Optimization Tool for Gear Shift Strategy Control Design

D A V I D H U L T M A N

Master of Science Thesis Stockholm, Sweden 2014

Optimization Tool for Gear Shift

Strategy Control Design

D A V I D H U L T M A N

Master’s Thesis in Systems Engineering (30 ECTS credits) Master Programme in Aerospace Engineering (120 credits) Royal Institute of Technology year 2014

Supervisor at Volvo construction equipment was Björn Brattberg Supervisor at KTH was Per Engvist Examiner was Per Engvist TRITA-MAT-E 2014:63 ISRN-KTH/MAT/E--14/63--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

Modern construction vehicles are required to perform very well in tough conditions.

It is therefore of great importance that they have well developed gear shift strategies

to manage with these requirements. These strategies are updated often and they are

normally based on empirical knowledge and extensive testing. It is of interest to know

if software tools can be used to provide theoretical analysis to further modernize and

improve the gear shift strategy design process.

This thesis investigates this possibility by creating an optimization tool that can

provide various optimized simulations. These simulations finds the theoretical optimal

velocities and gear shifts for driving specific distances. It is done by creating a vehicle

model that is used to form an optimal control problem. This optimal control problem

uses a dynamical programming algorithm as a solver and the results are optimized

simulations.

The optimization tool has the possibility to investigate ways to minimize travel

time, fuel consumption and durability losses. It is concluded that there are ways to

drive that can reduce the fuel consumption and durability losses by a large amount

while not affecting the travel time very much.

ii

Sammanfattning

Moderna byggfordon har hoga krav pa att prestera val i tuffa miljoer. Det ar darfor

viktigt att de har val utvecklade vaxlingsstrategier sa att dessa krav kan uppfyllas.

Strategierna ar uppdaterade ofta och bygger normalt pa empirisk kunskap samt nog-

granna tester. Det ar utav intresse att undersoka om mjukvaruverktyg kan anvandas

for att gora teoretiska analyser som kan modernisera och forbattra utvecklingspro-

cessen utav vaxlingsstrategier.

Det har arbetet undersoker den har mojligheten genom att skapa ett optimer-

ingsverktyg som kan bidra med diverse optimerade simuleringar. Dessa simuleringar

gors genom att hitta de teoretiskt basta hastighetsprofilerna samt vaxlingar for att

kora specifika strackor. Detta mojliggors genom skapandet utav en fordonsmodell

som sedan anvands till att formulera ett optimeringsproblem. Optimeringsproblemet

loses genom en dynamisk programmeringsalgoritm.

Det resulterande optimeringsverktyget kan undersoka satt att minimera kortid,

bransleforbrukning samt hallbarhetsforluster. Det visas att det finns satt att kora som

kan minska bransleforbrukningen och hallbarhetsforlusterna mycket medans kortiden

bara okas lite.

iii

Acknowledgements

This work has been performed at Volvo Construction Equipment in Eskilstuna. I

have gotten extensive support from several individuals here and I have made some

great friends. I am very grateful for all the help I have received and the friendly

environment that exists here. Firstly I would like to thank my supervisor at Volvo,

Bjorn Brattberg for always providing the assistance I have needed. I would like to

thank Anders Froberg for the invaluable tips and feedback and for providing me with

the opportunity to come here. Others that have helped me and deserve a thanks are

Anton Hugo, David Berggren and Mats Akerblom. A thanks goes out to the entire

department at Controls Driveline and everyone else located at the floor for making

me feel so welcome. I am also very grateful from the help I have gotten from KTH

through the work of my supervisor there, Per Enqvist.

I would also like to thank my sisters Marta and Natalia for their support, and

lastly, Magdalena Sierant for brightening my days.

David Hultman

Eskilstuna, October 2014

iv

Contents

1 Introduction 2

1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Construction Equipment 4

2.1 Articulated Haulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Wheel Loaders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Motor Graders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Modeling 8

3.1 Powertrain Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 The Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.2 The Torque Converter . . . . . . . . . . . . . . . . . . . . . . 10

3.1.3 The Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.4 Gear Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.5 Total Powertrain Ratio and Inertia . . . . . . . . . . . . . . . 14

3.1.6 The Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.7 Resisting Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.8 Longitudinal motion . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Criterion Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Travel Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.3 Durability Losses . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.3.1 Fatigue of rolling bearings . . . . . . . . . . . . . . . 18

v

3.2.3.2 Clutch Wear . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3.3 Total Durability Losses . . . . . . . . . . . . . . . . 19

4 The Optimal Control Problem 20

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Designing and choosing the weight and scale parameters . . . 21

4.2 Dynamical Programming . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 The Inverse Method . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . 24

4.2.3 State Space Representation . . . . . . . . . . . . . . . . . . . 25

4.2.4 Limiting the Search Space . . . . . . . . . . . . . . . . . . . . 26

5 Simulations and Analysis 27

5.1 Model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Gear Shift Model Analysis . . . . . . . . . . . . . . . . . . . . 28

5.2 Optimized Acceleration From Standstill . . . . . . . . . . . . . . . . . 30

5.2.1 Minimizing Travel Time . . . . . . . . . . . . . . . . . . . . . 31

5.2.2 Minimizing Fuel Consumption and Durability Losses . . . . . 32

5.3 Pareto Fronts of Criteria Trade-off’s . . . . . . . . . . . . . . . . . . . 34

5.3.1 Travel Time versus Fuel Consumption . . . . . . . . . . . . . 34

5.3.2 Varying All Criteria Weights . . . . . . . . . . . . . . . . . . . 36

6 Conclusions 37

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7 References 40

1

Chapter 1

Introduction

Creating intelligent gear shift schedules is of great importance to the design of any

vehicle using an automatic gearbox. It is a key part in generating desired and effi-

cient performance, and much effort is put into finding the best gear shifting strategies.

These strategies highly affect important properties like fuel consumption and travel

time, but also some less noticeable ones, like gearbox lifetime.

The conventional way to implement such strategy schedules into the automatic

gearboxes is to do it through the use of shift maps. These maps consist of a schedule

of points which tells the transmission when to shift up or down based mainly on the

velocity and power output of the vehicle. The traditional way to create these is to do

it based on empirical knowledge and extensive testing [1]. Passenger cars often have

3 different gear shifting maps, corresponding to the balanced, eco and sport modes.

Off-road construction vehicles which drive in a more complex way in unpredictable

terrain can have up to as many as 10 different maps.

Research is being done in several fields to find ways to improve and optimize the

gear shifting strategies. It is often focused towards passenger cars or long-range trucks

with focus on reducing fuel consumption. Previous work includes [2], where a Dy-

namical Programming (DP) algorithm is developed to numerically solve optimization

problems on fuel consumption in trucks. A similar algorithm is used in [1] to create

a gear shift schedule for a passenger car that results in lowering its fuel consumption

by up to 11.2%. In [3], the concept is further developed to take into consideration

2

not only fuel consumption, but also driveability.

This thesis will look into the possibility of using a DP algorithm as help in the

design process of creating gear shifting maps for off-road construction vehicles. A

software tool will be developed that can find the theoretically optimal ways to drive

the vehicles over specific distances and terrain based on certain criteria. The criteria

will cover not only fuel consumption and travel time, but also durability losses of the

gearbox and other components.

Extending the lifetime of different vehicle parts is something that is of high interest

for both manufacturers and users, but is often disregarded when designing gearshift

maps. In this work, a model for durability losses is implemented. This gives the

possibility to analyse the impact that different drive styles have on vehicle lifetime.

The vehicles covered in the scope of this thesis are articulated haulers, wheel

loaders and motor graders.

1.1 Outline of the thesis

The chapters of this report are organized as following: Chapter 2 introduces the three

different construction vehicles which are covered in the scope of this work and de-

scribes their areas of usage. It is followed by a chapter where a generalized vehicle

model is created. This model is made to be able to simulate all three different vehicle

types described previously. The way in which the different optimization criteria are

modeled is also presented. In Chapter 4, an optimal control problem is formed and

the dynamical programming algorithm that is used as a solver is described. This

control problem uses the model created in the precedent chapter and is able to sim-

ulate various different ways to drive the construction vehicles in an optimal way. A

description of the numerical complexity is also given. In Chapter 5 various example

simulations are presented and some analysis is done. A model verification is made to

show how well the model of this work compares to previous test data. This is followed

by the last chapter which cover conclusions and recommendations for future work.

3

Chapter 2

The Construction Equipment

The optimization tool which is created in this thesis is applicable to 3 different con-

struction equipment vehicles: haulers, loaders and graders. Their work tasks and

objectives differ from each other and this chapter will give a description of each of

the vehicle types.

2.1 Articulated Haulers

A hauler is similar to a dump truck, but is designed to be driven off-road over rough

terrain. Its objective is to transport material from one point to another and its

most common loads include gravel, stone and soil. The vehicle consists out of two

parts, the front tractor including the driver cab and the back trailer with the dump

body. The two parts are steered relative to each other by articulation, providing good

path-tracking which is desired in off-road driving.

4

Figure 2.1: A Volvo A40F Articulated Hauler

2.2 Wheel Loaders

The objective of a wheel loader is mainly to load material. See Figure 2.3 for an

illustrated scheme of one of the most common drive cycles: the short loading cycle.

The wheel loader can be equipped with a wide range of attachments including buckets,

forks, grapples and more. The most common material it loads include gravel, stone

and soil, but it can also be used for timber and much more. A wheel loader often

loads material into a hauler which then transports it.

5

Figure 2.2: A Volvo L250H Wheel Loader

Figure 2.3: A short loading cycle, picture from [4]

6

Figure 2.4: A Volvo G940B Grader

2.3 Motor Graders

A grader is a type of vehicle which is used mainly for creating gravel roads, preparing

the ground before creating an asphalt road or, in some cases, for snow ploughing and

similar activities. Some variations in equipment exist, but most commonly it has one

large blade which it uses to grade the road.

7

Chapter 3

Modeling

In this chapter, a mathematical model of the necessary parts needed to derive the

longitudinal motion of the vehicles is created. This is equal to modeling a powertrain

and the external forces acting on it. The resulting equation of the longitudinal motion

is what will be used to simulate the driving of the vehicles and it is created with

a level of detail which will suit the complexity of the optimization problem. The

modeling process will be similar to that described in [5] with addition to a torque

converter. The torque converter equations will be created like in [6] and [7] together

with experimental data taken from Volvo CE.

3.1 Powertrain Modeling

A Powertrain is a combination of all the parts that are used to generate power and

transfer it to the ground. It consists of the engine, shafts, torque converter, gear-

box, axles and the wheels. An illustration of the layout of a powertrain is show in

Figure 3.1.

3.1.1 The Engine

A combustion process inside the engine consumes fuel and generates torque to an

outgoing shaft. This generated torque is dependent on the current angular rotation

8

Figure 3.1: A Powertrain

speed of the engine, ωe, and the amount of fuel combusted, uf . It is assumed that

at any given point, any desired torque level can be reached within the limitations of

the engine by adding the related amount of fuel. Transient behaviour is disregarded.

This allows the outgoing engine torque to be described as

Te = f(ωe, uf ) (3.1)

A typical map of the maximum torque available as a function of engine speed is

shown in Figure 3.2

9

Figure 3.2: An example of engine performance

3.1.2 The Torque Converter

A torque converter (TC) is a type of fluid coupling with the ability to multiply torque.

It is positioned between the engine and the gearbox and it is used in vehicles with

automatic transmissions as a replacement of the traditional clutch used for manual

transmissions. Where such a clutch normally has friction plates to connect the shaft

going out from the engine to the shaft going in to the gearbox, a TC instead connects

the two through a hydraulic fluid. It consists of a container of oil where both shafts

meet and transfer torque through the fluid. The engine shaft going into the TC is

connected to an impeller which makes the oil rotate. The shaft going out of the TC,

towards the gearbox, is connected to a turbine that gains a rotation because of the

10

movement of the oil. This construction allows the two shafts to have different rotation

speeds and therefore prevents engine stalls at low vehicle speeds. It removes the need

of disconnecting the engine and the transmission every time the vehicle stops.

The speed difference is defined as

Φ =ωTC

ωe

(3.2)

where ωTC is the rotation speed of the shaft going out of the TC.

Inside the TC, there is a stator which ensures that the flow of the returning fluid

from the turbine is directed into the same direction as the impeller is rotating. This

ensures that the returning fluid does not slow down the impeller and cause energy

losses. Another effect of this is that at a large difference in rotation speeds of the two

shafts, the transferred torque is multiplied.

As described in [6], the speed difference Φ plays a big role in how much torque is

multiplied. The converter output torque TTC is determined by the relation

TTC = ξ(Φ) · Te (3.3)

where ξ is a function that must be experimentally calculated for each converter

and will be modeled using data from Volvo CE. A typical graph of how much torque

is being multiplied as a function of Φ is shown in Figure 3.3

11

Figure 3.3: An example of torque converter performance

The TC has a built-in lockup clutch system which gives the possibility of locking

the two rotating shafts together into a fixed one and transfer the torque directly

instead of through the fluid. This is because in some cases this gives a better efficiency.

In general the TC is most useful at low vehicle speeds and in situations where frequent

stops and starts occur. At higher vehicle speeds the two rotating shafts reach almost

the same speed and then it is often preferred to lock them together to remove slippage

and additional losses caused by the TC. Lockup is engaged at different situations in

different vehicles. Some engage it directly after start and some never do it.

The lockup will be modeled as a simple switch which can change between leading

12

the power through the TC or through a locked shaft. TTC = Te if Lockup is engaged

TTC = ξ(Φ) · Te if Lockup is disengaged

3.1.3 The Gearbox

The next component in the powertrain is the gearbox. It consists of a number of

gears of different sizes and enables a discrete choice of the ratio between ingoing

and outgoing angular velocity of the shafts connected to it. This allows the speed

conversion between the engine shaft and the wheel axles to be controlled, making

it possible to provide more accurately desired torque values to the wheels at a wide

range of vehicle velocities.

The gearbox is modeled as a discrete choice of the gear ratio igear(g), where g is

the current gear. The corresponding torque and angular rotation speed equations are

given by

TGB = igear(g) · TTC (3.4)

ωGB =ωTC

igear(g)(3.5)

where TGB and ωGB are the output torque and angular rotation speed from the

gearbox.

3.1.4 Gear Shifts

The process of changing the gear ratio is done during a gearshift. This is traditionally

done by disconnecting the gearbox axle at a mechanical clutch and then connecting

it again when the gear ratio has been changed. Such a disconnection creates an

undesired drop in the transferred torque and in modern automatic gearboxes a so

called powershift technique is used instead. This way of shifting is done in a way

in which the current gear is being disconnected at the same time as the new one is

connected. This is a more efficient technique than to completely disconnect the axles

while shifting and produces a smaller reduction of the transferred torque.

In this work, the shift process will be modeled as a one second time period during

13

which the transferred torque is reduced by 50%. At the end of the one second period,

the gear ratio is changed.

The value of 50% was chosen after comparing different ones in a gear box model

test, in which the one of 50% was shown to give closest results to that of existing test

data. The results of this test is shown in section 5.1.1.

3.1.5 Total Powertrain Ratio and Inertia

Besides the gearbox and the TC, some other parts in the powertrain also have a ratio

of the transferred torque and angular rotation speed. These are fixed and are modeled

simply as a single total ratio between the gearbox and the wheels. It will here be

denoted iaxle. The corresponding equations become

Tw = iaxle · TGB (3.6)

ωw =ωGB

iaxle(3.7)

where Tw and ωw are the wheel torque and angular velocity.

The inertias of the powertrain are modeled into two parameters. One is the engine

inertia Je and one is a lumped inertia of the other components combined, denoted Jw.

Jw is the sum of the inertias of the wheels, the shafts and the gearbox components.

3.1.6 The Wheels

Slippage is disregarded and the wheels are modeled with a constant effective radius

rw. The velocity of the vehicle then becomes

v = ωw · rw (3.8)

3.1.7 Resisting Forces

The external forces acting on the vehicles in the longitudinal direction are air re-

sistance, Fa, rolling resistance, Fr, and gravity, Fg [8]. Air resistance is estimated

as

Fa(v2) =

1

2cwAv

2 (3.9)

14

Where cw is the air drag coefficient of the vehicle and A is the cross section area of

the vehicle front.

Rolling resistance comes from the fact that the rubber wheels of a vehicle are

elastic and get slightly deformed at the contact surface with the road. This creates

some resistance which can be modeled as

Fr = crmg cos(α) (3.10)

where cr is the rolling resistance coefficient, m the vehicle mass, g the gravitational

acceleration and α the road slope.

The resisting force that comes from the gravitational forces is

Fg = mg sin(α) (3.11)

The expression for the total resisting forces, Fres, becomes

Fres = Fa + Fr + Fg (3.12)

3.1.8 Longitudinal motion

The motion of the vehicles is derived from Newton’s second law. The acceleration is

described by

mdv

dt= Fw − Fres (3.13)

where Fw is the driving force on the wheels and Fres = Fa +Fr +Fg is the sum of the

resisting forces.

The dynamics of the wheels are

Jwωw = Tw − Tb − rwFw (3.14)

where Tb is the brake torque.

The dynamics of the engine are

Jeωe = Te − TTC (3.15)

15

By putting together equations 3.3 - 3.14 an expression for the acceleration can be

determined to be

dv

dt=

rwJw +mr2w + itot(g)2Je

(itot(g)ξ(Φ)Te(ωe, uf )− Tb(ub)− rw(Fres)) (3.16)

where itot(g) = igear(g) · iaxle and Tb(ub) is the brake torque depending in the brake

control signal ub.

With lockup engaged it becomes

dv

dt=

rwJw +mr2w + itot(g)2Je

(itot(g)Te(ωe, uf )− Tb(ub)− rw(Fres)) (3.17)

3.2 Criterion Modeling

The optimal control problem which is formed in Chapter 4 is based on minimizing

the following 3 criteria: travel time, fuel consumption and durability losses. These

are modeled in this section. The control problem uses a discretization in step length

of the distance travelled and all the criteria costs are calculated at each step. The

total number of steps is denoted as N and the step index as k.

3.2.1 Travel Time

The time it takes to drive a certain step distance ds is calculated as

t(k) =ds

v(k)(3.18)

where v(k) is the vehicle velocity at that step.

The travel time for the entire distance is the sum of the time for each step

ttot =N∑k=1

t(k) (3.19)

3.2.2 Fuel Consumption

The amount of fuel required to drive the vehicles is calculated from a fuel map which

is built up from test data from Volvo CE. At each step, the fuel consumption Mf (k),

16

Figure 3.4: An example of a fuel consumption map

is extracted from the map based on engine rotation speed and the generated torque.

It is measured in liters consumed for step k.

Mf (k) = f(ωw, Te) (3.20)

The amount of fuel used for the entire distance is summed up as

Mf,tot =N∑k=1

Mf (k) (3.21)

A typical fuel map is shown in figure 3.4

17

3.2.3 Durability Losses

The method used to estimate the durability loss in each simulation assumes that

there are only two causes of losses: fatigue of rotating parts, and a discrete loss

each time a clutch is opened and closed during a gear shift. All other failure modes

are disregarded. Engine durability loss is also disregarded and the modelling will

approximate the rest of the driveline losses to only occur in the gearbox and the

rolling bearings at its ingoing and outgoing shafts.

The losses will be measured in the estimated price in SEK that the wear has

costed.

3.2.3.1 Fatigue of rolling bearings

The estimation of durability losses of the two rolling bearings are based on the SKF

formula for rolling bearing life [12], which states that their lifetime L is inversely

proportional to their stress as1

L∝ ωT 10/3 (3.22)

To be able to compare different simulations to eachother, the durability loss of

each rolling bearing will be measured as a duty value D for each step as

D(k) = t(k)ωT 10/3 (3.23)

This gives an estimate of how much damage the rolling bearing has taken at each

step.

The concept is further developed to estimate the price cost of the wear. It is done

by taking the price of a gearbox and dividing it by the average amount of duty value

that the rolling bearings reach in their lifetime. This associates a unit of duty value

to a price κ in SEKduty

. It is then possible to estimate the price of the wear of each

bearing at every step as

Pbearing(k) = κ ·D(k) (3.24)

For the two different bearings, the prices become

Pbearing,in(k) = κ · t(k)ωTCT10/3TC (3.25)

18

Pbearing,out(k) = κ · t(k)ωGBT10/3GB (3.26)

3.2.3.2 Clutch Wear

Each time a clutch is opened and closed during a gear shift, some durability loss

occurs. The magnitude of the loss is related to the amount of torque that is being

transferred at the moment and also to the rotation speed difference of the two parts.

A large torque and a large speed difference give a large loss and vice versa.

The clutch wear will here be modelled as a fixed price for making a gear shift. The

price will be set to an estimated average calculated as the price of a clutch divided

by the average amount of gear shifts in a clutch lifetime, denoted Pgearshift.

3.2.3.3 Total Durability Losses

The total durability loss at a step k is calculated as the price of the bearing wear plus

the gear shift price times a help variable gs(k) that tells if a gear shift has occurred

or not.

P (k) = Pbearing,in(k) + Pbearing,out(k) + Pgearshift · gs(k) (3.27)

where

gs(k) =

1 if a gear shift has occured at step k

0 otherwise(3.28)

The total price of the wear caused during the entire distance is summed up as

Ptot =N∑k=1

P (k) (3.29)

19

Chapter 4

The Optimal Control Problem

This chapter covers the formulation of the optimal control problem, which is the

core of the optimization tool that is being created. The objective function which is

the target to minimize and optimize is presented, and the dynamical programming

algorithm used as a solver is analysed. The problem complexity is discussed and

methods used to lower the computation time are explained.

4.1 Problem Formulation

The objective of the optimization tool is to be able to run calculations that will result

in data of the optimal way to drive over a given distance. Optimal refers to the ’best’

way to drive, and the definition of that will in this work be decided by the 3 critera

mentioned earlier: travel time, fuel consumption and durability losses.

Having 3 criteria to minimize leads to a multiobjective optimization problem and

to formulate the objective function, the so called scalarization technique and a weight-

ing method will be used. The concept of this method is to summarize the values of

each criteria cost function into one scalar. Since it can be desirable to prioritize the

various criteria differently, each of them is multiplied by a weight and scale factor qi.

The objective function cost C is then calculated as

C = q1ttot + q2Mf,tot + q3Ptot (4.1)

20

where ttot, Mf,tot and Ptot are calculated as in Equations (3.19), (3.21) and (3.29).

The optimal control problem is to minimize this cost while obeying the limitations

of the vehicle dynamics modeled in Equation (3.16). The problem can be formulated

as

minimize q1ttot + q2Mf,tot + q3Ptot

subject to dvdt

= rwJw+mr2w+itot(g)2Je

(itot(g)Te(ωe, uf )− Tb(ub)− rw(Fres))

Te ≤ Te,max

ωe ≤ ωe,max

Tb ≤ Tb,max

g = {1, ..., gmax}

(4.2)

4.1.1 Designing and choosing the weight and scale parame-

ters

The weight and scale parameters qi can be designed in various different ways and

opinions on which is the best way differ. This is discussed in [9] and there, several

approaches are presented. It is clear that the parameters should be scaled to get an

easier overview of the problem and to create unit-less costs that makes more sense

to summarize. A suggestion is made that each criteria should be scaled with its

ideal value. This would put the ranges of the costs to between 0 and 1, but is only

applicable to problems with non-zero ideal cost values, which is not the case here.

Instead, the approach used in this problem will be to scale each criteria with their

average values. These average values are calculated based on driving a step with

cruising speed of the particular vehicle.

qi =wi

qi,scale(4.3)

Here, wi are weighting parameters which decide how much each of the criteria will be

prioritized. They will be normalized to give a sum of 1.∑wi = 1 (4.4)

The interpretation of wi is that it is the relative importance of the criteria it is

associated with. While qi,scale are fixed in the optimization tool, wi are allowed to be

21

changed. By changing these weights, the priority of the criteria functions are changed

and the optimization will result in a different solution of the ’best’ way to drive. This

opens up the possibility to analyse many different drive styles.

4.2 Dynamical Programming

The algorithm which will be used to solve the optimal control problem (4.2) is dy-

namical programming. It breaks down the problem into subproblems and is suitable

for finding the optimal solution of a complex computation. The algorithm follows

the principle of optimality which states that for any initial state and decision, each

following decision must obey an optimal policy with regard to the initial state. [10]

For dynamical programming to be applicable to (4.2), the equations and simula-

tions will first be further discretized and re-written into a deterministic multi-stage

decision problem. This is done by dividing the driving distance into N steps with

fixed length, (the step index will be denoted k), and also quantizing the allowed ve-

hicle velocities and engine rotation speeds into a state matrix X(k) together with the

already quantified gear choices. The goal now becomes to find a series of decisions

that takes the inital state x0 through each step k to the end state xN via an optimal

path.

The method of the algorithm is to go backwards from the last state while calcu-

lating the cost to take each step in every possible way. The running-cost for taking

the step k is denoted ξk(x, u) and is based on the objective function cost (4.1). The

costs of each passed step are then added together and the lowest cost from the current

point, denoted Jk(x), is stored. This creates subproblems at each grid point in the

state matrices and the lowest cost of driving through the entire distance, which will

be J0(x0), is found when the algorithm has gone through all steps and reaches the

initial state. The standard dynamical programming algorithm as stated as in [11] is

1. Let JN(xN) = 0

2. Let k = N − 1

22

3. Let Jk(x) = minu∈Uk{ξk(x, u) + Jk+1(C(x, u))} x ∈ Xk

4. Repeat (3) for k = N − 2, N − 3, ..., 0

5. The optimal cost is J0 and the sought control vector is the optimal control from

the initial state to the final state

Here, u is the control signal between the states and Uk is the set of allowed,

dizcretized controls. With this standard approach it is not certain that the discrete

controls will make the states transition exactly into other grid points in the state

matrices, creating the need for interpolations to be made. This interpolation need

and the fact that the number of calculations increases exponentially with the size

of Uk has made an alternative method more preferable. In this work, the so called

inverse method which is presented in [2] is used.

4.2.1 The Inverse Method

In this method, instead of going through all possible controls in Uk and interpolating

the results, only the required control to exactly transition between each grid point

in Xk to each point Xk+1 are calculated. If the running cost to make the transition

between xi ∈ Xk and xj ∈ Xk+1, which is denoted ξi,jk , is found to be not feasible,

the cost will be set to infinity. By using this inverse method, the algorithm becomes

1. Let JN(x) = 0

2. Let k = N − 1

3. Let Jk(xi) = minxj∈Xk+1{ξ(i,j)k + Jk+1(xj)} xi ∈ Xk

4. Repeat (3) for k = N − 2, N − 3, ..., 0

5. The optimal cost is J0 and the sought control vector is the optimal control from

the initial state to the final state

23

in Figure 4.1 an illustration of the algorithm is made where the discretized driving

distance is shown with the state matrices at each step.

Figure 4.1: An illustration of the algorithm

4.2.2 Computational Complexity

The total number of calculations that is performed using the dynamical programming

with the inverse method will be

Q = NK2X (4.5)

where KX is the number of gridpoints inside the state matrix X. This is the result

of that at each of the N steps, the trajectory from each state in Xi to each state in

Xj is computed. Therefore the number of calculations increase quadratically with the

number of gridpoints in the state matrix, and linearly with the number of steps.

24

4.2.3 State Space Representation

2 states, (vehicle velocity and gear choice), are required to do the optimizations while

driving with the torque converter lockup engaged, and 3 states (vehicle velocity,

gear choice and engine rotation speed) are required if the torque converter lockup is

disengaged. (Generally, only the computations of the wheel loaders require the use of

3 states, the hauler and grader computations will most often only require 2 states).

Finding a suitable number of gridpoints when quantifying the state space is of

high importance. Because the algorithm only finds trajectories from point-to-point,

it is important to have a dense enough space with gridpoints as close to eachother

as possible to get realistic results. The drawback of having many points is that

the computation time increases quadratically with the number of points, therefore

picking a number that provides good results while not taking unnecessarily long time

to compute is important.

The number of gears is already quantized. For most of the gearboxes this is 9 and

will be representing one dimension in the state matrix. Quantizing the velocity and

engine rotation speed is more complex and a priori knowledge of the drive pattern

currently being analyzed is very helpful. If for example a hauler driving on a flat road

is being analyzed, the velocity range will be large and probably reach values between

0 and 50 km/h. Therefore a large number of points is needed to get a dense space

and somewhere around 200 steps in the velocity dimension has been tested and found

suitable. Because of the locked up torque converter, only two states are needed and

the final amount of gridpoints in the state matrix in this example is 9× 200.

If a wheel loader is being analyzed, 3 states are required. A priori knowledge

will say that the velocity probably does not reach more than 25 km/h, making a

suitable quantification of the velocity and engine rotation speed dimensions to have

100 steps each. This makes the amount of gridpoints in this example to be chosen to

9× 100× 100

25

4.2.4 Limiting the Search Space

The calculation time can easily become quite long, especially in the cases where 3

states are being used. Therefore some further methods to limit the computational

complexity can be used.

One way that will be used is to limit the so called search space. The search space

is the amount of points in the to-go state matrix Xk+1 that trajectories are being

calculated to. Normally this covers all of the state matrix at that step, but a priori

knowledge tells that many of these are not feasible and can be skipped beforehand.

One of the largest limitations to make is to only compute trajectories to the reachable

velocity points. This is done by checking the possible accelerations dvds

for the current

state xi ∈ Xk and see which states xj ∈ Xk+1 that the maximum and minimum

acceleration can reach, then limit the calculations to cover only gridpoints in between

these values. A visualization for this is shown in Figure 4.2 where only the points

between the maximum and minimum velocity change dv are allowed.

Figure 4.2: Search space limitation

26

Chapter 5

Simulations and Analysis

This chapter will present various results of analysis made about the optimization tool

as well showing example simulations.

A model verification is made where the vehicle model in the optimization tool is

compared with existing test data to see how well it simulates a real vehicle, different

example optimizations are presented on various ways to drive a hauler and an analysis

is made about what happens if the weights of the different optimization criteria are

varied. Pareto fronts are used to visualize the results of the optimizations with varied

weights. (A pareto front is a collection of optimal points found from making many

simulations with varying criteria weights.)

5.1 Model verification

To check that the vehicle model that is created in section 3.1 and used in the optimal

control problem behaves in a realistic way, a comparison is made with a Volvo CE

in-house Simulink model which is known to behave similarly to a real vehicle. The

case that is tested is to accelerate a hauler from standstill for 200m on a flat road with

full throttle. The simulations are all made with an unloaded hauler with a mass of

30 tonnes. The optimal control model is in the comparison limited to make the same

gear choices as the simulink model and it is set to prioritize only travel time. This

means that the weight associated to travel time is set to 1 while the other weights

27

are set to 0. The result is shown in figure 5.1.

It is seen that the two models have very similar results, with the only exception be-

ing that the simulink model has a slightly bigger loss of vehicle speed at the gearshift

between gear 7 and 8. This is due to complicated behavior of the mechanical compo-

nents in the planetary gearbox that occurs in real life but which is not modelled in

the optimal control model.

Figure 5.1: A comparison between the optimal control model and a Volvo CE in-house

simulink simulation

5.1.1 Gear Shift Model Analysis

As described in section 3.1.4 the gearshift is modeled as a 1 second time period during

which the transferred torque is reduced by 50%. To find this value, some different

gear shift models are tested in the same driving case as in section 5.1 where an

unloaded hauler is accelerated on flat ground. 3 different shift models is tested with

the transferred torque being 50%, 75% and 100%. The results are shown in Figure 5.2

28

where it can be seen that the simulation with the model with 50% transferred torque

comes closest to the existing test data and therefore best represents a real vehicle.

Figure 5.2: Gearshift model analysis

A 2D-pareto front is also created for each of the different gear shift models, where

the two critera, fuel consumption and travel time, are compared. There, the weighting

parameter for the travel time in the optimization, w1, is varied from 1 to 0 and

the weighting parameter for the fuel consumption, w2 is varied from 0 to 1. The

weighting parameter for durability losses ,w3, is kept at 0. This means that the cases

that are tested are first with full prioritization of travel time, then this is gradually

shifted towards fuel consumption and in the last case, the fuel consumption has full

prioritization. A point is plotted with the value of the optimal time taken and fuel

used for each of the simulations. This creates a front of points with the optimal values

for different weights. They are connected by a line for better visualization and this is

repeated for each of the three different gear shift models, resulting in three different

29

fronts. They are simulated in a case of full acceleration from standstill for 60 meters

and the results are shown in figure 5.3.

Figure 5.3: Pareto fronts for different gear shifting models

What can be concluded from this is that the different gear shift models tested give

simulation results that vary by up to about 9% (for the point where only travel time

is prioritized). This means that it is important to choose a model that is very similar

to the test data. The model that best does this is the one with 50% transferred torque

and is the one that is chosen.

5.2 Optimized Acceleration From Standstill

In this section, some further tests are made on the same case as previous, where a

a hauler accelerates from standstill. The best ways to drive and to shift gears are

investigated based on different weights on the criteria. The optimal control model

and its dynamical programming optimization is from now on allowed to freely start

30

at any gear and make any gear shifts possible. It is not limited to make the same

gear shifts as the Volvo CE Simulink model.

5.2.1 Minimizing Travel Time

To find the optimized way to accelerate based on only minimizing the travel time, the

weights are kept at 1 for the travel time and 0 for fuel consumption and durability

losses. With the model now being free to take any gear shifts, the theoretically fastest

way to accelerate based on this model is found.

In Figure 5.4 the results are shown, plotted against the Simulink model and the

limited optimal control model from the previous section.

Figure 5.4: Acceleration from standstill

What can be concluded from this simulation is that the optimal way to shift in this

specific situation, according to the optimization algorithm, is to start at the fourth

31

gear and then go to sixth, eighth and finally ninth gear. This gives the simulation with

the fastest acceleration and it shows that for an empty hauler as used in this case,

taking fewer gear shift steps and skipping more gears gives a better result. Skipping

gears in this way is often the best case for any vehicle that has a powerful engine and

is unloaded.

The same case was also tested for a fully loaded hauler that in total has a mass

of 60 tonnes which is twice as much as an empty one. The results of this simulation

is shown in

5.2.2 Minimizing Fuel Consumption and Durability Losses

In this section, two simulations are made to investigate the optimal ways to drive

when also taking into consideration either fuel consumption or durability losses. The

case is still to accelerate a hauler from standstill and the results show the differences

in how to drive to save fuel compared to saving durability.

While making tests on these other criteria, it is of interest to not only find the

best gear shifts, but to also investigate the engine torque and engine RPM. When

minimizing travel time it is quite straight forward that the optimal way to drive is to

always use the maximum available power and engine torque, but when considering

the other criteria it is not obvious how much power should be used. To investigate

this, the optimal engine torque and engine RPM are also plotted in these tests.

In Figure 5.5 the two new simulations are plotted together with the previous one

which prioritizes only travel time. The two new ones consist of one with the weights

being 0.85 for travel time and 0.15 for fuel consumption, and one with the weights

being 0.85 for travel time and 0.15 for durability losses. In Table 5.1 the values of the

total time, fuel consumption and durability losses for the three different simulations

are listed.

32

Figure 5.5: Acceleration from standstill

The results show that the optimal engine torque and optimal engine RPM differ

a lot between the two simulations. It is seen that to save fuel, the engine should

be driven with low RPM and high torque. To save durability, the engine should be

driven with high RPM and low torque. The reason why it is good to drive in such

a way to save durability is because the durability losses of the rolling bearings are

high when a high torque is used and therefore a low torque is preferred. This is the

opposite strategy compared to saving fuel where instead high torques are preferred

due to the engines being more fuel efficient in those situations.

33

Criteria weights Time Taken [s] Fuel Cons. [l] Dur. Loss [SEK]

100% Time 20.22 0.3816 0.7126

85% Time 15% Fuel Cons. 20.62 0.3083 0.2051

85% Time 15% Dur. Loss 21.00 0.3134 0.1514

Table 5.1: Results from 3 simulations with different criteria weights

5.3 Pareto Fronts of Criteria Trade-off’s

One of the major objectives of the optimization tool is to be able to compare how

different ways of prioritizing the different optimization criteria results in different

optimal ways of driving. In this section, further analysis of this is made by creating

Pareto fronts which will show how the different criteria are correlated and what trade-

off’s can be made between them.

5.3.1 Travel Time versus Fuel Consumption

By only looking at travel time and fuel consumption, it is possible to get a good idea

of how these two affect each other. By prioritizing only travel time, one can find

what can be described as the performance driving mode which gives the fastest way

to drive. By giving increased prioritization towards the fuel consumption it is possible

to find various ways of balanced driving modes and also more eco-focused modes.

To do a more detailed investigation of what happens when these prioritization

weights are varied, one can look at a Pareto front like the ones in figure 5.3. As

described in Section 5.1.1, the same simulation is run with different criteria weights.

The results are then plotted and by fitting a line through the optimal points, a pareto

front is created. in Figure 5.6 a pareto front of travel time versus fuel consumption

for the final vehicle model is shown. The caste tested is to accelerate a hauler from

standstill for 60 meters.

34

Figure 5.6: Pareto front of travel time versus fuel consumption

What can be gained from this front is a better understanding of how the trade-off

of one criteria affects the other. It also gives some information which can be of value

when designing gear shift maps. It is for example seen that in the area of maximum

speed, which is the point in the top left end of the front, the slope of the front is

very steep. This means that by just changing the drive style a little bit, a lot of fuel

can be saved with a small trade-off in travel time. By comparing some points in that

region of the front, it is found that a reduction in fuel consumption by about 20%

can be made while only lowering the travel time with about 2%.

It is of course of high importance that one keeps in mind what exactly has been

simulated. The case tested in Figure 5.6 is acceleration from standstill for 60 meters.

The good trade-off that can be achieved by moving along the steep area of the front

might not be the same in other driving conditions. It is also needed to make a further

analysis of what causes the good trade-off. It could for example be to drive slower,

or it might be that the engine RPM and torque is adjusted to follow a path in the

fuel consumption map that is more fuel economical.

35

5.3.2 Varying All Criteria Weights

In this section, the same case as in Section 5.3.1 is investigated, but this time all

3 criteria are active. Many combinations of the the 3 weights are used to create a

three-dimensional Pareto front which will show what trade-off’s can be made when

taking all criteria into consideration . The front is show in Figure 5.7

Figure 5.7: Pareto front for all 3 critera

A front like this is more complicated, but will still give information in what di-

rections good trade-off’s can be found. Again, the most interesting areas will be the

ones with steepest slopes. There, a lot can be gained in one criteria while only losing

a little in the others.

36

Chapter 6

Conclusions

In this chapter the work is summarized and the most important conclusions are pre-

sented and discussed. A suggestion on what future work can be done as a continuation

is given.

6.1 Summary

This thesis work started with an introduction of different types of construction vehi-

cles and the standard way of implementing gear shift strategies were presented. This

was followed by an in-depth chapter of how a general vehicle model was created by

mainly modeling a powertrain and its components. This powertrain model was used

to find expressions for the vehicle dynamics in a longitudinal direction which were

then implemented in a optimal control problem.

The optimal control problem uses a dynamical programming algorithm as a solver

to find the optimal way to drive a distance that the user inputs. The optimal way

is found by calculating the optimal velocity and gear shifts based on the criteria of

travel time, fuel consumption and durability losses. These criteria can be weighted

as the user wants.

The work also include some example simulations and analysis where driving situ-

ations with varying criteria weights are investigated.

37

6.2 Discussion

The goal of the thesis was to create an optimization tool which is capable of making

theoretical analysis that can help when designing gear shift strategies. This has been

accomplished and the optimal way to drive in many different situations can be found

by using this optimization tool.

One of the most interesting things that the tool can be used for is to vary the

criteria weights to find the best trade-off’s. This can be useful when creating shift

maps because the vehicle can be designed to drives in ways that are particularly

beneficial. For example, it was seen in the example simulations that accelerating in a

way that took 2% longer time than the fastest way saved about 20% fuel. Similarly,

about 70% durability could be saved by accelerating in a way that also only took

about 2% longer time. Creating vehicles that drive in these ways could be of interest.

The optimization tool opens up for various interesting other analysis to be made.

It must however always be remembered that the result of the optimal way to drive

by the tool only holds for that specific situation that has been used as an input.

To create a shift strategy based on this tool, it is needed to analyse many different

situations. One must also keep in mind that the durability model is a based on big

generalizations and only take into consideration a few components. To get a more

reliable estimation of the durability losses, the model for that needs to get further

developed.

6.3 Future Work

To further develop the optimization tool, the main thing that will be suggested to

work on is the model for durability losses. Durability is a complex concept and

many layers of details can be added to the current model that will improve it. More

components can be added to the model and a further study can be made about which

other causes exist which create durability losses on the vehicles.

Something else that can be added is to create a model for using the hydraulic

work functions of the vehicles, for example to use and move the arm and bucket of a

38

wheel loader or to tilt the trailer of a hauler. This would open up the possibility to

investigate more complex driving situations and work cycles.

39

Chapter 7

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