

2000 1000 0

mf [g]

15 10 5 0

Figure 5.8: Detailed results for case 1.

5

3

x 10

14

2.5 12

engine power [Nm*rad/s]

2 10

fuel lines [g/s] constant speed optimized speed

1.5

1

8

0.5 6 0 4

−0.5

−1 110

115 120 engine speed [rad/s]

125

Figure 5.9: Operating points in the engine map.

Rolling resistance 1.12 [MJ]

Air drag 0.96 [MJ]

27

Braking 0.08 [MJ]

5.2 Performances

Fuel 5.62 [MJ] Engine losses 3.46 [MJ]

Rolling resistance 1.12 [MJ]

Air drag 0.96 [MJ]

Figure 5.10: Energy flows for the reference.

Fuel 5.34 [MJ] Engine losses 3.26 [MJ]

Figure 5.11: Energy flows for PCC.

Chapter 6

Conclusions and Recommendations 6.1

Conclusions

The goal of this assignment was to investigate the potential of a new way to save fuel: predictive cruise control, which allows small deviations from the set speed. A DP algorithm, developed by Sundstr¨ om and Guzzella [10], was used to determine the optimized speed profile en the corresponding fuel savings. Simulations show that the fuel saving is realized by more efficient use of the engine and minimization of brake use. It appears that fuel saving is therefore achieved by going downhill at optimized speeds. Decelerating or selecting a low speed is preferred for uphill road sections, whereas acceleration is preferred for downhill sections. The average fuel savings for different scenario’s in this assignment 4.76%, which corresponds with the findings in literature. Because of the complexity of the DP algorithm, many assumptions had to be done, but if future work manages to simplify the algorithm, many alternatives are possible. One could think of gear shifting and larger road profiles.

6.2

Recommendations

The first and most important recommendation is to simplify the provided DP algorithm. If there is a way to stop the algorithm from calculating every single possible step, the needed memory and processor load improvements are immense. This could also involve writing a new algorithm, being either DP or something similar. Calculation times with the current files are too long to deal with properly. Selecting or writing a new algorithm should take into account real-time performance: it is preferable to calculate the optimum online (in-vehicle). Furthermore, one should investigate how the signals can be provided to the algorithm. The vehicle data and the road profile data should be available and accessible. Also, one should consider implementing more vehicle sensors or state estimators: it seems to be beneficial to know the total vehicle weight, as this has consequences for the ideal speed profile.

28

Bibliography [1] A. Fr¨ oberg, E. Hellstr¨ om, L. Nielsen: Explicit Fuel Optimal Speed Profiles for Heavy Trucks on a Set of Topographic Road Profiles, 2006 SAE World Congress - Detroit, Michigan - SAE Technical Paper 2006-01-1071. [2] E. Hellstr¨ om, A. Fr¨ oberg, L. Nielsen: A Real-Time Fuel-Optimal Cruise Controller for Heavy Trucks Using Road Topography Information, 2006 SAE World Congress - Detroit, Michigan SAE Technical Paper 2006-01-0008. [3] E. Hellstr¨ om, Master’s Thesis: Explicit Use of Road Topography for Model Predictive Cruise Control in Heavy Trucks, Link¨ opings Universitet - Reg. nr. LiTH-ISY-EX–05/3660–SE. [4] E. Hellstr¨ om, M. Ivarsson, J. Slund, L. Nielsen: Look-ahead Control For Heavy Trucks to Minimize Trip Time and Fuel Consumption, Control Engineering Practice, Volume 17, Issue 2, February 2009 - pages 245-254 [5] Scania press release on Ecocruise: http://www.scanianewsroom.com/2009/02/23/master-of-thehills, February 2009. [6] F. Lattemann, K. Neiss, S. Terwen, T. Connolly: The Predictive Cruise Control - A System to Reduce Fuel Consumption of Heavy Duty Trucks, Commercial Vehicle Engineering Congress and Exhibition - Rosemount, Illinois USA - SAE Technical Paper 2004-01-2616. [7] Neiss et al.: Predictive Speed Control for a Motor Vehicle, United States Patent, no.: US 6,990,401 B2. [8] R. Walker Sentience - Using Electronic Horizon Data to Improve Hybrid Vehicle Fuel Economy, Transport Research Laboratory, 2008 - ISBN: 978-0-86341-920-1 (via IEEE). [9] M. Barth, G. Scora, T. Younglove: Intelligent Off-Board Management of Vehicle operating Parameters, University of California, 2003 - IEEE: 0-7803-8125-4/03. [10] O. Sundstr¨ om, L. Guzzella: A Generic Dynamic Programming Matlab Function, 18th IEEE International Conference on Control Applications - Saint Petersburg, Russia, 2009 - ISBN: 9781-4244-4602-5/09 (via IEEE). [11] R. Bellman: Dynamic Programming, Dover Publications, 2003 - ISBN: 0-486-42809-5. [12] D. Berstekas: Dynamic Programming and Optimal Control, vol. 1, Athena Scientific, 1995 - ISBN: 1-886529-12-4. [13] R. Giannelli, E. Nam, K. Helmer, T. Younglove, G. Scora, M. Barth: Heavy-Duty Diesel Vehicle Fuel Consumption Modeling Based on Road Load and Power Train Parameters, SAE International 2005 - Paper Offer #: 05CV-3.

29

BIBLIOGRAPHY

30

[14] L. Guzzella, A. Sciaretta: Vehicle Propulsion Systems: Introduction to Modeling and Optimization, Springer 2005 - ISBN: 3-540-25195-2. [15] L. Guzzella, C. Onder: Introduction to Modeling and Control of Internal Combustion Engine Systems, Springer 2004 - ISBN: 3-540-22274-3.

Appendix A

Powertrain Modeling A.1

Engine model

A Willans curve methodology is used to model the engine’s friction and efficiency parameters. The engine specifications and some measurement data for the MX340 engine are provided by DAF. Below, four different approximations are described, each having a different number of parameters. These parameters are optimized with the fminsearch routine in MATLAB, minimizing the squared difference between the model and the measurements. This way, opposing errors cannot compensate each other and big deviations are punished harder than small deviations. Because of the confidential nature of parameter values, these are left out. However, the mean squared error and graphic results are given.

A.1.1

Model 1

Equation 3.13 follows the model of Guzzella [15], but uses a slightly different notation. A rather complex model with many parameters is provided, but first, a simple model is tested: only the two main parameters (ηi and pml ) are optimized:

m ˙f =

A.1.2

ωe V d (pme + pml ). 4πLHV ηi

(A.1)

Model 2

Model 2 follows the paper of Giannelli [13]. The fuel consumption is described as a function of friction (depending on engine speed) and engine efficiency (constant). The main difference is that this approach has a higher order dependence on the engine speed. Note that this is not in S.I. units: Giannelli uses [L], [kPa], [kJ/kg] and [kW]. The three parameters are the

31

A.1 Engine model

32

indicated engine efficiency and two friction parameters: (ηi , k0 and k1 )   kne Vd,t 1 4000πne Te m ˙f = + 1000 Nr LHV Nr LHV ηi

[kg/s],

(A.2)

with the friction parameters: k = k0 + k1 ne .

A.1.3

(A.3)

Model 3

Model 3 is basically the same as model 2, but uses a more extensive definition of the friction. Here, also a dependency on the mean piston speed is included, which makes the model use four parameters: k2 is added.

2

k = k0 + k1 ne + k2 U p .

A.1.4

(A.4)

Model 4

Model 4 is an extension of model 1 and uses five parameters, describing the indicated engine efficiency, some friction coefficients and a gas exchange component. Note that the parameter numbers do not match Guzzella’s definition, as some of them can be combined: ηi , k1 , k2 , k3 and pmlg . m ˙f =

ωe V d (pme + pml ) 4πLHV ηi

[kg/s],

(A.5)

with pml = pmlf + pmlg

(A.6) r

= k1 + k2 s2 ωe2 Πe,max

A.1.5

k3 + pmlg . B

Comparison

The models are compared using the mean value of the squared error, which is given below. Figure A.1 shows the error distribution. Note that this is not the squared error, since that would not show whether the model more often results in a higher or lower fuel consumption. It appears that model 4 matches the measurements best, although it has the tendency to result in a slightly smaller fuel consumption. Figure A.2 shows this in a different way; the power input and power output from the engine are displayed here, also clearly showing that model 4 matches the measurements best.

A.1 Engine model

33

Table A.1: Engine model performance. Model

Mean squared error

Model Model Model Model

0.1498 0.1455 0.1340 0.0436

1 2 3 4

Model 1

Model 2 60 Frequency

Frequency

60 40 20 0 −1

−0.5

0 error Model 3

0.5

−0.5

0 error Model 4

0.5

1

−0.5

0 error

0.5

1

60 Frequency

Frequency

20 0 −1

1

60 40 20 0 −1

40

−0.5

0 error

0.5

1

40 20 0 −1

Figure A.1: Error distribution for all models.

A.1 Engine model

34

Model 1

800

800

700

700

600

600

500 400 300

400 300 200

100

100 −50

0

50

100 150 200 Engine Power [kW]

250

300

0 −100

350

Model 3

900 800

800

700

700

600

600

500 400 300

100 50

100 150 200 Engine Power [kW]

250

300

350

100 150 200 Engine Power [kW]

250

300

350

250

300

350

Model 4

300 200

0

50

400

100 −50

0

500

200

0 −100

−50

900

Fuel Power [kW]

Fuel Power [kW]

500

200

0 −100

Model 2

900

Fuel Power [kW]

Fuel Power [kW]

900

0 −100

−50

0

50

100 150 200 Engine Power [kW]

Figure A.2: Willans lines for all models.

A.2 Transmission

A.2

35

Transmission

12 AS 2540 TD

ZF-AS Tronic (Truck)

The DAF XF that is used as an example in this report uses a 12-speed automatic ZF AS-Tronic transmission. This transmission is from the company ZF, type 12 AS 2540 TD (depicted in Automatic transmission heavy Figure A.3). The final drivesystem ratio forfor this trucktrucks is 2.69 and a constant transmission efficiency of 95% is assumed (including differential).

Figure A.3: ZF AS-Tronic automatic 12 AS 2540 TD - Basic transmission 12 AS 2541 TD - Transmission with Intarder 12 AS 2545 TD - Transmission with clutch-dependen t PTO

transmission (AMT).

Table A.2: Gear ratio’s for the ZF 12 AS 2540 TD. 1 2 3 Transmission type 9.57 15.86 12.29

4 7.41

5 5.89

6 4.57

7 3.47

8 2.69

9 2.10

10 11 12 AS 25401.29 TD 1.62

12 1.00

R1 14.68

R2 11.38

max. 2500 Nm

Input torque Weight * without auxiliaries

≈ 256 kg

Oil capacity for initial filling

≈ 13 dm 3 as specified in ZF List of Lubricants TE-ML 02

Oil grade

integrated

Clutch bell housing SAE 1

≈ 70 kg

Additional weight for ZF-Intarder

≈ 10 dm 3

Additional oil quantity for ZF-Intarder *without oil fill, clutch and clutch release compon

ents

2

3

4

5

6

7

8

9

10

11

12

R1

R2

12.29

9.57

7.41

5.89

4.57

3.47

2.69

2.10

1.62

1.29

1.00

14.68

11.38

see also Data Sheet for Intarder: IT_AS_Tronic_Truc

Subject to technical change without notice.

k_EN

ZF Friedrichshafen AG • Lkw-Antriebstechnik • 88038 Friedrichshafen/Germany Tel. +49 7541 77- 0 • Fax +49 7541 77- 908000 • www.zf.com/truck-transmission

EN

1 15.86

2008-09

Standard ratios in gear

Appendix B

Sensitivity Analysis In order to determine the effect of vehicle speed, road slope and vehicle mass, calculations on the fuel consumption are performed here. First, the governing equations are given. In order to determine the fuel consumption, also the engine speed is needed, but that is directly dependent on the vehicle speed ( ωe = vrg rf /Rw ). A fixed gear is assumed: 12th gear (highest) and the three variables are seen as inputs here and have standard values v = 80[km/h], α = 0[rad] and m = 40[tonnes]. Equation 3.13 is repeated here:

ωe V d (pme + pml ) 4πLHV ηi   ωe Vd 4πTe + p = 4πLHV ml ηi Vd

m ˙f =

(B.1)

Equations 3.8 to 3.9 are substituted in the equation for the engine torque (3.11): Te = =

Tdr Fdr Rw = rg rf ηtrm rg rf ηtrm (mg sin(α)+mgCr cos(α)+0.5ρair Cw Av2 )Rw rg rf ηtrm

36

(B.2)

B.1 Vehicle speed sensitivity

B.1

37

Vehicle speed sensitivity

The velocity dependence is determined (standard slope and weight). It appears that the dependency is of third order. Figure B.1 shows the result.
mgCr + 0.5ρair Cw Av 2 Rw (B.3) Te (v) = rg rf ηtrm After combining this with B.1, this yields

!
4πRw mgCr + 0.5ρair Cw Av 2 + pml Vd rg rf ηtrm

vrg rf Vd m ˙ f (v) = 4πRw LHV ηi = V1 v + V2 v 3

(B.4)

with constants 1 V1 = ηi LHV

Vd rg rf pml mgCr + ηtrm 4πRw



0.5ρair Cw A ηi ηtrm LHV

8 7 6 fuel flow rate [g/s]

V2 =



5 4 3 2 1 0

0

10

20

30

40 50 velocity [km/h]

60

70

80

90

Figure B.1: Vehicle speed sensitivity of fuel consumption.

B.2 Slope sensitivity

B.2

38

Slope sensitivity

The slope dependence is determined (standard velocity and weight). It appears that the fuel consumption depends on the sine and cosine of the road angle. However, since the angles are small, the dependency seems linear, as can be seen in Figure B.2. Substitution of B.2 in B.1:

vrg rf Vd m ˙ f (α) = 4πRw LHV ηi

!
4πRw mg sin(α) + mgCr cos(α) + 0.5ρair Cw Av 2 + pml Vd rg rf ηtrm

= A1 + A2 cos(α) + A3 sin(α)

(B.5)

with constants v A1 = LHV ηi

0.5ρair Cw Av 2 rg rf Vd pml + ηtrm 4πRw



A2 =

vmgCr LHV ηi ηt rm

A3 =

vmg LHV ηi ηt rm



(B.6)

14

12

fuel flow rate [g/s]

10

8

6

4

2

0 −2

−1.5

−1

−0.5

0 slope [%]

0.5

1

1.5

2

Figure B.2: Slope sensitivity of fuel consumption.

B.3 Mass sensitivity

B.3

39

Mass sensitivity

The weight dependence is determined (standard velocity and slope). It appears that the dependency is of first order. Figure B.3 shows the result. vrg rf Vd m ˙ f (α) = 4πRw LHV ηi

!
4πRw mg sin(α) + mgCr cos(α) + 0.5ρair Cw Av 2 + pml Vd rg rf ηtrm

= M1 + M 2 m

(B.7)

with constants v M1 = LHV ηi

0.5ρair Cw Av 2 rg rf Vd pml + ηtrm 4πRw



vgCr LHV ηi ηtrm (B.8)

6

5.5

fuel flow rate [g/s]

M2 =



5

4.5

4

3.5 10

15

20

25 mass [tonnes]

30

35

40

Figure B.3: Mass sensitivity of fuel consumption.

Appendix C

Nomenclature Symbol

Unit

Description

A a Cr Cw Fbrk Fdf Fdr Feng Fext Fsl g h Jf uel k LHV m m ˙f ne Nr pme pml Rw rf rg sr Te Tt Up v

[m2 ] [m/s2 ] [−] [1/m2 ] [N ] [N ] [N ] [N ] [N ] [N ] [m/s2 ] [m] [L/100km] [−] [J/kg] [kg] [kg/s] [1/s] [−] [P a] [P a] [m] [−] [−] [m] [N m] [N m] [m/s] [m/s]

Vehicle frontal area Vehicle acceleration Rolling resistance coefficient Air drag resistance coefficient Brake force Drag and friction force Drive force Engine force External forces Force due to slope Gravitational acceleration Road height Cost function for fuel consumption Willans line parameter Fuel lower heating value Total vehicle weight Fuel flow rate Engine speed in rotations per second Number of revolutions per cycle Mean effective pressure Mean pressure loss Driven wheel radius Final drive ratio Gear ratio Traveled distance Engine torque Transmission torque Mean piston speed Vehicle speed 40

41 Symbol

Unit

Description

Vd Vf yr

[m3 ] [L] [m]

Displacement volume Volume of used fuel vertical road length

α ηi ηtrm ρair ρf ΣF ωe ωt

[rad] [−] [−] [kg/m3 ] [kg/m3 ] [N ] [rad/s] [rad/s]

Road profile angle Indicated engine efficiency Overall transmission efficiency Air density Fuel density Sum of all longitudinal forces acting on the vehicle Engine speed Transmission speed