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Fuel optimal control of hybrid vehicles

Citation for published version (APA):Keulen, van, T. A. C. (2011). Fuel optimal control of hybrid vehicles. Eindhoven: Technische UniversiteitEindhoven. https://doi.org/10.6100/IR712657

DOI:10.6100/IR712657

Document status and date:Published: 01/01/2011

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https://doi.org/10.6100/IR712657

https://doi.org/10.6100/IR712657

https://research.tue.nl/en/publications/fuel-optimal-control-of-hybrid-vehicles(2312e3c5-1a6b-4851-a265-cdf53eea2a90).html

Fuel Optimal Control of Hybrid Vehicles

The research leading to these results has received funding from the ENIAC Joint Undertakingand from Agentschap NL under Grant Agreement number 120009, and is part of a moreextensive project in the development of advanced energy management for urban distributiontrucks which has been made possible by TNO Automotive.

A catalogue record is available from the Eindhoven University of TechnologyLibrary

Fuel optimal control of hybrid vehicles / by Thijs van Keulen. – Eindhoven :Technische Universiteit Eindhoven, 2011Proefschrift. – ISBN: 978-90-386-2502-7NUR 978Copyright c©2011 by T.A.C. van Keulen. All rights reserved.

Cover design: Oranje Vormgevers, Eindhoven, the Netherlands.Reproduction: Ipskamp Drukkers B.V., Enschede, the Netherlands.


PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op dinsdag 14 juni 2011 om 16.00 uur

door

Thijs Adriaan Cornelis van Keulen

geboren te Valkenisse

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. M. Steinbuch

Copromotor:

dr.ir. A.G. de Jager

Contents

Summary vii

Nomenclature ix

1 Introduction 1

1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 What is a hybrid vehicle . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Control applied in hybrid vehicles . . . . . . . . . . . . . . . . . . 6

1.1.3 Route information and estimation of the vehicle conditions . . . . 9

1.2 Problem statement and objectives . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Contributions and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Optimal Trajectories for Vehicles with Energy Recovery Options 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Derivation of the cost function . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Necessary conditions for optimality . . . . . . . . . . . . . . . . . . . . . 21

2.4 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Structure of the solution . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Construction of a nonlinear program . . . . . . . . . . . . . . . . 27

2.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Optimal Power Split Control for Predefined Trajectories 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Root finding algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 The unconstrained solution . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 Iterative loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Evaluation of RASS-OCP . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

vi Contents

3.5.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5.2 Computational effort . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Real-time Power Split Control in Hybrid Vehicles 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Problem formulation and necessary conditions for optimality . . . . . . . 59

4.2.1 The power split control problem . . . . . . . . . . . . . . . . . . . 61

4.2.2 Necessary conditions of optimality . . . . . . . . . . . . . . . . . . 62

4.3 Real-time implementable energy management strategies . . . . . . . . . . 63

4.3.1 Real-time estimation of the multiplier function . . . . . . . . . . . 63

4.3.2 Real-time minimization of the Hamiltonian . . . . . . . . . . . . . 65

4.4 Component characteristics and implemented strategy . . . . . . . . . . . 67

4.4.1 Component characteristics . . . . . . . . . . . . . . . . . . . . . . 68

4.4.2 Implemented strategy . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.2 Tuning of the multiplier estimation . . . . . . . . . . . . . . . . . 75

4.5.3 Strategy comparison . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Conclusions and recommendations 85

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1.1 Optimal trajectories for vehicles with energy recovery options . . 86

5.1.2 Optimal power split control for predefined trajectories . . . . . . . 86

5.1.3 Real-time power split control in hybrid vehicles . . . . . . . . . . 87

5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1 Advanced battery modeling . . . . . . . . . . . . . . . . . . . . . 88

5.2.2 Towards integrated power train control . . . . . . . . . . . . . . . 89

5.2.3 Towards autonomous driving . . . . . . . . . . . . . . . . . . . . . 90

5.2.4 Numerical solutions for state constrained optimal control problems 90

A Singular solution 91

Bibliography 99

Samenvatting (in Dutch) 101

Dankwoord (in Dutch) 103

Curriculum Vitae 105

Summary


Hybrid vehicles have, at least, two power converters. Usually a prime mover, which can

provide tractive power, consuming fuel with an irreversible proces, and secondary power

converter(s), which convert tractive power, reversibly, into a power quantity suitable for

a storage device, or visa versa. The fuel optimal control of hybrid vehicles involves the

control of vehicle velocity, transmission ratio, power split between the prime mover

and secondary power converter(s), and stop-start of the prime mover. The potential of

hybrid vehicles has not been fully realized due to a lack of control methods that can

cope with the unknown future power requests, can be embedded in industry standard

hardware, and can obtain fuel use close to a global minimum.

The control objective is to drive the vehicle to the next destination with a minimum of

fuel subject to a time constraint. The combined control of vehicle velocity, transmission

ratio and power split is approximated with a piecewise continuous scalar control signal

-the combined power request- and optimized with non-smooth optimal control theory.

The stop-start of the prime mover and capacity boundaries of the storage device are

hereby neglected. Using data from an onboard navigation system, providing information

for the upcoming route, e.g., road curvature, road grade, and velocity limitations, the

optimal power request, vehicle velocity, transmission ratio and power split trajectories

for the upcoming route are obtained. The optimal velocity and transmission ratio

trajectories can be used as set points for the real-time velocity (cruise) control and

gearshift strategy, also for non-hybrid vehicles.

The optimal power request and velocity trajectory can be applied to more involved op-

timization methods that obtain the optimal power split trajectory, including stop-start

of the prime mover, subject to constraints on the storage device capacity boundaries. In

case the power split cost function can be approximated with a convex function and there

is a monotonically increasing relation between the storage power and the output power

of the secondary power converter, a novel numerical approach is applied which is based

on observations obtained with the -in optimal control theory well known- Pontryagin

vii

viii Summary

Maximum Principle. The resulting optimal power split trajectory can be used as set

point for the real-time power split controller which gives a robustness against errors

in the predicted trajectories. Optimal trajectories can also be used to benchmark and

design real-time implementable power split controls, or to derive optimal technology,

topology, and component sizes in the design of hybrid vehicle drive trains. In this thesis

the optimal hybridization ratio for a long-haul truck is derived for a 513 km long input

trajectory.

The design of a real-time implementable strategy takes advantage of the results obtained

from the necessary conditions of optimality from the previously mentioned Maximum

Principle, and boils down to: i) estimation of a multiplier function, that adjoins the

energy stored in the storage device to the fuel cost, using real-time available information,

and ii) optimization of a locally approximated Hamiltonian like function, given the

limited available onboard computational capacity. The optimal control based real-time

power split control estimates the multiplier function using linear feedback on an adaptive

set point which is based on the energy currently stored in the storage device and the

actual kinetic and potential energy of the vehicle. The strategy is implemented in a

hybrid electric truck on standard industry hardware. This control is evaluated with

experiments on a chassis dynamometer. The controller is easy to tune and obtains a

fuel consumption, without a priori knowledge of future power requests, within 1.5%

of the global optimum on routes where the capacity boundaries of the storage device

are not reached. In case the storage device boundaries are reached, optimal power

split trajectories, obtained with data coming from navigation systems, can enhance the

performance to become close to optimal.

The calculation of optimal trajectories, based on information from a navigation system,

the novel numerical solution for scalar optimal control problems with state constraints,

and the implemented power split controller adaptive for vehicle mass, vehicle velocity

and elevation, together with the observations when predictive information is beneficial,

can be seen as the main results of this research.

Nomenclature

Acronyms

ACC adaptive cruise control

AMT automated manual transmission

CAN controller-area network

CC cruise control

DP dynamic programming

ECMS equivalent consumption minimization strategy

ECU electronic computation unit

EMS energy management strategy

EV electric vehicle

FF feed forward

FTP federal test procedure

GIS geographic information system

GPS global position system

HEV hybrid electric vehicle

MP maximum principle

NLP nonlinear programming

OCP optimal control problem

PID proportional integral derivative

PWA piecewise affine

RASS-OCP rootfinding algorithm for scalar state

constraint optimal control problems

SOC state-of-charge

SOE state-of-energy

UDDS urban dynamometer driving schedule

ix

x Nomenclature

Roman uppercase

Symbol Description Unit

Cd dynamo road load control –

Cv velocity controller –

D Discriminant –

D Discriminant –

E energy J

Ec theoretical battery capacity J

Ed dynamo energy consumption J

Eeq equivalent fuel energy J

Ef fuel energy J

Es energy in storage device J

E0s initial energy in storage device J

Efs final energy in storage device J

Es storage device state vector J

F force N

Fa inertial force N

Frl road load force N

H Hamiltonian –

I current A

Id dynamo control current A

Is battery storage current A

K feedback gain –

K1 linear state-of-energy feedback gain –

KE kinetic energy recovery feedback gain –

Kh potential energy recovery feedback gain –

L Lagrangian –

P power W

Pb battery electrical power W

Pd brake power W

Peq equivalent fuel power W

Pf fuel power W

Pf,i fuel power during idle W

Pf,p fuel power during traction W

Pk service brake power W

Pm motor/generator mechanical power W

Pm r motor/generator power request W

Pp prime mover mechanical power W

Pp r prime mover power request W

Nomenclature xi


(continued)

Ppen penalty for engine idle loss power W

Pq recovery power W

Pr power request W

Pr,s power request at the singular solution W

Ps storage power W

Ps ch storage power during charging W

Ps dis storage power during discharging W

Pv power reserve W

Pw power at the wheels W

Ps control vector W

R resistance Ohm

S clutch position –

SOC state-of-charge –

SOE state-of-energy –

SOEr reference state-of-energy –

T torque Nm

Td torque at dynamo drum circumference Nm

Tm motor/generator torque Nm

Tp prime mover torque Nm

Tr torque request Nm

Tset dynamo torque set point Nm

U voltage V

U0 voltage of a depleted battery V

UR over potential voltage V

Uoc open circuit voltage V

Ut terminal voltage V

U set of admissible controls –

X state matrix –

Roman lowercase


a polynomial coefficient –

a1 polynomial coefficient –



xii Nomenclature


(continued)


b polynomial coefficient –

c polynomial coefficient –

c0 rolling resistance N

c1 velocity dependent drive train resistance Ns m−1

c2 aerodynamic resistance Ns2m−2

f general function –

g constraints function –

ga acceleration due to gravity m s−2

h elevation m

switching variable –

i segment number –

j number –

m mass kg

vehicle mass kg

me effective vehicle mass kg

n number of data points –

p multiplier function vector –

p0 multiplier initial guess –

q polynomial coefficient m3 s−1

r radius m

slack variable –

rd dynamo drum radius m

rg gear ratio –

rs power split ratio –

s distance m

strip trip distance m

t time s

v velocity m s−1

va initial velocity of a subarc m s−1

vb final velocity of a subarc m s−1

vd circumferential drum velocity m s−1

vr reference velocity m s−1

x state variable –

z1−3 roots of cubic equation –

Nomenclature xiii

Greek


α road slope rad

γ conversion characteristic –

γm,0 electric motor conversion characteristic –



γp,0 prime mover conversion characteristic –

γp,1 prime mover conversion characteristic –

γw well-to-tank conversion characteristic –

ε1 allowed error of end point constraint –

εk allowed error of state constraint –

ϑ gas pedal position %

µ jump parameter –

ν velocity set m s−1

σ piecewise constant –

φ incremental voltage V/-

ω rotational velocity rad s−1

ωm rotational velocity of the electric machine rad s−1

ωp rotational velocity of the prime mover rad s−1

Superscripts

Symbol Description

+ positive, right going− negative, left going? optimum

ˆ estimate0 initial value1 final value

xiv Nomenclature

Symbols and Operations

Symbol Description

a minimum value of set a a = min(a)

a maximum value of set a a = max(a)

|a| complex modulus (magnitude)√

Re(a)2 + Im(a)2

int(a) interior point a of a set X a ⊂ X

a averaged value 1N

∑N1 ak

∂a(x,t)∂x

partial derivative ∂(bx+ct)∂x

= b

a time derivative ∂(bx+ct)∂t

= c+ bx

∂x Clarke’s generalized sub-differential: a vector v ∈ Rn is said to be a

sub-gradient of a convex continuous function f at a point x if the

inequalityf(z) ≥ f(x) + vT (z − x) holds for all z ∈ Rn. The set of

all sub-gradients of f(x) at the point x, called the sub-differential at

the point x, is denoted by ∂f(x)

Chapter one

Introduction

Abstract / In this chapter, a general introduction of control in hybrid vehicles is presented.From a description of hybrid drive trains, the opportunities for control are discussed and theresearch objectives are formulated.

1.1 General introduction

Mobility forms one of our basic needs. Ever since people started specializing in one

profession, they relied on the products and knowledge of others, requiring a form of

mobility. Transport cost, of people and goods, in time and effort, is, therefore, directly

related to economic growth. Throughout history, faster and more efficient ways of

transport are searched.

A milestone formed the discovery of petroleum oil, as a cheap and widely available fuel,

and the invention of the internal combustion engine propelled vehicle (Benz, 1886), as

a light and efficient transporter. An on-demand travel with flexibility, average velocity,

and reach that soon surpassed previous means of transportation became possible and

was one of the motors of the enormous increase in prosperity over the last century.

Although petroleum oil is cheap to produce and widely available, it is clear that the

fossil fuel reserves are not inexhaustible. Besides, the combustion of fossil fuels results

in the emission of carbon dioxide (CO2) which is considered to be a main contributor

to global warming. Also, the air quality in large urban areas is becoming a major issue.

To sustain or increase mobility in western countries and to offer to developing countries

a prospect of similar levels of mobility, while oil reserves are decreasing and emission

legislation becomes more stringent, research for other energy sources than petroleum

oil and cleaner and more efficient means of fuel use in road transport is necessary.

Energy carriers other than petroleum oil that are suggested are, e.g., hydrogen gas,

natural gas, bio fuels, and electrochemical storage in batteries. The energy density

1

2 1 Introduction

characteristics of the different energy carriers are depicted in Fig. 1.1. It can be seen that

petroleum related fuels, diesel, gasoline, etc, have excellent energy-volume properties.

The good energy density property of petroleum oil related fuels lead to the expectation

(International Energy Agency, 2007) that internal combustion engines have a dominant

role in the next decades. This holds especially for heavy-duty vehicles that require a

large range in combination with a low mass. For a detailed comparison of the energy

carriers, the reader is referred to, e.g., (Edwards et al., 2006).

0 50 100 1500

5

10

15

20

25

30

35

40

45

50

Diesel

Ethanol

Gasoline

Hydrogen Gas (700 bar)

Hydrydrogen liquid

Natural Gas liquidLPG PropaneLPG Butane

Natural Gas (250 bar)

Lithium−Ion Battery

[MJ/kg]

[MJ/

l]

Figure 1.1 / Energy volume density vs lower heating value of several energy carriers.

Improving the overall efficiency of petroleum oil powered vehicles is thus a relevant

research topic. One possibility to improve fuel consumption, without compromising on

the performance of the vehicle, is to equip the conventional vehicle with the possibility

to recover energy during braking or driving downhill by adding a motor/generator.

Vehicles with energy recovery options are commonly referred to as hybrid vehicles. The

main benefits of hybrid vehicles are outlined in Subsection 1.1.1.

A second opportunity for enhancing fuel consumption efficiency, is to operate the vehicle

more efficient than the average human driver would do, e.g., by controlling the hybrid

drive train components automatically. The application of control in hybrid vehicle drive

trains is the main topic of this thesis. A brief introduction is given in Subsection 1.1.2.

Finally, new communication and information systems become available. For instance,

the availability of information of the route ahead could be used to compute a prediction

of the future conditions of the vehicle. This prediction could improve the operation of

the vehicle, see Subsection 1.1.3 for additional comments.

1.1 General introduction 3

1.1.1 What is a hybrid vehicle

To understand the benefits of hybrid vehicles let us first describe the power required to

propel a vehicle (Guzzella and Sciarretta, 2005, p. 14), using an elementary model:

Pw = mvv︸︷︷︸(1)

+mvga sinα︸︷︷︸(2)

+ c0mvga cosα︸︷︷︸(3)

+ c2v3︸︷︷︸

(4)

, (1.1)

where Pw is the power request at the wheels, m the vehicle mass, v the vehicle velocity

(where we consider only positive values), v the vehicle acceleration, ga the gravitational

constant, α the road angle, c0 a coefficient for the rolling resistance, and c2 a coefficient

for the aerodynamic losses. Several components that contribute to the required power

can be distinguished:

(1) inertial power, the power to overcome the inertia of the vehicle, to acquire kinetic

energy,

(2) gravitational power, the power to overcome elevations in the route, to acquire po-

tential energy,

(3) rolling losses, the power required to overcome the resistance of the tires,

(4) aerodynamic losses, the power required to overcome the aerodynamic resistance of

the vehicle.

One way to improve fuel consumption is to minimize the cumulative required power.

If we assume that the road angle α is prescribed and that parameters ga, m, c0 and c2

are positive and constant, then the only way to improve the cumulative required power

is to alter the vehicle velocity trajectory v. Note that v and α, and thus the inertial

power and gravitational power, could become negative, i.e. a negative power request is

possible Pw < 0.

In case the power request is negative then there is the possibility of energy recovery.

Using a generator in combination with a storage device, the energy can be recovered,

temporarily stored, and used at a later moment to provide tractive power such that

less power from the combustion engine is required. This is one of the major benefits of

hybrid vehicles.

Another way to improve fuel consumption is to provide the power request Pw in an

efficient manner. In a hybrid drive train, there are three possibilities to meet the power

request:

Pw = Pp + Pm + Pk, (1.2)

4 1 Introduction

in which Pp denotes the prime mover output power, Pm the output power of the hybrid

system consisting of the secondary power converter(s), and Pk the power delivered by

the service brakes.

The prime mover converts a fuel flow Pf from one of the fuel based energy carriers

of Fig. 1.1 with an irreversible process into output power Pp, therefore, Pf ≥ 0. The

secondary power converter can operate both as a motor and a generator, converting Pminto storage power Ps, or visa versa. The service brakes convert useful energy in the

service brakes into heat, so Pk ≤ 0.

The efficiency of the fuel conversion of the engine as well as the conversion efficiency of

the secondary power converter is in general a nonlinear function of the working points of

the device. The working points involve, for instance, power throughput and rotational

velocity. The hybrid system has the freedom to govern the working points of the drive

train in a way that the combined cost is minimized.

Moreover, the size of the combustion engine has influence on the efficiency of the fuel

conversion, a large engine has a large maximum power output, however, also a large

internal drag. If part of the power requirements for a certain performance of the vehicle

are covered by the secondary power converter, the engine can be downsized, such that

the fuel conversion efficiency is improved while the performance is maintained.

Furthermore, the hybrid system allows for stop-start of the prime mover. In case the

secondary power converter has enough tractive power, the prime mover can be stopped

such that idle losses of the prime mover are eliminated while the vehicle launch time of

the vehicle is not compromised. Using stop-start, the emission of the vehicle can also

be temporarily reduced, e.g., when driving in a city center.

To summarize, the main advantages of hybrid vehicles are:

• kinetic and potential energy can be recovered and stored, such that it can be used

at a later, more convenient, time to propel the vehicle,

• the working points of the prime mover can be improved,

• the prime mover can be downsized, such that the average fuel conversion efficiency

is improved,

• idle losses are eliminated and emissions are locally reduced due to stop-start of

the prime mover.

Sofar, the advantages of hybridizing a vehicle are outlined. A possible disadvantage

is the additional cost and complexity that the hybrid system brings along. Therefore,

a careful design and operation of a hybrid vehicle is required which involves several

choices regarding:


• technology,

• topology,

• component size,

• control strategy.

Firstly, technology choices can be made regarding i) the energy carrier of the prime

mover as shown in Fig. 1.1, and ii) the quantity in which the secondary power converter

converts the recovered energy. Possible technologies for the hybrid system are, electric,

hydraulic, mechanic, and pneumatic applications. The theory developed in this thesis is

mainly focused on a diesel powered internal combustion engine in combination with an

electric machine as secondary power converter. However, the results can also be applied

to other technologies.

Secondly, the topology of the drive train has to be designed. Two types of hybrid

topologies that can be found are i) parallel and ii) series, see Fig. 1.2. The main

difference between a parallel and a series topology is that, in the first situation, there

is a mechanical coupling between the prime mover and the wheels, while in the series

topology there is a coupling in the quantity of the hybrid system. In a series topology,

the clutch and gearbox can be omitted on the other hand an additional generator is

required. Advantage of the series topology is that the prime mover can operate in

its optimal working point, and transmission losses in the gearbox are omitted. The

efficiency improvement of the prime mover should out weight the added losses due to

the energy conversion in the two motor-generators.

primemover

Pr

Pf Pp

Ef

EsPm

generator

Ps

primemover

storagedevice

Pr

Pf Pp

Ef

EsPm

S

motor-generator

Ps

gearbox

motor-generator

storagedevice

rg

Pw

Pw

Pk

Pk

Figure 1.2 / Parallel (top) hybrid vs series (bottom) hybrid drive train topology.

6 1 Introduction

More involved topologies can also be found, e.g., a combination of parallel and series

hybrid topology. These parallel-series topologies have both a mechanical and electrical

connection between the prime mover and the wheels. This adds additional control

freedom, however, also requires an increased number of components and cost.

For trucks, that drive mostly on highways, the engine has a good working point during

cruise speed if the engine has a direct connection to the wheels. So, in that application,

the parallel topology generally offers the lowest fuel consumption potential. If other

objectives are imposed, such as noise reduction and comfort, a series hybrid topology

could be favored.

A topology often used for heavy-duty vehicles is depicted in Fig. 1.3. It is acknowl-

edged that both Fig. 1.2 and 1.3 depict only a possible layout. The position of several

components can be reordered, for example, the gearbox and clutch can be placed before

or behind the secondary power converter. Moreover, the number of secondary power

converters is not necessarily limited to one. For a detailed description of hybrid vehicle

topology design, the reader is referred to, e.g., Hofman (2007).

engineelectric

clutch

gearbox final

power battery wheels

service

machinedrive

electronics

brakes

Figure 1.3 / Parallel drive train of a hybrid electric vehicle.

The third design choice is the size of the components, i.e., the maximum output power

of the prime mover and secondary power converter(s) and the capacity of the storage

device. The optimal component size requires the balance between investment cost,

operational cost and performance. An important design parameter is the hybridization

ratio describing the maximum recovery power as the ratio of the total available tractive

power. The storage capacity is also a design variable.

Finally, the design of a control algorithm that governs the hybrid drive train components

is required. The main focus of this thesis is on the control strategy which is discussed

in the next section.

1.1.2 Control applied in hybrid vehicles

The main control objective in this thesis is fuel consumption reduction, i.e., to minimize

the cumulative fuel use Ef =∫Pfdt. Control problems that deal with optimization as


objective, are generally referred to as optimal control (Pontryagin et al., 1962; Clarke,

1983; Vinter, 2000; Geering, 2007).

To achieve a good fuel economy the hybrid vehicle has several control inputs that can

be used, in case of a parallel hybrid topology this involves, e.g.,

• power of the primary power converter Pp,

• power of the secondary power converter(s) Pm,

• power of the service brakes Pk,

• gear ratio rg,

• clutch position S,

see again Fig. 1.2 for descriptions.

In Subsection 1.1.1, it is outlined that fuel consumption can be reduced by i) reduced

cumulative power requirements, and ii) an optimal division of the power request over

the different power converters. Therefore, it is useful to define the following control

signals that are directly related to the objectives:

• the combined power output of the power converters Pw = Pp + Pm + Pk,

• the power split, the division of power request over the primary and secondary

power converter rs = Pm

Pr= 1− Pp

Pr, with Pr = Pp + Pm.

The control of the tractive power at the wheels Pw to achieve a velocity is known as cruise

control (CC). The supervisory control algorithm, dealing with the balanced generation

and re-use of the stored energy, using the power split rs such that fuel consumption is

minimized, is called Energy Management Strategy (EMS).

The control strategy has only limited freedom, the control actions are subject to several

constraints:

• average velocity (travel time), maximum velocity of the vehicle, and distance

covered by the vehicle,

• velocity limitations of the power converters, the rotational velocity of the prime

mover and secondary power converter is bounded,

• power limitations of the power converters, the power throughput of the power

converters is bounded,

• storage device energy level,

8 1 Introduction

• temperature limitations of the prime mover, secondary power converter and stor-

age device.

Besides the constraints, challenges in the control of hybrid drive trains are:

• the hybrid nature of clutch and gear operation as well as the nonlinear description

of the power converters results in a non-convex cost function, complicating the

use of optimization methods,

• the future conditions, e.g., vehicle mass, road angles, and weather conditions, that

result in a power request Pw and velocity v, are not known a priori,

• real-time implementation requires the use of standard hardware with limited com-

putational power and storage capacity.

To deal with the constraints and challenges described above, the fuel optimal control

problem is often simplified such that part of the difficulties can be ignored. This requires

a trade off between the model description complexity and the number of control variables

of the problem that is considered. Different approaches can be found in literature, each

method with its own objective and dealing with the trade-off in its own way.

Algorithms dealing with the combined optimization of vehicle velocity and operation of

the hybrid drive train components is relatively new. An extensive literature overview

on vehicle velocity control combined with the hybrid drive train operation, is presented

in Chapter 2. Since traffic situations often impose very strict velocity limitations, the

optimization of the drive train operation can be separated from the velocity control.

To find an optimal technology, topology and component size, a common practice is to

use measured velocity and power requests, e.g., obtained with a conventional vehicle on

a specific duty cycle, to investigate the potential benefit of hybridizing the drive train.

A detailed literature overview for power split optimization for a predefined power and

velocity trajectory is given in Chapter 3.

Moreover, a real-time controller that has to deal with the limited computational power

and storage capacity is required. Input for the real-time control are the actual vehicle

velocity and a power request from the driver (gas and brake pedal position). To deal

with the computational power limitations in practice, often heuristic based strategies

are applied. Implementations of optimal control based strategies are scarce. A detailed

literature overview on real-time implementable power split control is given in Chapter 4.

It is also possible to estimate the future power and velocity trajectories based on in-

formation coming from a geographical information system or signals available in the

vehicle. This is further discussed in Subsection 1.1.3.

1.2 Problem statement and objectives 9

1.1.3 Route information and estimation of the vehicle conditions

Equation (1.1) describes the power flow that plays a role in the movement of a vehicle.

Especially for heavy-duty vehicles the vehicle mass m and aerodynamic resistance c2

can change considerably due to varying payload. Moreover, the power request Pw in

vehicles is influenced widely by route characteristics such as the elevation and velocity

limitations. However, an increasing number of sensors and information systems enable

prediction of the future power requirements such that the energy usage can be optimized.

Recently, advances are made in real-time estimation of vehicle mass and road angle

(Vahidi et al., 2005; Kolmanovsky and Winstead, 2006). Route information can be

derived from geographical information systems in combination with a routeplanner and

GPS. The information that could be obtained involves road angle, (dynamic) velocity

limitations, stopping points and road curvature (Beuk et al., 2006).

A special class of vehicles are vehicles that operate on a fixed route, e.g., garbage trucks

and busses (Bartholomaeus et al., 2008). By comparing velocity and power requests

measured in the past with the actual situation, a good prediction of future power and

velocities can be made, without the need of detailed topographical data.

1.2 Problem statement and objectives

The potential of hybrid vehicles has not been fully realized due to a lack of control

methods that can cope with i) the constraints and nonlinear description of the compo-

nents, ii) the unknown power requests, iii) the limited computation power of industry

standard hardware, and can obtain fuel use close to a global optimum.

The goal of this thesis can be formulated as:

Derive a methodology for the fuel optimal design and operation of a hybrid vehi-

cle, i.e., by application of predictive and adaptive control strategies. Implement

a real-time strategy on standard industry hardware and evaluate the strategy with

experiments.

From this research goal the following sub-problems are derived:

velocity trajectory optimization

The velocity trajectory optimization problem can be defined as: find an algorithm

for velocity trajectory optimization, taking advantage of satellite navigation, providing

10 1 Introduction

velocity constraints, road curvature and road slope, and estimated vehicle parameters,

that minimizes the fuel consumption in a hybrid vehicle subject to a time and distance

constraint.

It is beneficial to optimize the velocity trajectory in order to minimize the fuel con-

sumption in two ways i) to assist the driver in tracking an optimal velocity trajectory,

e.g., input to an (adaptive) cruise control, and ii) to estimate the future power request

trajectory which can be used to optimize the hybrid components use.

power split optimization for a predefined power request trajectory

This problem is defined as: find an algorithm to compute the optimal power split tra-

jectory for a predefined power and velocity trajectory with a computation time suitable

for real-time implementation and taking into account the storage device capacity con-

straints.

The results can be used by the real-time control. Besides, the algorithm for power split

optimization can be used to determine the optimal topology and component size and

to bench mark the real-time controller.

real-time power split optimization

The objective is defined as: find a real-time implementable energy management strat-

egy which minimizes fuel consumption, without exact knowledge of the future power

trajectory, however, if available, can take advantage of the predictive data outlined

in Subsection 1.1.3. Evaluate the performance of the algorithm with experiments on

a chassis dynamometer and determine to what extent this predictive information is

beneficial for the fuel consumption.

1.3 Contributions and outline

This thesis consists of three research chapters. Each research chapter is submitted

integrally for journal publication and is self contained. Nevertheless, an interconnection

between the different chapters can be found, see Fig. 1.4. The three chapters together

form a framework for the real-time application of predictive EMS. It is hereby assumed

that we can take advantage of the signals and estimations provided by geographical

information systems and parameter estimation as discussed in Subsection 1.1.3.

1.3 Contributions and outline 11

Chapter 2

Chapter 3

Chapter 4

p, Es

Pr, v

α, v, m

Pm, Pp, Pk, rg, S

power request andvelocity trajectory

multiplier andstate-of-energy

requested travel timeroute data andvehicle parameters

real-time control

trajectory optimization

power split forpredefined trajectories

real-timepower split

cruisecontrol

topologydesign

Figure 1.4 / Interconnection between chapters. A hat indicates a prediction. Legend:α is the road slope, v the velocity limitations, Pr the power request, vthe vehicle velocity, Ef the fuel consumption, p a Lagrange multiplierdenoting the equivalent cost of stored energy in fuel, Es the state-of-energy of the storage device, Pm the power output of the secondarypower converter, Pp the power output of the prime mover, Pk the powerof the service brakes, rg the gear ratio, S the clutch position.

Chapter 2

In Chapter 2, a novel algorithm is derived for the computation of optimal velocity, power

request, gear ratio and power split trajectories for vehicles with energy recovery options.

The novel algorithm for the computation of the optimal trajectories, takes advantage of

elevation-distance and velocity limitations, e.g., coming from a geographical information

system.

Chapter 3

Chapter 3 deals with the calculation of optimal power split control based on predefined

velocity and power request trajectories.

A novel numerical solution is derived for state constrained optimal control problems

with a scalar state. Application of the novel numerical solution for state constrained

problems to the power split control problem for a known power and velocity trajectory,

results in computation of the optimal state-of-energy (and multiplier) trajectory with

a computation time several orders lower than the often applied dynamic programming

algorithm.

12 1 Introduction

Due to the low computational effort, this algorithm is well suited for application in a

real-time setting where based on predicted power and velocity trajectories the optimal

state-of-energy trajectory is calculated and communicated to the real-time controller.

Also, it is possible to use the algorithm to compute the fuel consumption. This can

be used to derive the optimal technology, topology and component size for the hybrid

drive train.

Chapter 4

In Chapter 4, the design, implementation, and evaluation of a real-time optimal power

split control is discussed.

A new methodology for the design and implementation of a real-time optimal control

based power split and clutch control, adaptive for vehicle mass and elevations, on stan-

dard hardware is presented. An experimental evaluation indicates that a performance

close to the global optimum can be expected, also when only real-time available infor-

mation is used. The control design is able to take advantage of predictive information,

if available.

Chapter 5

In Chapter 5, conclusions are provided and recommendations for future research are

given.

1.4 Publications

In the research leading to this thesis, the following journal and conference papers are

published.

Refereed journal publications

• Van Keulen, T., De Jager, B., and Steinbuch, M. (2011). Optimal Tra-

jectories for Vehicles with Energy Recovery Options. Submitted for journal publi-

cation. Chapter 2.

• Van Keulen, T., Gillot, J., De Jager, B., and Steinbuch, M. (2011).

Optimal Power Split Control for Hybrid Vehicles for a Predefined Input Trajectory.

Submitted for journal publication. Chapter 3.

• Van Keulen, T., Van Mullem, D., De Jager, B., Kessels, J.T.B.A.,

and Steinbuch, M. (2011). Optimal Power Split Control in Hybrid Vehicles.

1.4 Publications 13

Submitted for journal publication. Chapter 4.

• Van Keulen, T., De Jager, B., Serrarens, A., and Steinbuch, M.

(2010). Optimal Energy Management in Hybrid Electric Trucks Using Route

Information. Oil and Gas Sci. & Technol., 65, 103–113.

Refereed conference contributions

• Van Keulen, T., De Jager, B., and Steinbuch, M. (2011). Optimal Tra-

jectories for Vehicles with Energy Recovery Options. In Proc. of the 18th IFAC

World Congress, Milan, Italy, (accepted).

• Van Mullem, D., Van Keulen, T., Kessels, J.T.B.A., De Jager, B.,

and Steinbuch, M. (2010). Implementation of an Optimal Control Energy

Management Strategy in a Hybrid Truck. In Proc. of the 6th IFAC Symposium

Advances in Automotive Control, Munich, Germany, (6 pages).

• Van Keulen, T., De Jager, B., Kessels, J.T.B.A., and Steinbuch, M.

(2010). Energy Management in Hybrid Electric Vehicles: Benefit of Prediction.

In Proc. of the 6th IFAC Symposium Advances in Automotive Control, Munich,

Germany, (6 pages).

• Van Keulen, T., De Jager, B., Foster, D., and Steinbuch, M. (2010).

Velocity Trajectory Optimization in Hybrid Electric Trucks. In Proc. of the

American Control Conference, Baltimore, United States, 5074–5079.

• Van Keulen, T., Naus, G., De Jager, B., Van de Molengraft, M.J.G.,

Steinbuch, M., and Aneke, N.P.I. (2009). Predictive Cruise Control in Hy-

brid Electric Vehicles. In Proc. of the EVS24 International Battery, Hybrid and

Fuel Cell Electric Vehicle Symposium, Stavanger, Norway.

• Van Keulen, T., De Jager, B., Serrarens, A., and Steinbuch, M.

(2008). Optimal Energy Management in Hybrid Electric Trucks Using Route

Information. In Proc. of the International Conference on Advances in Hybrid

Powertrains, Rueil Malmaison, France, session 2.

• Van Keulen, T., De Jager, B., and Steinbuch, M. (2008). Influence of

Driver, Route and Vehicle Mass on Hybrid Electric Truck Fuel Economy. In Proc.

of the 9th International Symposium on Advanced Vehicle Control, Kobe, Japan,

911–916.

• Van Keulen, T., De Jager, B., and Steinbuch, M. (2008). An Adaptive

Sub-Optimal Energy Management Strategy for Hybrid Drivetrains. In Proc. of

the 17th World Congress, Seoul, Korea, Democratic People’s Republic of, 102–107.

Chapter two

Optimal Trajectories for Vehicles withEnergy Recovery Options1

Abstract / This chapter deals with the fuel consumption optimization of state-of-the-art vehi-cles that have energy recovery options. A novel cost function is used which is piecewise affine.This function describes the influence of the automated manual transmission, the potential ofbrake energy recovery, and the vehicle velocity, on fuel consumption, with one control sig-nal. Non-smooth optimal control theory in combination with theory on singular extremals isinvolved to obtain a sequence of subarcs that fulfills the necessary conditions of optimality.Using the necessary conditions of optimality, the fuel optimal control of a vehicle with energyrecovery options is rewritten like a nonlinear programming problem, and numerical solutionsare obtained.

2.1 Introduction

Modern vehicles are often equipped with energy recovery options. This involves hybrid

vehicles that use a secondary power converter and a storage buffer to recover and store

energy during braking or driving downhill, and Electric Vehicles (EVs) that use an

electric machine both as engine and generator.

Control strategies play an increasing role in these modern vehicle drive trains, for in-

stance, local controllers implemented for Automated Manual Transmissions (AMTs),

for clutch and engine stop-start, for power split between different power converters in

a hybrid drive train, often referred to as Energy Management Strategy (EMS), and

for (Adaptive) Cruise Control ((A)CC) where the velocity of the vehicle is adjusted by

using, e.g., the combined power output of engine and electric machine. The different

control systems have a common objective, to minimize the energy consumption, while

1This chapter has been submitted for journal publication in the form: T. van Keulen, B. de Jager,M. Steinbuch, “Optimal Trajectories for Vehicles with Energy Recovery Options”, 2011.

15

16 2 Optimal Trajectories for Vehicles with Energy Recovery Options

satisfying constraints on the driveability, comfort, drive train components, and vehicle

velocity. Optimization of individual systems separately leads to suboptimal results.

Besides, the range of EVs is relatively small due to the cost and weight of batteries.

Therefore, a prediction of the energy consumption for the upcoming route, can be

valuable information for the driver. Indicating the driver to track an optimal velocity

trajectory can prevent depletion of the battery before the desired destination is reached

and minimize energy consumption.

Several contributions regarding velocity trajectory optimization for vehicles (includ-

ing trains) with an AMT and braking capabilities have been made (Monastyrsky and

Golownykh, 1993; Ko et al., 2004; Hellstrom et al., 2008; Vasak et al., 2009). In Hell-

strom et al. (2010) the velocity trajectory optimization is solved, accounting for the

energy recovery potential of hybrid vehicles. In the previously mentioned contributions,

the optimization problem is attacked using Dynamic Programming techniques, where

the time constraint is adjoined to the cost function with a penalty function.

In Schwarzkopf and Leipnik (1977) and Stoicescu (1995), necessary conditions for an

optimal velocity trajectory, for conventional vehicles, are derived using the Pontryagin

Maximum Principle (MP), however, the non-smoothness is not fully addressed. First

order conditions of optimality, for the optimal velocity and gearshift control of conven-

tional vehicles, are then used by Passenberg et al. (2009) to derive a boundary value

problem which is solved numerically. In Outrata (1983), the MP is used to derive a

nonlinear programming (NLP) problem for solving the optimal velocity trajectory.

In Van Keulen et al. (2010b), it is shown that the route information received from a

navigation system can be used to construct a velocity trajectory optimization problem

using a novel non-smooth description of the hybrid drive train cost function which

reduces the computational complexity since only one control signal is required instead

of three (control of AMT, hybrid drive train and vehicle velocity). The cost function

was based on engineering intuition only. The resulting optimal trajectories can serve as

set point for the real-time AMT, EMS, and (A)CC, and as a range estimator for EVs.

The main contribution of this chapter is the derivation of the optimal solution for the

cost function proposed in Van Keulen et al. (2010b). It is shown that describing the cost

function with piecewise affine relations allows the analytical derivation of the optimal

solution shape.

This chapter is organized as follows. Section 2.2 discusses the cost function derivation.

Next, Section 2.3 shows the derivation of a solution shape that fulfills the necessary

conditions for optimality for the velocity trajectory for vehicles with energy recovery

options. Section 2.4 deals with the structure of the solution shape and sketches the pos-

sibilities of real-time implementation. In Section 2.5, simulation results are presented.

Finally, in Section 2.6, the chapter is summarized with conclusions.

2.2 Derivation of the cost function 17

2.2 Derivation of the cost function

Finding a control that simultaneously optimizes the AMT, EMS and (A)CC is not trivial

due to i) the nonlinear characteristics of the drive train components, and ii) the large

number of control parameters that hamper the practical implementation of numerical

solutions with, e.g., Dynamic Programming. To reduce the computational complexity,

it is proposed to simplify the problem, by approximating the energy cost of the drive

train operation and the velocity of the vehicle with a scalar piecewise affine function

which is convex and continuous and has a scalar argument.

Using a cost function of this form has the advantage that the control appears linearly

in the Hamiltonian, which enables the use of non-smooth optimal control theory devel-

oped in Clarke (1983, 2005) and Vinter (2000) in combination with theory on singular

extremals developed in Johnson and Gibson (1963); Kelley (1965); Kopp and Moyer

(1965); Bell and Jacobson (1975). In the remainder of this section, firstly, the approx-

imation of the engine and AMT, and hereafter the hybrid system, is discussed and

motivated, secondly, a formal system description is presented.

It is suggested to approximate the fuel cost of a vehicle’s engine, or electric machine in

case of an EV, at rotational velocity ω, with an affine relation:

Pf = γp,0(ω) + γp,1(ω)Pp (2.1)

with Pf the fuel input power, the internal loss parameter γp,0 > 0, the weighting pa-

rameter also called incremental cost γp,1 > 1, and Pp the delivered engine power. This

relation will later be expanded to form the cost function for the different optimization

problems.

The cost of production, transport, refinery and distribution of fuel, or production of

electric energy and charging of the battery, can be incorporated in (2.1), by multiplying

with a factor γw > 1. This allows for a well-to-wheel efficiency analysis (Williamson and

Emadi, 2005) of vehicles with different energy sources.

It is also proposed here to approximate the control of the AMT, if present, with (in-

finitely) many gear settings, such that the rotational velocity of the power converter

can be chosen virtually independent of the velocity of the vehicle, and to base the gear

ratio selection on choosing the most efficient rotational velocity ω using a predefined

level of power reserve Pv ≥ 0,

Pp + Pv ≤ ωmax(Tp(ω))

with Tp(ω) the engine torque. The optimal gear setting is obtained from

minω

∫Pf (ω)dt. (2.2)


For Pv = 0, e-line tracking is approximated: the line connecting the engine optimal

operating points (rotational velocity and torque), for each power request. Figure 2.1

depicts the equivalent fuel consumption Peq, of a medium-duty truck, as a function of

the tractive power Pr and different levels of power reserve Pv. Here, above 58P p the

power reserve is linearly build off to become zero at P p. An affine relation like (2.1)

enables a proper fit.

Changing (2.1) to,

Pf = γp,0(Pv) + γp,1(Pv)Pp,

for P p(Pv) < Pp < P p with

P p(Pv) =−γp,0(Pv)

γp,1(Pv)

the engine drag power and P p the engine maximum output power is a viable way to

incorporate the gear selection strategy in the cost function.

−200 −150 −100 −50 0 50 100 150 200−200

−100

0

100

200

300

400

500

Pr [kW]

P eq [

kW]

HEV e−line with P

v > 0 [kW]

HEV e−line with Pv > 40 [kW]

HEV approximation at Pv>0

EV approximation

Figure 2.1 / Cost function, Peq is the equivalent fuel power, and Pr the tractivepower. Abbreviation (H)EV indicates (Hybrid) Electric Vehicle.

This cost function reflects an approximation of the engine and AMT use. The simplifi-

cation of discrete gear shifting with many gear settings has two main consequences; i)

the fuel to power conversion of the engine is too optimistic, due to the limited number

of gears, engine power cannot always be delivered at the preferred rotational velocity

of the engine, see Saerens et al. (2008); and ii) the average engine incremental fuel

cost might be improved by assisting the engine with the secondary power converter at

operating points (rotational velocity and torque) with relatively high incremental fuel

cost.

2.2 Derivation of the cost function 19

The energy recovery options can be incorporated in the cost function as well. The

power conversion characteristics, of the electric machine and battery combined, are

approximated as

Ps = max(γ+m,1Pm, Pm/γ

−m,1)

or visa versa

Pm = min(γ−m,1Ps, Ps/γ+m,1)

with the weighting factor γ−m,1 > 1 and γ+m,1 > 1. The power stored in the battery, Ps, is

modeled as a piecewise affine function of the mechanical power of the electric machine

Pm. In this chapter, an electric machine is used as secondary power converter. However,

a hydraulic, pneumatic or mechanical hybrid can be treated similarly.

The tractive power Pr is split over, or provided by, the two power converters as

Pr = Pp + Pm.

In case of hybrid vehicles, the power split is obtained from

minPm

∫Pf (Pm)dt (2.3)

such that the initial and final battery energy level is the same (or is a predefined

difference). Note that Pf ≥ 0, so for an affine approximation the cost function is

convex (γp,1 > 1) and piecewise affine. Problem (2.3) can be formally solved, but it is

also possible to obtain the solution based on physical insight alone.

The optimal strategy is to generate electricity and store it in the battery if Pr drops

below the level where Pf = 0, i.e., Pr < P p, this is referred to as energy recovery. Given

the losses associated with charging the battery, using the PWA model description, it is

not profitable to charge the battery at any other occasion. The energy stored in the

battery can then be re-used at any time where Pr > P p. Because the incremental cost

γp,1 and discharge/motor factor γ+m,1 are constant, it does not matter when the battery

is actually discharged, as long as constraints on the battery state-of-energy are met and

the electric machine is used to provide tractive power.

The optimal power split can be incorporated in the cost function, in order to derive

a function of the tractive power Pr alone, using recovered energy as “negative” fuel

consumption. This is described with an affine relation for Pr between the drag power

P p and the maximum regenerative power P q, P q ≤ Pr ≤ P p. The amount of fuel saved

by storing energy in the battery and using it later is obtained from γp,1

γ+m,1γ

−m,1

Pm, with

Pm = Pr − P p ≤ 0 for charging/generating.

The EMS problem is simplified in several ways: the route dependent influences of clutch

opening, and engine stop-start are disregarded, and the temperature dependent electric


machine overload capability is not accounted for, the maximum tractive power is limited

to the maximum engine output power Pr ≤ P p, and, finally, the component description

is simplified compared to the power split problem solved by, e.g., Sciarretta and Guzzella

(2007).

The application of the service brakes can be incorporated in the cost function as well.

Applying the service brakes does not consume or recover energy, and is, therefore,

modeled with a constant in the cost function for P d < Pr < P q, where Pd is the

available brake power.

Problems (2.2) and (2.3) can be combined in a non-smooth cost function, see again

Fig. 2.1, with a single control signal Pr, the mechanical power at the wheels, instead of

three (the output power of engine and electric machine and gearbox operation)

Peq(t, Pr) = max

(γp,1(Pr(t)− P p),

γp,1(Pr(t)− P p)

γ+m,1γ

−m,1

,γp,1(P q − P p)

γ+m,1γ

−m,1

), (2.4)

here, Peq is a piecewise affine function representing the fuel consumption power. The

optimal velocity trajectory follows from the vehicle dynamics and optimal tractive power

trajectory Pr.

The minimization of the equivalent fuel cost is formulated as a standard optimal control

problem

(P1)

minPr∈U

∫ t1t0Peq(t, Pr)dt

subject to: dxdt

= f(t, x, Pr),

g(t, x) ≤ 0,

K(x0, t0;x1, t1) = 0,

where x is the state vector, Pr the control variable, and function f describes the state

dynamics which involve the nonlinear dynamics of the vehicle:

v(t) = a1Pr(t)

v(t)− a2(t, s)− a3v(t)− a4v

2(t), (2.5)

s(t) = v(t), (2.6)

where v > 0 is the vehicle velocity, s the traveled distance, a1 = 1me

> 0 the reciprocal

effective vehicle mass (including rotating parts of the drive train), a2 = mmec0ga cosα +

mmega sinα a constant related to rolling resistance and gravitational force, here m is the

vehicle mass, ga the gravitational constant, α the road slope as a function of traveled

distance, a3 = c1me

> 0 a loss parameter proportional to velocity, a4 = c2me

> 0 the

parameter for (aerodynamic) losses quadratic to velocity.

Function g involves inequality constraints on the velocity state:

v(t)− v(t) ≤ 0, (2.7)

v(t)− v(t) ≤ 0. (2.8)

2.3 Necessary conditions for optimality 21

The control variable Pr is bounded to the set:

U = [P d, P p]. (2.9)

The function K is the endpoint component of the problem, and

v(t0) = v0, s(t0) = s0, (2.10)

v(t1) = ve, s(t1) = se, (2.11)

are endpoint variables. Where ve is the desired final velocity and se the distance to be

reached.

The velocity trajectory optimization problem can be defined as a fixed-time, fixed-end-

point, non-smooth optimal control problem, in which the fuel cost is described as a

PWA continuous function. It is assumed that the final time t1 is feasible, so a solution

exists.

2.3 Necessary conditions for optimality

In this section, a non-smooth Maximum Principle (MP) is applied to derive a set of

controls that fulfill the necessary conditions of optimality. Firstly, this comprises the

definition of the Hamiltonian. The velocity constraints are adjoined to the Hamiltonian

as pure state constraints. Hereafter, the necessary conditions are stated which includes

higher order necessary conditions for the singular control subarcs in situations where

the Hamiltonian does not explicitly depends on the control. Finally, the optimal control

subarcs are derived.

The fuel optimal velocity trajectory of a vehicle is related to the well known Optimal

Control Problem (OCP) example of the fuel optimal flight of a rocket (Goh, 2008).

When the cost function is affine on the control interval (zero trust to maximum thrust),

the Pontryagin MP, extended with theory on singular extremals (Johnson and Gibson,

1963; Kelley, 1965; Kopp and Moyer, 1965; Bell and Jacobson, 1975), and with theory

on state constraint problems (Maurer., 1977; Seierstad and Sydsæter, 1987; Hartl et

al., 1995) can be applied to solve the problem. The solution consists of the extremal

controls and a singular control arc where the velocity is constant and the trust is in

equilibrium with the aerodynamic losses, see, e.g., Geering (2007, p. 62) for details on

the solution.

The Pontryagin MP does not apply to non-smooth systems as it requires the underly-

ing data to be differentiable. Several extensions of the MP to the non-smooth case are

known, see, e.g., Clarke (1983, 2005) and Vinter (2000) for an overview. The require-

ments on the underlying system can be relaxed by considering generalizations of the


derivative, that is, the adjoint multiplier functions are described in terms of a differential

inclusion set rather than explicit differential equations.

Using a set of multiplier functions p(t), the Hamiltonian is given by:

H(t) = max

(γp,1(Pr(t)− P p),

γp,1(Pr(t)− P p)

γ+m,1γ

−m,1

,γp,1(P q − P p)

γ+m,1γ

−m,1

)+

p1(t)

(a1Pr(t)

v(t)− a2(t, s)− a3v(t)− a4v

2(t)

)+ p2(t)v(t). (2.12)

The velocity constraints are adjoined using a set of nonnegative multiplier functions

λ(t) leading to the Lagrangian:

L(t) = H(t) + λ1(t) (v(t)− v(t)) + λ2(t) (v(t)− v(t)) . (2.13)

The MP (Vinter, 2000, Theorem 9.3.1 on p. 329), states that if the control is optimal,

then there exists a nontrivial piecewise continuous multiplier function:p1(t)p2(t)λ1(t)λ2(t)

6≡

0000

(2.14)

such that the following necessary conditions are satisfied:

• the adjoint inclusion p ∈ ∂v,sL holds, in which ∂v,sL denotes the generalized

subdifferential2 of L. Since the dynamics in (2.5) and (2.6) are smooth this reduces

to differential equations on the adjoint multiplier functions:

p1(t) = −∂L∂v

= p1(t)

(a1Pr(t)

v2(t)+ a3 + 2a4v(t)

)− p2(t)− λ1(t) + λ2(t),

(2.15)

p2(t) = −∂L∂s

= p1(t)∂a2(t, s)

∂s, (2.16)

• complementary slackness condition:

λ1(t) = 0 for t ∈ [r : v∗(r) < v(r)], (2.17)

λ2(t) = 0 for t ∈ [r : v∗(r) > v(r)], (2.18)

2a formal definition of a subdifferential can be found on p. xiv of this thesis.


• condition on the adjoint multiplier, see also Hartl et al. (1995, Theorem 4, p. 186),

for ta < tb in [t0, t1]:

p1(t+b )− p1(t+a ) =

tb∫ta

p1(t)dt+

∫(ta,tb]

dξ1(t)−∫

(ta,tb]

dξ2(t), (2.19)

where ξ1 and ξ2 are of bounded variation, non-increasing, constant on intervals

where v < v < v, right continuous and have left-sided limits everywhere. Equation

(2.19) is modified such that multiplier trajectory p has a discontinuity given by

the following jump condition:

p1(τ+) = p1(τ−) + µ1(τ)− µ2(τ), (2.20)

with µ1 ≥ 0 and µ2 ≥ 0. Under the assumption that ξ1 and ξ2 have a piecewise

continuous derivative, it is possible to set

λ1(t) = ξ1(t), (2.21)

λ2(t) = ξ2(t), (2.22)

for every t for which ξ1 and ξ2 exist and

µ1(τ) = ξ1(τ−)− ξ1(τ+), (2.23)

µ2(τ) = ξ2(τ−)− ξ2(τ+), (2.24)

for all τ ∈ [t0, t1] where ξ1 and ξ2 are not differentiable.

• the Hamiltonian has a global minimum with respect to Pr:

P ∗r = arg minPr

H(v∗, s∗, Pr, p∗1, p∗2) (2.25)

where v∗ is the optimal velocity state trajectory, s∗ the optimal distance trajec-

tory, P ∗r the optimal power input trajectory, p∗1 and p∗2 the corresponding adjoint

multiplier functions.

For convenience, the Hamiltonian is written as an affine relation of the control param-

eter:

H(t) = g(t, v, s, p1, p2) + h(t, v, p1)Pr. (2.26)

Minimizing the Hamiltonian yields:

h(t)P ∗r ≤ h(t)Pr. (2.27)


Switching function h is of first order with respect to the control Pr, and described by:

h(t) =

p1(t) a1

v(t)+ γp,1 in int [P p, P p],

p1(t) a1

v(t)+ γp,1

γ+m,1γ

−m,1

in int [P q, P p],

p1(t) a1

v(t)in int [P d, P q].

(2.28)

Here “int” denotes the interior of the region.

A particular situation occurs when h becomes identically zero, h ≡ 0. In that case,

H does not depend upon Pr explicitly. Although the control arc satisfies the MP, the

optimal control cannot be found directly by minimizing H, it must satisfy additional

higher order necessary conditions for optimality, see Johnson and Gibson (1963); Kelley

(1965); Kopp and Moyer (1965); Bell and Jacobson (1975), so called singular control.

The necessary conditions require that all of the derivatives of h, along the optimal

trajectory, must vanish in this time interval as well, i.e., h ≡ 0, h ≡ 0 , h(3) ≡ 0, and so

on. A derivation of the conditions on h can be found in Appendix A.

The following necessary condition for optimality is obtained:

h(t) =a1p1(t)

v(t)+ σ ≡ 0, (2.29)

which leads to:

p∗1(t) ≡ −σv(t)

a1

, (2.30)

where σ is a piecewise constant:

σ =

γp,1 in int [P p, P p],

γp,1

γ+m,1γ

−m,1

in int [P q, P p],

0 in int [P d, P q].

(2.31)

The condition on the first derivative of h becomes:

h(t) = p1(t)a1

v(t)− p1(t)

a1v(t)

v2(t)≡ 0. (2.32)

Using (2.5) and (2.15), a condition on the multiplier p2 can be derived:

p∗2(t) = p∗1(t)

(a2(t, s)

v(t)+ 2a3 + 3a4v(t)

)− λ1(t) + λ2(t). (2.33)

The condition on the second derivative of h becomes:

h(t) = p1(t)

(a1a2(t, s)

v2(t)+

2a1a3

v(t)+ 3a1a4

)+ v(t)

(−2p1(t)a1a2(t, s)

v3(t)− 2p1(t)a1a3

v2(t)+p2(t)a1

v2(t)+λ1(t)a1

v2(t)− λ2(t)a1

v2(t)

)− λ1(t)

a1

v(t)+ λ2(t)

a1

v(t)− p2(t)

a1

v(t)+ a2(t, s)

p1(t)a1

v2(t)≡ 0


Using (2.5), (2.15), (2.16), (2.19), and (2.33) the condition above can be reduced to:

h(t) = p1(t)a1v(t)

(2a3

v2(t)+

6a4

v(t)

)≡ 0. (2.34)

Therefore, h can only vanish for the singular control v ≡ 0 which explicitly contains the

control variable Pr, so, v∗ is constant and by (2.5) it then holds that:

P ∗r (t) ≡ a2(t, s)v∗ + a3v∗2 + a4v

∗3

a1

. (2.35)

From the necessary conditions it can be seen that the singular control arcs have the

following features:

• the velocity v∗ is constant,

• the tractive power P ∗r is in equilibrium with the vehicle losses (2.35), the situation

of negative velocity is neglected,

• the costate variable p∗1 is constant and attains the value p∗1 = −σv∗a1

,

• the costate variable p∗2 attains the value p∗2(t) = p∗1

(a2(t,s)v∗

+ 2a3 + 3a4v∗)−λ∗1(t)+

λ∗2(t),

• if ∂a2(t,s)∂s

= 0, i.e., the road slope is constant, and the state constraints v and v are

constant, then the costate variable p∗2 and control variable P ∗r are also constant

(2.16), so, it is concluded that, under these particular conditions, only one of the

three singular arcs is in the solution structure.

Note that, the singular control subarc in [P q, P p], in practice, only occurs at a downhill

where the electric machine force, aerodynamic drag force, and rolling resistance are in

equilibrium with the gravitational force.

Furthermore, due to the non-triviality condition (2.14), the singular solution in [P d, P q]

can only occur if p∗2 ≡ −λ1 or p∗2 ≡ λ2, i.e., when a state constraint is active.

The optimal control has the following finite set of optimal control subarcs:

P ∗r (t) ∈

P p for p1 <−γp,1v

a1,

[P p, P p] for p1 ≡ −γp,1v

a1,

P p for −γp,1v

a1< p1 <

−γp,1v

γ+m,1γ

−m,1a1

,

[P q, P p] for p1 ≡ −γp,1v

γ+m,1γ

−m,1a1

,

P q for −γp,1v

γ+m,1γ

−m,1a1

< p1 < 0,

[P d, P q] for p1 ≡ 0,

P d for p1 > 0.

(2.36)


2.4 Numerical solution

To arrive at optimal trajectories P ∗r (t) and v∗(t) two problems remain to be solved:

i) the structure of the solution (regularity), i.e., the sequence of nonsingular and sin-

gular subarcs composing the optimal trajectory, and ii) the junction points between

nonsingular and singular subarcs describing the length of each subarc.

Under the assumption that the route can be divided into segments with constant road

slope and constant velocity limitations, it is shown that the structure of the solution

can be reduced to a set of possible solution shapes. The equivalent fuel cost and travel

time can then be analytically expressed in terms of the velocities at the junction points

of the solution structure, which enables the construction of an NLP problem.

2.4.1 Structure of the solution

It is assumed that it is possible to divide the route into n segments with:

• constant velocity limitations,

• constant road slope.

Under these assumptions, the multiplier p2 is constant and only one of the three singular

arcs is in the solution structure such that at each segment the structure of the solution

is limited to a number of sequences. It is stretched that due to the affine cost function,

reaching a certain elevation requires a constant amount of fuel that has to be delivered

in any case, as long as Pp ∈ [P p, P p]. The same reasoning holds for regenerative braking

if Pm ∈ [P q, P p]. Therefore, the road slopes can be aggregated if power requests are

known to remain in the intervals [P p, P p] and [P q, P p].

Several observations limit the number of possible solution shapes. Firstly, by the con-

tinuity of p1(t), only a switch from one control extremal to the neighboring extremal,

or a singular solution, is allowed. For instance during deceleration a switch from P p

followed by P q to P d is possible. However, the sequence P p → P d → P q is not allowed.

Furthermore, a brake arc can only occur at the end of the trajectory, unless a velocity

bound is reached and a jump occurs or on a steep down slope. For the brake arc, it is

required that p1 > 0, if p∗2 < 0 it follows from (2.15) that p1 ≥ 0. So, it is not possible

to go from a P d to P q unless λ1 6= 0 or a2 < 0 “sufficiently” large, i.e., (2.33) becomes

negative for small v.

The following possible solution shapes can be expected if the singular solution Pr ∈[P p, P p] and the above mentioned assumptions are satisfied. It is acknowledged that

this forms only a subset of all possible solution shapes in the general case.

2.4 Numerical solution 27

• a maximum power subarc, a constant velocity, and another maximum power sub-

arc: P p → Pr ∈ [P p, P p]→ P p,

• a maximum power subarc, a constant velocity, and a sequence of coasting, energy

recuperation and braking: P p → Pr ∈ [P p, P p]→ P p → P q → P d, see Fig. 2.2,

• coasting and energy recuperation, a constant velocity, and a maximum power

subarc: P q → P p → Pr ∈ [P p, P p] → P p,

• coasting and energy recuperation, a constant velocity, and a sequence of coasting,

energy recuperation and braking: P q → P p → Pr ∈ [P p, P p]→ P p → P q → P d,

distance s

velo

city

v

v0

v1 v2

ve

segment isegmenti− 1

segmenti + 1

v3

v

v

s0 se

Figure 2.2 / Solution shape with a maximum power subarc, a singular solution in theinterval Pr ∈ [P p, P p], and a sequence of coasting, energy recuperationand braking.

Four similar solutions structures can be expected if the singular solution Pr ∈ [P q, P p]

or Pr ∈ [P d, P q]. In the next subsection the possible solution structures are used to

setup nonlinear functions describing fuel consumption and travel time as a function of

the velocities at the subarc junction points.

2.4.2 Construction of a nonlinear program

The optimal shape and structure of the solution, derived in Section 2.3 and 2.4.1, can

be used to write the minimization of (P1) as an NLP problem.

Firstly, it is assumed that the route is divided into segments with constant road slope

and velocity limitations. Preliminary results of route segmentation can be found in

Van Keulen et al. (2010b) and are not further discussed here.

Secondly, for each segment one of the four solution structures presented in Section 2.4.1,

is used. A priori it is unknown which of the four solution shapes is optimal, i.e.,


whether the optimal solution starts and ends with positive or negative tractive power.

By introducing an additional variable, the number of arcs before and after the singular

solution becomes equal (note that the segment end velocity is the initial velocity of

the next segment). This allows for a general solution structure: P1 → P2 → Pr ∈[P p, P p]→ P3 → P4 → P5.

Thirdly, a nonlinear cost function and nonlinear constrained functions are derived based

on the general solution structure obtained in Section 2.4.1. The equivalent fuel con-

sumption and time to complete the route are written as nonlinear functions of the

initial and final velocities at the junction points of each subarc in the solution structure

in combination with additional constraints.

The problem (P1) is reformulated as an NLP problem:

(P2)

minν∑n

i=1 Eieq(ν

i),

subject to:

vis − vi ≤ 0,

−vis + vi ≤ 0,

C(ν) ≤ 0,

where Eieq is the nonlinear fuel cost function, ν is a set of velocities describing all junction

points of all subarcs, νi is a set of velocities describing the junction points of all subarcs

in segment i, vis is the singular velocity in segment i, and vi and vi the lower and upper

velocity constraint, respectively. The number of segments in the route is n, the size

of νi is 5. The segment end velocity is also an optimization variable, except at the

last segment in the route. Optimization parameters ν are subject to linear inequality

constraints. Function C(ν) involves the nonlinear constraint functions necessary to

obtain a feasible solution:

−∆ti,l(νi) ≤ 0, (2.37)

P ∗r,s − P p ≤ 0, (2.38)

P d − P ∗r,s ≤ 0, (2.39)n∑i=1

ti(νi) + t0 − t1 ≤ 0, (2.40)

−vib,l − zijvia,l − zij

≤ 0. (2.41)

Here, ∆ti,l is the length of subarc l in segment i, P ∗r,s is the power request of the singular

solution at segment i, via,l the initial velocity of subarc l at segment i, vib,l the end velocity

of subarc l at segment i, and zij is the jth root at the ith segment of a cubic equation

which is discussed below. Subarcs are allowed to have length zero.

Eq. (2.37) to (2.39) enforce a feasible singular solution; (2.37) is an inequality con-

straints on the length of the subarc. Besides, the power request of the singular solution

2.4 Numerical solution 29

P ∗r,s is constrained. Eq. (2.40) is a constraint on the total travel time which has to be

smaller than t1 − t0, see (2.11). Constraint (2.41) is discussed below.

The required equivalent fuel power in the nonsingular subarcs is predetermined, there-

fore, the energy consumption in these arcs follows directly from the length of the arc.

From the covered distance with the nonsingular subarcs, follows the distance to be cov-

ered with the singular subarc (constant velocity). Thus, condition K in P1 is implicitly

obtained with (2.37). Since the velocity of the singular arc is also known, the travel time

∆t of this arc is easily obtained, and the fuel cost of the singular arc can be calculated

from (2.35).

The nonlinear cost and constraint functions require a description of the nonsingular arc

length as function of the arc begin and end velocity. The length (time and distance) of

each nonsingular subarc can be analytically expressed in terms of the velocities at the

junction points between the subarcs:

∆t|tbta =1

a1

3∑j=1

zj ln(vb−zj

va−zj

)a2 + 2a3zj + 3a4z2

j

. (2.42)

Here, va is the initial velocity of the subarc, vb is the final velocity of the subarc, ta is

the initial time, tb is the final time, and zj is the jth root of the cubic equation:

−a1Pr + a2z + a3z2 + a4z

3 = 0. (2.43)

Note that (2.43) can be solved a priori. The covered distance ∆s|sb

sais calculated simi-

larly by multiplying (2.42) with zj. A positive argument is required in the “ln” function

in (2.42). The cubic equation with real coefficients (2.43) has either three real roots or

one real root and two imaginary roots, see Abramowitz and Stegun (1972, p. 17).

If z1, z2, z3 are the roots of the cubic equation the following conditions hold:

z1 + z2 + z3 = −a3

a4

, (2.44)

z1z2 + z1z3 + z2z3 =a2

a4

, (2.45)

z1z2z3 =a1Pra4

. (2.46)

Since a3 > 0 and a4 > 0 it follows from (2.44) that, at least, the real part of one root is

negative.

In case Pr > 0, it follows from (2.46) that the real part of two roots is negative and

one is positive and real. Then it depends on a2 whether the two “negative” roots are

complex valued. The roots with a negative real part result in a positive term in the

“ln” function. The positive real valued root z1 in the complex conjugated situation


provides the equilibrium velocity for power P p. Physically, it is not possible to have an

equilibrium velocity in between va and vb. A feasible solution is enforced by constraint

(2.41). Moreover, it is required that va < vb < z1 or va > vb > z1. This is enforced by

(2.37).

In case Pr < 0 and a2 ≥ 0 it follows from (2.44) to (2.46) that the real part of all

roots is negative, i.e., on a flat or uphill road the vehicle decelerates during coasting,

regenerating and braking. The “negative” roots result in a positive term in the “ln”

function anyway.

Examining (2.44) to (2.46) when Pr < 0 and a2 < 0 results in one negative valued

root, and two roots with a positive real part which provide two equilibrium velocities.

The smallest positive root is an instable solution, below this velocity the vehicle will

decelerate to standstill, therefore, this velocity is implemented as lower constraint v in

P2. Between the two equilibrium velocities the vehicle will accelerate, on the other

hand when the velocity of the vehicle is above the upper equilibrium point, the vehicle

decelerates. To enforce a feasible solution, again, constraint (2.37) is required.

In addition to (2.37) to (2.41) one could add a constraint on the energy recovered Eiq

during the regenerative braking arc in segment i to account for the limited battery

capacity:

Es −Eiq(νi)

γp,1γw≤ 0, (2.47)

here, Es is the storage device capacity. However, optimality cannot be guaranteed with

this constraint.

2.5 Simulation results

From onboard navigation systems one can acquire the data, e.g., road curvature, road

slope, velocity limitations, and stopping points where the vehicle has to stand still, for

instance at a traffic light. This data can then be used to set-up an optimization problem

described above. A road slope trajectory is depicted in Fig. 2.3, and is subject to a

time constraint of t1 = 500 s.

The vehicle parameters used in this simulation example are given in Table 2.1. The

Hybrid Electric Vehicle (HEV) parameters are taken from Van Keulen et al. (2011),

while the EV parameters are taken from Besselink et al. (2010). The well-to-tank

parameters γw are calculated from Edwards et al. (2006, pp. 15–18, and pp. 48–51),

where the production cost of electric energy is based on the European average. Note

that, using this European average, the EV has, more or less, a similar well-to-wheel

energy efficiency as the diesel powered heavy-duty HEV.

2.5 Simulation results 31

0 2 4 6 8 10−100

−80

−60

−40

−20

0

20

distance [km]

elev

atio

n [m

]

Figure 2.3 / Distance vs elevation input trajectory.

0 2 4 6 8 100

5

10

15

20

25

Distance s [km]

Vel

ocity

v [

m/s

]

HEV unconstrainedHEV constrainedEV unconstrained

Figure 2.4 / Optimal velocity vs distance trajectories.

The optimal velocity v∗ and power P ∗r trajectories, as function of distance, are pre-

sented in Fig. 2.4 and 2.5, respectively. The figures show three lines indicating the

unconstrained optimal solution for the HEV in black, the constrained optimal solution,

with a velocity constraint v = 22 m/s for the HEV in dark gray, and the optimal solution

for the EV in light gray.

Both the unconstrained and constrained HEV solution have a coast arc in front of the

downhill segment, which is continued during the first part of the downhill segment, and

is followed by a regenerative braking arc. At the middle segment the “high” velocity

is maintained and then, in the unconstrained case, the vehicle is coasted to reach the

maximum velocity with which it is possible to drive uphill with a constant velocity

using maximum available power. In the constrained case, the vehicle keeps the boundary

velocity until the start of the climb, then maximum power is applied while the velocity is

gradually decreasing. The vehicle is accelerated in the final segment to reach a constant

velocity which is identical to the constant velocity at the first segment.

Since the slopes in the input trajectory of Fig. 2.3 are not steep enough for the EV to


Table 2.1 / Vehicle parameters.

Name Description HEV EV Unit

Es battery storage capacity 9.0 97.2 MJ

P d brake power -180 -80 kW

P p maximum engine power 142 50 kW

P p engine drag power -4.6 0 kW

P q regenerative power -44 -50 kW

c0 rolling resistance 0.006 0.0085 -

c1 drive train resistance 3.4 0 Ns/m

c2 aerodynamic resistance 3.33 0.35 Ns2/m2

ga gravitational constant 9.81 9.81 m/s2

m vehicle mass 13000 1190 kg

me effective vehicle mass 13375 1230 kg

γp,1 fuel cost factor 2.734 1.235 -

γ+m,1 chemical to mechanical cost 1.253 1.235 -

γ−m,1 mechanical to chemical cost 1.253 1.235 -

γw well-to-tank cost 1.162 2.857 -

reach P ∗r,s > P p, or P ∗r,s < P q, the EV solution is different. The velocity is kept constant

on the entire route, except an acceleration at the start and a deceleration at the end.

Although, 5 segments where used as input here, the solution structure converged well to

the solution structure that can be expected if the route was described with 1 segment

with averaged road slope.

The equivalent fuel consumption results are presented in Fig. 2.6. The fuel consumption

of the constrained solution is only 1.4% higher than the unconstrained solution, although

the nature of the solution is considerably different. After 5.5 km, the constrained solu-

tion has a lower equivalent fuel consumption than the unconstrained solution. However,

the unconstrained solution obtains a higher velocity during the downhill which creates

a time advantage. This time advantage then results in a lower velocity and longer coast

arc at the end of the route such that the final fuel consumption is lower.

2.5 Simulation results 33

0 2 4 6 8 10−200

−150

−100

−50

0

50

100

150

Distance s [km]

Tra

ctiv

e po

wer

P r [kW

]


Figure 2.5 / Optimal tractive power vs distance trajectories.

0 2 4 6 8 100

10

20

30

40

50

60

70

80

Distance s [km]

Ene

rgy

Eeq

[M

J]


Figure 2.6 / Equivalent fuel consumption vs distance trajectories.


2.6 Conclusions

This chapter has been concerned with the fuel optimal control of vehicles with energy

recovery options. The main contribution is the optimal control solution for a novel cost

function description which approximates the control of gear shift, energy recovery, and

the velocity of a vehicle with an affine piecewise continuous function. This approach

reduces the complexity of the problem considerably as only one control signal is used

instead of three. Describing the cost function with a piecewise affine function enables

an analytical derivation of optimal control subarcs that fulfill the necessary conditions

of optimality.

It is shown that the control structure, under constant road slope and velocity limita-

tions, contains only one of the three singular control arcs and can be reduced to only

a few possible sequences of subarcs. Each sequence can be analytically described with

the velocities at the junction points of the subarcs. With this observation the initial

control problem is rewritten as a nonlinear optimization. This reduces the computa-

tional complexity of the problem compared to direct methods, since these methods have

difficulties to converge to singular solutions. The feasibility of this approach is indicated

with a numerical example.

It is acknowledged that, at this point, optimality of the solution cannot be guaranteed.

Firstly, convexity of the resulting nonlinear programming problem is not proven which

might result in a local minimum. Secondly, only necessary conditions of optimality

are derived by the Maximum Principle. Future work should focus on finding a suitable

numerical method for solving the nonlinear programming problem and to check whether

the resulting trajectories satisfy the sufficient conditions for an optimum.

Chapter three

Optimal Power Split Control forPredefined Trajectories1

Abstract / This chapter presents a numerical solution for state constrained optimal controlproblems that does not appear to have been demonstrated previously. The approach is appliedto the power split control for hybrid vehicles for a predefined power and velocity trajectory andcompared with a Dynamic Programming solution. The computational time is 250 times fasterthan the Dynamic Programming algorithm for the same accuracy. The novel algorithm takesadvantage of the observations obtained from the Pontryagin Maximum Principle i) the opti-mal solution is described with a piecewise continuous multiplier which has a jump if the stateconstraints are hit, and ii) there is a monotonic decreasing relation between the initial mul-tiplier value and the end state of the unconstrained optimal power trajectory. Incorporatingbattery state dependent losses is possible; however, this reduces the computational advantage.The proposed methodology is illustrated with a case study. The effect of hybridization isdetermined for a long-haul truck at a duty profile with a length of 513 km.

3.1 Introduction

Hybrid Electric Vehicles (HEVs) employ an electric machine, in combination with a

battery, to recover and store energy and also to optimize the operating condition of the

prime mover. The stored energy can be used at a later time to assist the prime mover

to provide tractive power such that fuel consumption is reduced. Fuel consumption can

be optimized by controlling the power split between the prime mover and the electric

machine, using a power request from the driver, and, e.g., the rotational velocity of the

power converters, as input trajectories.

1This chapter has been submitted for journal publication in the form: T. van Keulen, J. Gillot,B. de Jager, M. Steinbuch, “Optimal Power Split Control for Hybrid Vehicles for a Predefined InputTrajectory”, 2011.

35

36 3 Optimal Power Split Control for Predefined Trajectories

Although HEVs offer a fuel consumption benefit, the additional cost of an electric

machine and especially a battery are considerable. A careful balancing of the expected

(fuel) savings and the additional cost is, therefore, required. The optimal component

sizing of electric machine and prime mover maximum power output and of battery

capacity is a challenging task. This is due to the interdependency of the component

sizes with the control algorithm that governs the power split between the prime mover

and the secondary power converter. Therefore, a fast computation of the optimal power

split trajectory, based on prescribed velocity and power request trajectories, is useful

for the optimization of drive train topology and sizing of the components (Rizzoni et

al., 1999; Filipi et al., 2004; Lukic and Emadi, 2004; Hofman et al., 2007; Sundstrom

et al., 2010b; Rotenberg et al., 2011). Besides, the optimal power split trajectory can

be used to evaluate and bench mark real-time control algorithms (Lin et al., 2003), or

to provide optimal control set points, for the real-time power split control based on

predicted power trajectories (Van Keulen et al., 2010a).

In literature, several approaches are known to tackle the power split problem for pre-

scribed velocity and power trajectories. The problem can be seen as a two point bound-

ary value problem which can be solved “directly” with, e.g., multiple shooting (Bock

and Plitt, 1984), where the original problem is reformulated as a sequence of finite

(non)linear programming problems by a parametrization of the controls and states.

In Tate and Boyd (2000), the power split problem is approximated and rewritten as

a linear programming problem, where the power conversion of prime mover and sec-

ondary power converter are approximated with a piecewise affine function. Dynamic

Programming (DP) (Bellman, 1957; Bertsekas, 2000) is another approach often applied

to optimize the power split for prescribed trajectories since it can deal with non-convex

component characteristics and constraints, and guarantees an optimal solution (with a

quantization error) consult, e.g., Kirschbaum et al. (2002); Koot et al. (2005a); Kessels

(2007); Sundstrom et al. (2010a) for details on DP in this context. DP is known to be

computationally heavy, however.

As a computationally more efficient alternative for direct multiple shooting algorithms

and DP, in Delprat et al. (2002) an indirect method is pursued which derives the neces-

sary conditions for optimality for the power split problem using the Pontryagin Maxi-

mum Principle (MP) (Pontryagin et al., 1962). The Pontryagin MP adjoins the system

dynamics to the fuel cost with a multiplier function, leading to a function referred to as

the Hamiltonian. The necessary conditions of optimality lead to the following observa-

tions under certain modeling assumptions: i) when the battery State-Of-Energy (SOE)

boundaries are not reached the multiplier function becomes continuous, ii) the multiplier

function jumps in case the SOE boundaries are reached, and iii) given the optimal mul-

tiplier function, the optimal control is obtained by minimizing the Hamiltonian locally

at each time instant.

3.2 Model description 37

In Delprat et al. (2002), the necessary conditions of optimality are used to set up

an initial value problem, using the constant multiplier function as scalar optimization

variable, calculating the unconstrained optimal SOE trajectory with a root finding

algorithm. Drawback of the algorithm presented in Delprat et al. (2002) is that the

algorithm cannot deal with state constraints and convexity of the underlying system is

required (Geering, 2007, p. 4). In Hofman (2007), it is proposed to use a combination

of heuristic rules with the root finding problem of (Delprat et al., 2002) to include

also non-convex behavior, such as stop-start of the prime mover. However, in that

case optimality cannot be guaranteed. In optimal control theory (Jacobson and Lele,

1967; Fabien, 1996), extensions to the state constrained optimal control case can be

found. These solutions involve a penalty function which takes a high value if the state

constraints are violated and a low value if the constraints are met. The penalty function

comes with the cost of an increased state dimension, however.

This chapter presents a numerical approach for scalar optimal control problems with

state constraints which appears not to have been demonstrated previously. The ap-

proach is applied to the power split control problem and adopts the method presented

in Delprat et al. (2002) and extends it to include also state constraints without the

introduction of a penalty function. A proof for optimality of the solution is included.

The algorithm is bench marked with a DP algorithm. The influence of battery voltage

increase as a function of the state-of-charge is also evaluated. Because the MP is used,

convexity of the underlying system is required.

The chapter is organized as follows. In Section 3.2, the power split problem is sketched.

Section 3.3 presents a DP algorithm as a bench mark solution for the power split prob-

lem, and Section 3.4 presents the novel numerical algorithm based on the MP. A com-

parison of the novel algorithm with DP results, on accuracy and computation time, is

presented in Section 3.5. In Section 3.6, a case study for the design of a hybrid drive

train for a long-haul truck is shown. Conclusions can be found in Section 3.7.

3.2 Model description

The objective for the power split problem is, given component size and characteristics

of the hybrid drive train, to minimize fuel consumption for a known power request Prand rotational velocity trajectory of the power converters ω, i.e., the gearshift strategy

is also known.

A requirement for the Maximum Principle, as will be further discussed later, is that the

underlying system is convex (Geering, 2007, p. 4). The suggested numerical approach

requires a monotonic relation between the adjoined multiplier and the control parameter.

Below a model description is presented which fulfills these requirements. Describing the

drive train with an affine relation for the engine and piecewise quadratic functions for the


electric machine allows for an analytic solution which is computationally advantageous,

however, not necessarily required.

A feasible input trajectory is assumed such that there is always sufficient power available

to match the power request, i.e., the mechanical output power of the prime mover Ppand electric machine Pm matches the power request:

Pr(t) = Pp(t) + Pm(t). (3.1)

The fuel cost of the prime mover, at rotational velocity ω, can be locally approximated

with a piecewise affine relation, sometimes referred to as a Willans approximation:

Pf (t) = max(0, γp,0(t, ω)P p + γp,1Pp(t)), (3.2)

here, Pf is the fuel power, parameter γp,0 > 0 describes the velocity dependent engine

drag loss, ω the rotational velocity of the drive train, P p the maximum power output of

the prime mover, fuel conversion parameter γp,1 > 1. For the component sizing problem

it is convenient to incorporate the maximum power output of the power converters in

the cost functions. At zero fuel consumption Pf = 0, the primer mover drag power P p

is given by:

P p(t) =−γp,0(t, ω)P p

γp,1. (3.3)

The conversion characteristics of the electric machine are approximated with two piece-

wise quadratic functions with a convex union:

P+b (t) = γm,0(t, ω)Pm + γ+

m,1Pm(t) +γ+m,2

Pm

P 2m(t), (3.4)

for Pm > 0 (motoring), and

P−b (t) = γm,0(t, ω)Pm + γ−m,1Pm(t) +γ−m,2

Pm

P 2m(t), (3.5)

for Pm < 0 (generating), where Pb is the electric power, and electric machine parameters

γm,0 > 0, γ+m,1 > 1, γ+

m,2 > 0, γ−m,1 < 1, γ−m,2 > 0, and Pm the maximum motoring power

of the electric machine.

Since the vehicle velocity and gearshift trajectory is prescribed, the rotational velocity

ω and parameters γp,0, and γm,0 are known as well. It is assumed that the electric

machine is always connected to the vehicle wheels such that the drag power of the

electric machine γm,0 is present anyway, and can, therefore, be incorporated in the

power request Pr.

The power converters have rotational velocity dependent power limitations:

Pm(t) ∈ U(t, ω), (3.6)

3.2 Model description 39

with

U(t, ω) = [max(Pm(ω(t)),−P p(ω(t))+Pr(ω(t))),min(Pm(ω(t)),−P p(ω(t))+Pr(ω(t)))],

(3.7)

the set of admissible controls, where Pm the maximum generator power of the electric

machine.

The battery open circuit voltage Uoc is approximated with a linear function of the

state-of-energy:

Uoc(t) = U0 + φEs(t), (3.8)

where U0 is the voltage of a fully discharged battery, φ > 0 is the battery open circuit

voltage increase factor, and Es the energy stored in the storage device. It is suggested

to approximate the battery loss power with an internal resistance model (Pop et al.,

2008):

Ps(t) = IsUoc(t, Es) = I2s (t)R + Pb(t, Pm), (3.9)

where Ps is the storage power, Is the battery current, and R the internal resistance

which, for simplicity, is here assumed to be constant. Current Is is solved from (3.9):

Is(t) =Uoc(t, Es)−

√U2oc(t, Es)− 4RPb(t, Pm)

2R. (3.10)

where U is such that U2oc > 4RPb(Pm), i.e., the battery terminal voltage is always

positive.

To protect the battery from under or overcharging, the battery energy levels are con-

strained:

Es(t)− Es ≤ 0, (3.11)

Es − Es(t) ≤ 0, (3.12)

here, Es is the maximum state-of-energy of the storage device, and Es is the minimum

state-of-energy. Parameter Pm can be used as control variable such that the state

constraints (3.11) and (3.12) are of first order, meaning that the first time derivative

contains the control explicitly.

The objective for power split control is to minimize the fuel consumption Ef for a known

power request and velocity trajectory with length t1 − t0:

Ef (t) =

t1∫t0

Pf (t, Pr, Pm)dt, (3.13)


subject to the state dynamics:

Es(t) = −Ps(t, Pm, Es), (3.14)

and constraints on the drive train components (3.6), (3.11) and (3.12). Here, (3.1) is

used to express Pf as a function of the control parameter Pm. The begin and end state

of the storage device are constrained:

Es(t0) = Es0, Es(t1) = Es1. (3.15)

3.3 Dynamic programming

In this section, a DP algorithm is presented as a bench mark solution for the power split

problem as described in Section 3.2. The prescribed power and velocity trajectories are

in discrete time with step size ∆t and length Nt. Dynamic programming involves then a

discretization of the state and control variables which allows the construction of a state

matrix, a cost matrix, and computation of the cost-to-go matrix. Note that, instead

of model descriptions (3.2) to (3.9), any static model description can be incorporated,

including more involved non-convex descriptions, without increasing the computational

complexity.

In Section 3.2, it was shown that the only state of the system is the battery energy level

(3.14). In the application of DP, the state variable is restricted to take a value on a

finite grid, resulting in a vector Es with a length NEs . The state matrix X has a size

Nt ×NEs .

The set of admissible controls in U can be approximated with a vector Ps, this allows

construction of the cost matrix C with a size Nt ×NP . Here, it is convenient to match

the grid size of the control vector with the state vector by:

∆Ps =Es − Es

NEsdt, (3.16)

such that interpolation of the state on the state grid is not required (Koot et al.,

2005a). From the control vector Ps, a vector describing the mechanical output power

of the electric machine can be defined Pm, using the component models of (3.4), (3.5)

and (3.9). Vector Pm should respect the time dependent control constraints defined by

(3.6). Infeasible controls lead to infinite cost in the cost matrix. Further details of the

algorithm can be found in Bertsekas (2000); Koot et al. (2005a) and Kessels (2007).

3.4 Root finding algorithm

It is known (Delprat et al., 2002) that an unconstrained solution (ignoring the state

constraints (3.11) and (3.12)) for the power split problem, as described in Section 3.2,

3.4 Root finding algorithm 41

can be found relatively easy. Using the observations obtained from the necessary con-

ditions of optimality derived from the Maximum Principle an initial value problem can

be constructed which can be solved with a root finding algorithm.

Our approach, which is denoted as Recursive root finding Algorithm for Scalar State

constrained Optimal Control Problems (RASS-OCP), will adopt the unconstrained so-

lution and extends it for the state constrained case. The procedure is briefly outlined:

• the unconstrained optimal trajectory is calculated,

• if state constraints are exceeded, the problem is split in two subproblems at the

time where the unconstrained solution exceeds the state constraint the most,

• the subproblem for times before the time where the maximum exceeding is reached,

has an endpoint constraint at the boundary that is reached, while the subproblem

for times after the time where the maximum is reached has an initial condition at

the boundary,

• this procedure is repeated until non of the sub-trajectories exceeds a bound (re-

cursion).

In the remainder of this section, the Maximum Principle for the power split problem

with state constraints is presented, in Section 3.4.1. The formulation of an initial value

problem for the unconstrained power split problem is described in Section 3.4.2 which is

solved with a root finding algorithm. In Section 3.4.3, the recursive loop which results

in the constrained optimal solution is proposed and optimality is proven.

3.4.1 Maximum principle

In Section 3.2, the power split problem is written as a convex optimal control prob-

lem. The non-smooth MP in combination with theory on control and state constraints

(Maurer., 1977; Seierstad and Sydsæter, 1987; Hartl et al., 1995) is required to derive

necessary conditions of optimality.

Application of a non-smooth MP requires the definition of sub-gradients as a general-

ization of the gradient for non-differentiable convex functions.

Definition 3.4.1. A vector v ∈ Rn is said to be a sub-gradient of a convex continuous

function f at a point x if the inequality

f(z) ≥ f(x) + vT (z − x) (3.17)

holds for all z ∈ Rn. The set of all sub-gradients of f(x) at the point x, called the

sub-differential at the point x, is denoted by ∂f(x).


The kernel of the cost function integral (3.13), the system dynamics (3.14), and the

multiplier function p(t), leads to the Hamiltonian:

H(t) = Pf (t, Pm, Pr) + p(t)Ps(t, Es, Pm). (3.18)

The state constraints (3.11) and (3.12), together with the multiplier functions λ1(t) and

λ2(t), result in the Lagrangian:

L(t) = H(t) + λ1(t)(Es(t)− Es) + λ2(t)(Es − Es(t)). (3.19)

Applying the non-smooth MP as in Theorem 9.3.1 of (Vinter, 2000, p. 375) the control

is optimal if there exists a nontrivial piecewise continuous multiplier function:

p(t) 6≡ 0, (3.20)

such that the following conditions are satisfied:

• differential equation on the adjoint multiplier function:

p(t) = −∂EsL(t, Es, p, Pm) (3.21)

where ∂EsL denotes the generalized sub-differential of L. The generalized sub-

differential is the set of all sub-gradients in [Es, Es]. Since Pm = 0, if Ps = 0, it

follows that L is smooth with respect to the state Es, such that (3.21) reduces to:

p(t) =∂L

∂Es= p(t)

∂Ps(Es, Pm)

∂Es+ λ1(t)− λ2(t). (3.22)

Often (Delprat et al., 2002) it is assumed that ∂L∂Es

= 0 such that p = 0 if λ1 =

λ2 = 0,

• complementary slackness condition:

λ1(t) = 0 for t ∈[r : E∗s (r) < Es

], (3.23)

λ2(t) = 0 for t ∈ [r : E∗s (r) > Es], (3.24)

• condition on the adjoint multiplier, see also Hartl et al. (1995, Theorem 4, p. 186),

for ta < tb in [t0, t1]:

p1(t+b )− p1(t+a ) =

tb∫ta

p1(t)dt+

∫(ta,tb]

dξ1(t)−∫

(ta,tb]

dξ2(t), (3.25)

where ξ1 and ξ2 are of bounded variation, non-increasing, constant on intervals

where Es < Es < Es, right continuous and have left-sided limits everywhere.


Equation (3.25) is modified such that multiplier trajectory p has a discontinuity

given by the following jump condition:

p1(τ+) = p1(τ−) + µ1(τ)− µ2(τ), (3.26)

with µ1 ≥ 0 and µ2 ≥ 0. Under the assumption that ξ1 and ξ2 have a piecewise

continuous derivative, it is possible to set

λ1(t) = ξ1(t), (3.27)

λ2(t) = ξ2(t), (3.28)

for every t for which ξ1 and ξ2 exist and

µ1(τ) = ξ1(τ−)− ξ1(τ+), (3.29)

µ2(τ) = ξ2(τ−)− ξ2(τ+), (3.30)

for all τ ∈ [t0, t1] where ξ1 and ξ2 are not differentiable,

• the Hamiltonian H has a global minimum with respect to control Pm:

P ∗m = arg minPm

H(E∗s , Pm, Pr, p∗), (3.31)

where E∗s is the optimal state trajectory, P ∗m the optimal power trajectory, p∗ the

corresponding adjoint multiplier function.

From the necessary conditions of optimality it can be seen that the optimal solution

has the following features:

• the optimal multiplier p∗ is a piecewise continuous function, where a jump occurs

if the state boundaries are reached,

• the optimization (3.13) subject to (3.14) is reduced to a stationary problem (3.31).

3.4.2 The unconstrained solution

The solution trajectory P um, that fulfills the necessary conditions of optimality, for

λ1 = λ2 = 0, is denoted as the unconstrained solution. Solving the unconstrained

optimal control problem, ignoring the state constraints (3.11) and (3.12), is relatively

easy (Jacobson and Lele, 1967) since i) the multiplier is continuous, and ii) no “guesses”

have to be made regarding the number and location of the junction points, i.e., a well

defined two point boundary value problem is obtained which is discussed next.

In this subsection, it is shown that under certain conditions, the unconstrained optimal

solution can be calculated numerically by solving a two point boundary value problem.

This requires, e.g., a shooting algorithm in combination with a root finding algorithm,

see Ascher et al. (1988). For the numerical solution the following properties are required.


Lemma 3.4.2. Let H = Pf (Pm) + pPs(Pm) with p > 0, Pf (Ps) a convex function and

Ps(Pm) monotonically increasing, then the solution P ∗m of (3.31) is a monotonically

decreasing function of p∗.

Proof. Since Ps(Pm) is monotonically increasing it follows directly from the multiplica-

tion between p and Ps(Pm) that the optimum of H by (3.31) is monotonically decreasing

with p∗: P ∗m(pa) ≤ P ∗m(pb) if pa > pb.

Lemma 3.4.2 is less restrictive than the convexity requirement on H for the MP since

it allows also the evaluation of non-convex behavior, e.g., in hybrid vehicles introduced

by the operation of the clutch or start-stop.

Lemma 3.4.3. Assume the differential equation p = A(p, t), with A(p, t) locally Lips-

chitz on a domain defined by U . Let there be a monotonic decreasing relation between

multiplier p∗ and control parameter Pm, and multiplier p(t) > 0, then there is a mono-

tonic decreasing relation between the initial value of the multiplier p(t0) and the final

state Es(t1).

Proof. The proof uses the existence and uniqueness of the solution of the differential

equation (3.22). If A(p, t) is Lipschitz, it follows that p(t) is a unique solution of (3.22)

(Khalil, 2002, Theorem 3.1), and crossing of solutions p(t) is not possible, hence p0a(t) >

p0b(t) if pa(t0) > pb(t0) where p0 denotes the trajectory resulting from p(t0). Given

Lemma 3.4.2, the monotonic relation between Pm and Ps, and by (3.14), a monotonic

decreasing relation is found between p(t0) and Es(t1).

The first part of Lemma 3.4.3 is similar to Seierstad and Sydsæter (1987, Appendix A3,

Theorem A6, p. 416) which is also generalized for vector differential equations.

A two point boundary value problem for nonlinear first order differential equations of

the form:

y = f(t, y), t0 < t < t1, (3.32)

subject to:

g(y(t0), y(t1)) ≤ 0, (3.33)

is derived. Here y = [p Es] and f is a nonlinear function described with (3.14), (3.22),

and (3.31). The constrained function g becomes:

|Es(t1)− Es1| − ε1 ≤ 0, (3.34)

where ε1 is an allowed error for the end point constraint.


Given a prescribed power and velocity trajectory and an initial multiplier value of p,

(3.31) can be solved at each time step. The initial value problem (3.32) subject to (3.34)

can be solved with a single shooting algorithm (Ascher et al., 1988, p. 137). If we denote

by f(t, E0s , p

0) the solution of (3.32) subject to the initial conditions Es(t0) = Es0 and

p(t0), the problem reduces to finding p(t0) = p∗ which solves (3.34). A standard root

finding algorithm can be used to obtain p∗, e.g., bisection, see Quarteroni et al. (2000,

p. 248).

The problem presented in Section 3.2 fits the requirements of Lemma 3.4.2 and 3.4.3,

within a restricted domain. Using the models of Section 3.2, (3.22) becomes:

p =φp(t)

2R

[(Uoc(t)−

√U2oc(t)− 4RPb(t)

)+ Uoc(t)

(1− Uoc(t)√

U2oc(t)− 4RPb(t)

)]+ λ1(t) − λ2(t). (3.35)

Note that, p ≤ 0 for all 4RPb < U2oc with p = 0 if Pb = λ1 = λ2 = 0.

Lemma 3.4.3 applies if it can be checked that Eq. (3.22) is locally Lipschitz continuous

in p and continuous in t. According to Vinter (2000, Theorem 4.9.1) to check Lipschitz

continuity of A(p) only the boundedness of the derivative remains to be tested at points

where they are defined.

It can be checked that, the multiplication of the piecewise smooth functions ∂A∂Pb

, ∂Pb

∂Pm

and ∂Pm

∂pat points where the derivative is defined, results in piecewise smooth function

that is Lipschitz on the domain U , i.e., that

∂A(p)

∂p= B + p

∂A

∂Pb

∂Pb∂Pm

∂Pm∂p

(3.36)

is bounded on this domain.

Using model description (3.2) to (3.9) with the piecewise affine fuel cost of the prime

mover, piecewise quadratic conversion cost of the electric machine, and quadratic con-

version cost of the battery, the minimization of the Hamiltonian H (3.31) requires:

∂H

∂Pm= −γp,1 − pUoc

γm,1 + 2γm,2

PmPm√

U2oc − 4R(γm,1Pm + γm,2

PmP 2m)

= 0. (3.37)

Since H is convex, condition (3.37) is also a sufficient condition for optimality. Solving


(3.37) for Pm provides an analytical expression for the optimal control value:

P ∗m,H =−Rγm,1γ2

p,1 − p2U2ocγm,1

γm,2

Pm

2γm,2

Pm

(Rγ2

p,1 + p2U2ocγm,2

Pm

) −√R2γ2

m,1γ4p,1 +Rγ2

m,1γ2p,1p

2U2ocγm,2

Pm+Rγm,2

Pmγ4p,1U

2oc + p2U4

oc

γ2m,2

P2m

γ2p,1

2γm,2

Pm

(Rγ2

p,1 + p2U2ocγm,2

Pm

) . (3.38)

Here, P ∗m,H is the argument of the minimum of H, hereby ignoring the control bounds

for the moment. Equation (3.38) needs to be solved for both charging variables γ−m,1and γ−m,2 and discharging variables γ+

m,1 and γ+m,2. If the solution of (3.38) is outside the

control set, the boundary value is used which is optimal when H is convex. Parameters

γp,0 and U are allowed to vary in time. This is illustrated in Fig. 3.1. The optimal

H

Pm

Pm = 0

P−∗m,H(p, Es)P+∗

m,H(p, Es)

U(t, Pr)U(t, Pr)

Figure 3.1 / Piecewise quadratic Hamiltonian, where the solid line indicates the fea-sible region, and the dashed line indicates the infeasible region. Theextremals are depicted with circles, in this case Pm = 0 is the globaloptimal solution.

control, taking control constraints into account, is computed by:

P+∗m (t) =

[max

(min

(P+∗m,H ,U

), 0)], (3.39)

denoting the optimum of H for Pm > 0, and

P−∗m (t) =[min

(max

(P−∗m,H ,U

), 0)], (3.40)

the minimum of H for Pm < 0, resulting in

P ∗m(t) = arg minP+∗

m , P−∗m

(H(P+∗

m ), H(P−∗m )), (3.41)

here, U = min(U), and U = max(U). Equations (3.4), (3.5), and (3.9) are then used

to calculate P ∗s . Note that, Pm = 0 is always in U . The state trajectory is found by

solving (3.14).


Since (3.4), (3.5), (3.9), (3.38) and (3.41) are analytical expressions, the unconstrained

power split problem can be solved with low computational effort. The computational

effort can be further reduced if the battery state-of-energy dependent losses are neglected

(p = 0). Then the integration of (3.32) reduces to a summation.

In this application, a shooting grid size of 1 s is used which is identical to the grid size

of the input trajectories Pr and ω. It is acknowledged that, more advanced algorithms

than single shooting and bisection can be applied such as multiple shooting (Ascher et

al., 1988, p. 175) and Newton’s method (Quarteroni et al., 2000, p. 251). However, no

numerical problems are encountered and the bisection algorithm converges fast.

3.4.3 Iterative loop

It is known from the state constrained Pontryagin MP that the optimal multiplier

becomes piecewise continuous, where a jump occurs in case a state constraint is reached,

see Section 3.4.1. If each boundary interval or contact time is known, the optimal

multiplier trajectory p∗(t) is found using the unconstrained solution method outlined

in Section 3.4.2. Next, a method is proposed to find each boundary interval or contact

time based on the times where the constraint is exceeded the most in the unconstrained

optimal trajectory such that the problem can be split in two subproblems that can

again be solved with the algorithm of Section 3.4.2. This procedure is repeated with a

recursive scheme until all constraints are met.

Theorem 3.4.4. The optimal multiplier p∗ and state E∗s trajectory for the state con-

strained optimal control problem with scalar state (e.g., as presented in Section 3.2) is

found by the following sequence:

• compute the unconstrained optimal solution, i.e., solve the initial value problem

(3.32) s.t. (3.34), if a state constraint is exceeded,

repeat

• find the time instance τn where the state boundary is exceeded the most,

τn = arg maxt∈[t0 t1]

(Es − Es(t), Es(t)− Es), (3.42)

where t0 and t1 are the initial and final time of the sub-trajectory, and n is the

nth iteration of this recursive scheme,

• split the initial (sub-)trajectory in two sub-trajectories: [t0 τn] and [τn t1]. Calcu-

late the unconstrained optimal solution at both sub-trajectories using:


– in case the upper capacity constraint is exceeded Es(τn) := Es as end point

constraint for the interval [t0 τn] and as initial condition for the interval

[τn t1],

– in case the lower capacity constraint is exceeded Es(τn) := Es as end point

constraint for the interval [t0 τn] and as initial condition for the interval

[τn t1],

until max(Es − Es(t), Es(t)− Es) ≤ 0 for all n+ 1 sub-trajectories.

Proof. The proof is based on the following two observations:

• from the jump conditions of the MP (3.25) the following proporties of the optimal

solution are obtained: p(τ+n ) > p(τ−n ) if the upper constraint is reached, p(τ+

n ) <

p(τ−n ) if the lower constraint is reached,

• using the monotonic increasing relation between p(t0) and Es(t1) described in

Lemma 3.4.3 it follows that the time at which the boundary is exceeded the most

is a contact point or part of the boundary interval, and, therefore, part of the

optimal constrained solution.

Let us denote by ∆Es the difference between the unconstrained optimal state trajectory

and the two state trajectories of the sub-problems with Es(τn) at the boundary:

∆Es(t) = E∗s (t, p∗τn)− E∗s (t, p∗u), (3.43)

with p∗u the multiplier of the unconstrained optimal solution, and p∗τn the multiplier

trajectory of the state trajectories with Es(τn) at the boundary, splitting the ini-

tial trajectory into two sub-trajectories at the point τn. It follows that |∆Es(τn)| =

max(Es − Es(t, p∗u), Es(t, p

∗u) − Es) and since we obey the initial condition and end

point constraints ∆Es(t0) = 0 and ∆Es(tn) = 0, by Lemma 3.4.3, it follows that

p∗τn(τ−n ) < p∗u < p∗τn(τ+n ) if the upper constraint is reached, and visa versa for the lower

constraint. So, the necessary conditions for optimality (3.25) are met.

The state trajectory Es(t, p∗τn) has not necessarily a global maximum at contact time

τn. It can happen that a local maximum at τn+1 of the unconstrained solution with:

Es < Es(τn+1, p∗u) ≤ Es(τn, p

∗u), still exceeds the bound in the sub-trajectory: Es <

Es(τn+1, p∗τn).

In that case another iteration is necessary, splitting the original trajectory into three

parts. Since Es(τn+1, p∗u) ≤ Es(τn, p

∗u), it follows again from Lemma 3.4.3 that

pτn+1(τ−n ) < pτn(τ+

n ), i.e., fulfilling the necessary conditions of optimality.

3.5 Evaluation of RASS-OCP 49

It can also happen that the state trajectory exceeds the lower constraint, for instance

at τn+2. However, since the above reasoning holds also on all the sub-trajectories,

optimality can be proven similarly by setting:

∆Es(t) = E∗s (t, p∗τn)− E∗s (t, p∗τn−1

), (3.44)

and so on until all constraints are satisfied.

An infeasible situation occurs if the state constraints are exceeded despite that maximum

control is applied in the whole sub-trajectory. In that case, the control is saturated on

the boundary that is exceeded.

With each iteration the length of each sub-trajectory is at least one time step smaller

than the initial trajectory. So, since the input grid has a finite size there is also a finite

number of iterations possible, so the algorithm will finish.

Note also that, if the upper constraint is reached, by the jump condition pτn(τ−n ) > pτn−1

and pτn(τ+n ) < pτn−1 , and visa versa for the lower constraint, the initial conditions of

the root finding algorithm is adjusted accordingly and computation time is enhanced.

3.5 Evaluation of RASS-OCP

In Section 3.4, RASS-OCP is presented as a novel algorithm to calculate the optimal

power split given a prescribed power request and velocity trajectory. In this section,

the RASS-OCP algorithm is evaluated on accuracy and computation time. The DP

algorithm presented in Section 3.3 is used as benchmark.

The vehicle parameters used in this simulation example are given in Table 3.1. The

power and velocity input trajectories are displayed in Figs. 3.2(a) and 3.2(b). This data

is obtained from chassis dynamo meter measurements with a 9 ton hybrid electric truck,

where the power was reconstructed from torque and rotational velocity measurements

at the dynamo drum. The measured data is corrected for rolling resistance and drive

train losses such that it represents the power requests at the crankshaft. The velocity

depicted in Fig. 3.2(b) is directly measured at the crankshaft, therefore, the gear shift

strategy can be excluded from the power split optimization.

The functioning of RASS-OCP is demonstrated in Fig. 3.3(a) and 3.3(b). The con-

strained optimal solution, for the state independent battery model, is calculated in 3

iterations, as can be seen in Fig. 3.3(a). In Fig. 3.3(b) the optimal multiplier values

are depicted. It can be seen that the unconstrained solution (red) exceeds both the

upper and lower state constraint. The lower constraint is exceeded the most, therefore,

the first iteration (blue) has a contact point on the lower constrained. In one of the


Table 3.1 / Vehicle parameters.

Name Description HEV Unit

Es lower battery storage capacity bound 1.8 MJ

Es upper battery storage capacity bound 4.1 MJ

P p maximum engine power 126.0 kW

R battery internal resistance 0.32 Ω

U0 discharged battery voltage 260 V

γ+m,1 electric to mechanical cost 1.12 -

γ+m,2 electric to mechanical cost 0.13 -

γ−m,1 mechanical to electric cost 0.88 -

γ−m,2 mechanical to electric cost 0.22 -

γp,1 incremental fuel cost 2.734 -

ε1 allowed relative end point error 3.34 0.01 -

sub-trajectories the upper constrained is exceeded, thus a second iteration (black) is

required. The calculated multiplier fulfills the jump condition (3.25).

In the next subsection, the calculated optimal constrained trajectory is compared on

accuracy and computational effort with DP results.

3.5.1 Accuracy

The RASS-OCP is evaluated by comparing the results with the DP algorithm presented

in Section 3.3. The computed state-of-energy and multiplier results, for the input

trajectories presented in Fig. 3.2(a) and 3.2(b), are depicted in Figs. 3.4(a) and 3.4(b).

Both the state dependent (blue) and state independent (green) RASS-OCP results are

compared with the state dependent DP solution (red-dashed).

The DP results overlap the results of the RASS-OCP algorithm (red-dashed on top of

blue line). The state dependent model has a state trajectory which is slightly higher

than the state trajectory resulting from the state independent model. It follows from

(3.8) that the open circuit voltage Uoc increases with state-of-energy Es, thus, a power

request Pb at a slightly higher state-of-energy requires a lower current than at low state-

of-energy. Therefore, the optimal multiplier will decrease such that the state-of-energy

is directed to higher values.

From the DP solution it is possible to provide an upper and lower bound on the multi-

plier values

Pf (P∗s )− Pf (P ∗s + ∆Ps)

∆Ps≤ p ≤ Pf (P

∗s )− Pf (P ∗s −∆Ps)

∆Ps, (3.45)

3.5 Evaluation of RASS-OCP 51

0 200 400 600 800 1000 1200 1400 1600−200

−150

−100

−50

0

50

100

150

Time t [s]

Pow

er r

eque

st P r [k

W]

(a)

0 200 400 600 800 1000 1200 1400 16000

500

1000

1500

2000

2500

Time t [s]

rota

tiona

l vel

ocity

ω [

rpm

](b)

Figure 3.2 / (a) power request input trajectory, and (b) rotational velocity inputtrajectory.

see Fig. 3.4(b). Here, the red squares indicate the upper bound and the green circles

the lower bound. In case control constraints are active than there is no direct relation

between∆Pf

∆Psand p.

The difference in fuel consumption between the state dependent and state independent

model is tiny: < 0.1%. Therefore, it is concluded that neglecting battery state-of-energy

dependent losses results in a good approximation of the fuel consumption. However,

the approximation of the optimal state-of-energy trajectories and multiplier trajectory

is less accurate. Hence, for the optimal component sizing in hybrid drive trains, the

simplified model can be used since the achievable fuel consumption is of most interest,

while for the evaluation of real-time implemented strategies it is desired to account for

the state dependent losses since the optimal control action can then be evaluated more

accurately.

3.5.2 Computational effort

The computation time of the DP algorithm depends on the grid size of the cost-to-go

matrix, see Fig. 3.5 for DP computation times for different grid sizes. The computa-

tion time of the proposed RASS-OCP algorithm depends on the number of times the

constraints are reached.

Calculation of the unconstrained solution, for the trajectories in Figs. 3.2(a) and 3.2(b)

with the RASS-OCP algorithm takes approximately 0.08 s for the simplified model

without battery state dependency and approximately 0.8 s for the model that accounts

for the battery state dependency. All computations where performed on a standard


0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time t [s]

Ene

rgy

Es [

MJ]

unconstrained solutioniteration 1iteration 2: optimal solution

(a)

0 200 400 600 800 1000 1200 1400 16002

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Time t [s]

Mul

tiplie

r p

[−]

unconstrained solutioniteration 1iteration 2: optimal solution

(b)

Figure 3.3 / (a) optimal state trajectory E∗s with intermediate iterations to achieveconstraints, and (b) optimal multiplier trajectory p∗ with intermediateiterations to achieve constraints.

laptop with a 2.00 GHz Intel dual core chip using Matlab 2007a.

Comparing the computation times of the RASS-OCP algorithms with the ones presented

in Fig. 3.5, the RASS-OCP algorithm is up to 250 times faster than the DP algorithm

with a grid size of 1000 [W] (length Ps = 80 and Es = 2280). In the worst case the

computation time of the RASS-OCP algorithm increases linearly with the number of

times a constraint is reached. In general the length of each sub-trajectory reduces after

each bound that is reached, such that the computation time increases less than linearly.

3.6 Case study

In this section, the algorithm presented in Section 3.4 is used to derive the fuel optimal

engine, electric machine, and battery size for a long-haul truck driving on a high way

trajectory in a hilly environment. The route is driven with a non-hybrid test vehicle of

40 ton and a maximum engine capacity of 340 kW. The measured vehicle velocity and

crank shaft velocity are sampled with 1 Hz and are shown in Figs. 3.6 and 3.7. The

power request trajectory, see Fig. 3.8, is estimated based on rated engine output of the

test vehicle in combination with estimated road load forces based on the velocity and

road slope information. The road slope along this trip is depicted in Fig. 3.9. During

descents the vehicle velocity increases while coasting (Pf = 0), such that the recovery

of potential energy is rather limited for this type of vehicle.

Next, the effect of hybridizing the drive train and downsizing the engine is evaluated.

The drag torque of the engine of the test vehicle γp,0(ω) is known. The incremental cost

3.6 Case study 53

0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time t [s]

Ene

rgy

E s [MJ]

RASS−OCP: state independentRASS−OCP: state dependentdynamic programming: state dependent

(a)

0 200 400 600 800 1000 1200 1400 16002

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Time t [s]

Mul

tiplie

r p

[−]

dynamic programming: ∆ Pf/∆ P

s

dynamic programming ∆ Pf/−∆ P

s

RASS−OCP: state independentRASS−OCP: state dependent

(b)

Figure 3.4 / (a) optimal state-of-energy trajectories, and (b) optimal multiplier tra-jectories.

parameter γp,1 is then scaled such that for Pp = Pr, Pf according to (3.2), Ef according

to (3.32), matched the measured fuel consumption of the test vehicle on the trajectory.

By downsizing, the engine parameter γp,0(ω) and γp,1 in (3.2) where kept constant while,

obviously, P p scaled accordingly.

Moreover, it is assumed that the electric machine maximum (dis)charge power matches

the battery maximum power throughput. It is assumed that the battery cells are con-

nected in series, such that the battery power and state limitations and open circuit

voltage Uoc scale proportionally with the battery capacity (number of cells), while the

battery internal resistance remains constant. Furthermore, parameters γ+m,2 and γ−m,2 are

kept constant and the electric machine cost are are scaled with the maximum electric

machine power according to (3.4) and (3.5).

The sizing of the hybrid drive train is, therefore, described with a single parameter, the

hybridization factor:

rh =Pm

P p + Pm

=Pm

340 [kW], (3.46)

the ratio between the electric motor power and the total power of prime mover P p and

electric motor Pm combined.

For this sizing problem, the battery losses are state independent, so p = 0. Calculation

of the constrained optimal trajectory for the 6.5 hr long trajectory required 1 to 1.5 s

depending on the number of times the constraints where hit which ranged from 9 to 19

times. The optimal state-of-energy trajectory, for a hybridization factor rh = 0.18 is

depicted in Fig. 3.10. The fuel consumption as a function of the hybridization factor is


0 500 1000 1500 20000

50

100

150

200

250

300

350

Grid size dPs [W]

Com

puta

tion

time

[s]

Figure 3.5 / Computation time vs grid size of dynamic programming for the inputtrajectories.

0 50 100 150 200 250 300 350 400 450 500 5500

10

20

30

Distance [km]

Vel

ocity

[m

/s]

Figure 3.6 / Velocity vs distance.

shown in Fig. 3.11. The shape of the solution is that the fuel consumption is decreasing

with increasing hybridization factor.

Using hybridization factors rh larger than 0.2, at some part of the route all potentially

recoverable energy is recovered. Therefore, adding more generator power does not lead

to lower fuel consumption at that point of the route. Consequently, the fuel consumption

reduction for rh > 0.2 is less pronounced.

The absolute fuel consumption reduction, for long-haul trucks, is rather limited though,

the reduction is approximately 6.2% for a hybridization factor of 0.4. Nevertheless, due

to the high milage (> 1 · 106 km) and fuel consumption (> 3 · 105 l) during the lifetime

of the vehicle, a hybrid drive train might still be cost effective. A profound cost analysis

is outside the scope of this chapter, however.

3.7 Conclusions 55

0 1 2 3 4 5 6 70

500

1000

1500

2000

2500

Time [hr]

Rot

atio

nal v

eloc

ity [

rpm

]

Figure 3.7 / Rotational velocity vs time.

0 1 2 3 4 5 6 7−600

−400

−200

0

200

400

Time [hr]

Pow

er r

eque

st [k

W]

Figure 3.8 / Power request vs time.

3.7 Conclusions

In this chapter, the optimal control of power split in hybrid vehicles based on a known

power and velocity trajectory has been addressed.

A novel algorithm called the RASS-OCP is presented which calculates the global optimal

power split trajectory with a computational effort up to 250 times smaller than Dynamic

Programming which is often applied in this context. The novel algorithm can also

include state dependent battery losses, in that case some of the computational advantage

is lost.

The simple model, neglecting the state dependent losses, is useful for the component

sizing problem, since it is able to predict the fuel consumption results fast and accurately.

The model including battery state dependent losses is more useful for the evaluation of

(optimal control based) real-time strategies, since it calculates the optimal multiplier

and state trajectory more accurately, and for the evaluation of storage devices that

strongly depend on the state-of-energy, e.g., super-capacitors.

As an example optimal hybrid component sizes were determined for a long-haul truck us-

ing a 513 km input trajectory. The fuel consumption decreases for increasing hybridiza-

tion factor. For this route, the fuel consumption reduction is 6.2 % at a hybridization

factor of 0.4.


0 50 100 150 200 250 300 350 400 450 500 550−200

0

200

400

600

800

Distance [km]

Ele

vatio

n [m

]

Figure 3.9 / Elevation vs distance.

0 50 100 150 200 250 300 350 400 450 500 5500.2

0.4

0.6

0.8

Distance [km]

Sta

te o

f Ene

rgy

[−]

Figure 3.10 / Optimal state-of-energy trajectory vs distance for a hybridization fac-tor rh = 0.18.

0 0.1 0.2 0.3 0.4174

176

178

180

182

184

186

188

Hybridization factor [−]

Fue

l con

sum

ptio

n [L

]

Figure 3.11 / Fuel consumption vs hybridization factor.

Chapter four

Real-time Power Split Control inHybrid Vehicles1

Abstract / Hybrid vehicles require an algorithm that controls the power split between theprime mover and secondary power converter(s) and stop-start of the prime mover. A real-timeoptimal control based power split algorithm is implemented on a standard Electronic Com-putation Unit of a parallel hybrid electric truck. The implemented strategy is experimentallyvalidated on a chassis dynamo meter. The fuel consumption is measured on twelve differenttrajectories and compared with a heuristic and a non-hybrid strategy. The optimal controlstrategy has superior performance on all trajectories that are evaluated, except one.

4.1 Introduction

Hybrid vehicles have, at least, two power converters instead of one: usually a prime

mover which can provide tractive power, consuming fuel with an irreversible process,

and a secondary power converter which converts tractive power, reversibly, into a power

quantity suitable for a storage device, or visa versa. In a parallel hybrid drive train,

the prime mover and secondary power converter have a mechanical coupling. In a series

hybrid, the prime mover drives a generator and is coupled, e.g., electrically with the

storage device.

An important benefit of the secondary power converter is energy recovery during braking

or driving downhill. This energy can be used at a latter, more convenient, time to

propel the vehicle. The secondary power converter enlarges the maximum available

tractive power and enables load shifting and start-stop of the prime mover. Stopping

the prime mover is an interesting feature, because the prime mover consumes fuel during

1This chapter has been submitted for journal publication in the form: T. van Keulen, D. van Mullem,B. de Jager, J.T.B.A Kessels, M. Steinbuch, “Optimal Power Split Control in Hybrid Vehicles”, 2011.

57

58 4 Real-time Power Split Control in Hybrid Vehicles

“idling” (when the prime mover delivers zero tractive power), and emissions and noise

are temporarily reduced. Examples of prime movers are an internal-combustion engine

or a fuel cell. Examples of the secondary power converter and energy storage are,

electric, hydraulic (Filipi and Kim, 2009), pneumatic or mechanic (Van Berkel et al.,

2010) systems.

A supervisory control algorithm, i.e., the Energy Management Strategy (EMS), deals

with the balanced generation and re-use of stored energy, using the prime mover start-

stop, and the power split between primary and secondary power converter as control

variables. Several contributions discuss the EMS design for hybrid vehicles, see Sciar-

retta and Guzzella (2007) for an overview.

Given a power and velocity trajectory, strategies using, e.g., necessary conditions for

optimality obtained with the Pontryagin Maximum Principle (MP) (Delprat et al., 2002,

2004; Wei et al., 2007; Serrao et al., 2009; Bernard et al., 2010; Ambuhl et al., 2010),

or the related Equivalent Consumption Minimization Strategies (ECMS) (Paganelli et

al., 2001, 2002; Sciarretta et al., 2004; Musardo et al., 2005; Rodatz et al., 2005), can

be used to obtain the global optimal solution.

Strategies that have no knowledge of the future power and velocity trajectories, and that

use only information available at the actual time, are referred to as causal strategies.

They rely, e.g., on heuristic rules (Schouten et al., 2003; Lin et al., 2004; Hofman et al.,

2007) or on optimization that is based upon the observations obtained with the MP and

is solved in real-time (Johnson et al., 2000; Kleimaier and Schroder, 2002; Delprat et al.,

2004; Koot et al., 2005a; Van Keulen et al., 2008; Borhan et al., 2009; Ripaccioli et al.,

2009; Bernard et al., 2010; Van Mullem et al., 2010; Ambuhl et al., 2010). The design

of optimal control based real-time implementable EMS boils down to i) estimation of a

multiplier function that adjoins the fuel cost to the energy stored in the storage device

using real-time available information, and ii) optimization of a locally approximated

Hamiltonian.

Often the characteristics of the secondary power converter require a non-smooth mod-

eling, where a non-smooth continuous function models the charging and discharging.

Since the Pontryagin MP requires the underlying system to be convex and, at least,

once continuously differentiable (Geering, 2007, p. 4), a non-smooth version of the MP

is required.

Frequently, the estimation of the multiplier is based on feedback on the current battery

state-of-energy using a constant reference (Kleimaier and Schroder, 2002; Delprat et

al., 2004; Koot et al., 2005a; Bernard et al., 2010; Ambuhl et al., 2010). Particulary

in situations where the recoverable energy is large compared to the battery capacity a

constant reference is restrictive since recovery and storage of energy leads to deviation

in the state-of-energy.

4.2 Problem formulation and necessary conditions for optimality 59

Simulation results indicate the fuel consumption advantages of optimal control based

strategies over rule-based strategies (see, e.g., Pisu and Rizzoni, 2007). Nevertheless,

it is believed that, in most commercially available hybrid vehicles, rule-based strategies

are implemented, although it is hard to find out exact figures. One reason industry

prefers rule-based strategies is that requirements on available computational power are

believed to hinder the application of real-time optimization.

This chapter is concerned with the design, implementation, and evaluation of optimal

power split control in hybrid vehicles. The contributions are i) the application of a

non-smooth MP to the power split control problem, ii) a methodology for the design of

optimal control based real-time implementable EMS, with a multiplier estimation which

is adaptive for future energy recovery potential by accounting for the current kinetic

energy and potential energy, iii) an implementation of optimal control based EMS, on

a standard Electronic Computation Unit (ECU) in a hybrid electric truck, and iv) an

experimental validation of the implemented EMS.

The chapter is organized as follows. In Section 4.2, the power split control problem is

stated (Section 4.2.1), and non-smooth optimal control is involved to derive necessary

conditions of optimality for systems where the charging and discharging of the storage

device is modeled with a piecewise continuous function (Section 4.2.2). Section 4.3 deals

with the design of a real-time implementable strategy, which amounts to estimation of

the multiplier function (Section 4.3.1), and approximation and optimization of a local

Hamiltonian function (Section 4.3.2). In Section 4.4, the component characteristics of

the test vehicle and implemented strategy are discussed. Experimental results are pre-

sented in Section 4.5, including an overview of the test setup (Section 4.5.1), tuning of

the real-time multiplier estimation (Section 4.5.2), and evaluation of the performance

at different trajectories, bench marked with a heuristic and a non-hybrid strategy (Sec-

tion 4.5.3). Finally, conclusions and recommendations are in Section 4.6.

4.2 Problem formulation and necessary conditions foroptimality

In this section the power split control problem is outlined. The necessary conditions of

optimality, obtained with a non-smooth version of the MP, are presented.

A schematic overview of a hybrid vehicle drive train is depicted in Fig. 4.1. The con-

version of fuel power Pf to the prime mover output power Pp is modeled as a function

of the power throughput:

Pf (t, S, Pp) =

Pf,i for S = 0,

max(0, Pf,p(t, Pp)) for S = 1,(4.1)


primemover

storagedevice

Pr

Pf Pp

Ef

EsPb Pm

S

secondarypower

converter

Ps

Figure 4.1 / Schematic overview of a hybrid drive train.

where S is a switch for clutch opening, Pf,i is the fuel power during idling, and Pf,p the

fuel power if tractive power is delivered. If S = 0 than Pp = 0. In case start-stop of the

prime mover is possible Pf,i = 0, see Fig. 4.2a.

In many hybrid vehicle applications, the characteristics of the secondary power converter

and storage device require a non-smooth modeling, where charging and discharging is

modeled with a non-smooth continuous function, see Fig. 4.2b. The combined conver-

sion of storage power Ps to the electric power Pb, and of Pb to the secondary converter

output power Pm is, therefore, modeled as a non-smooth function of the power through-

put:

Ps(t) =

Ps ch(t, Pm, Es) for Pm < 0,

Ps dis(t, Pm, Es) for Pm ≥ 0,(4.2)

in which Ps ch is the storage power during charging of the storage device, Ps dis is the

storage power during discharging of the storage device, see Fig. 4.2b. The conversion

could depend on the stored energy in the battery Es. It is assumed that the influence

of other variables on the conversion efficiency, e.g., rotational velocity of the electric

machine, temperature, and ageing are known, and are incorporated in the cost function

at time t. Note also that the drag power of the secondary power converter is always

present (since decoupling the secondary power converter from the wheels is not possible)

and can be incorporated in the power request Pr such that Pm = 0 if Ps = 0.

Pf

Pp

Ps

PmP p

P fP s dis

PmP p

P s ch

Pm

S=0

S=1

a) b)

Figure 4.2 / Schematic cost functions of a) prime mover and b) secondary powerconverter and storage device.

It is assumed that Pr is a known and feasible driver input from the gas pedal only, such

4.2 Problem formulation and necessary conditions for optimality 61

that the operation of the service brakes can be ignored and the system can be modeled

with the independent scalar control Pm, and Pp becomes dependent by:

Pp(t) = Pr(t)− Pm(t). (4.3)

4.2.1 The power split control problem

Objective for the power split control is to minimize the fuel consumption Ef for a driving

mission with arbitrary length, subject to constraints on the storage device. This can be

written as a standard optimal control problem, with t1 the end time of the trajectory,

subject to the storage dynamics:

Ef (t) = Pf (t, S, Pm), (4.4)

and

Es(t) = −Ps(t, Pm, Es). (4.5)

Using (4.3), Pf becomes a function of Pm and t.

Several constraints are present in the control problem: the power converters have power

limitations as was already indicated in Fig. 4.2, the control is bounded:

Pm(t) ∈ U(t, Pr), (4.6)

with

U(t, Pr) = [max(Pm(t),−P p(t) + Pr(t)),min(Pm(t),−P p(t) + Pr(t))], (4.7)

the set of admissible controls. Where P p is the “drag” power of the prime mover

at zero fuel consumption, P p the maximum engine output power, Pm the maximum

regenerative power, and Pm the maximum motoring power.

The begin and end states of the storage device are constrained:

Es(t0) = Es0, Es(t1) = Es1. (4.8)

In case of hybrid electric vehicles, often a charge-sustaining solution is imposed, i.e.,

Es0 = Es1. There are situations where other endpoint constraints are preferred, however.

Before entering an inner city green-zone it is advantageous to charge the storage device

such that the vehicle drives fully electric in the green-zone and the emissions and noise

are reduced temporarily. Considering optimal control on the trajectory before the green-

zone results in an endpoint constraint above the initial battery level: Es1 > Es0.

For hybrid vehicles with a large battery capacity, it is appealing to recharge the battery

from the grid, these vehicles are referred to as plug-in hybrids (Gong et al., 2008; Stockar

et al., 2010). In that case, the endpoint constraint becomes Es1 ≥ Es, where Es is the

minimum energy level of the storage device.


4.2.2 Necessary conditions of optimality

In the previous section, the power split control problem, for systems with non-smooth

dynamics, is written as a standard optimal control problem. In this section, the nec-

essary conditions of optimality are derived using a non-smooth MP. The observations

provided by the MP can be used for the derivation of real-time implementable strategies.

During the last decades, a non-smooth version of the Pontryagin MP is developed, see

Clarke (1983, 2005) and Vinter (2000) for an overview on non-smooth optimal control.

The MP adjoins the storage device dynamics to the fuel cost with a multiplier function

p, leading to a function referred to as the Hamiltonian:

H(t, Es, p, S, Pm) = Pf (t, S, Pm) + p(t)Ps(t, Pm, Es). (4.9)

A convexity hypothesis is required for the application of the MP. If function

H(Es, S, Pm) fails to be convex, there may be no unique local minimizer (Vinter, 2000,

p. 92). Function H is referred to as convex depending on whether H(Es, S, Pm) has

the referred-to property for all Pm ∈ U and Es ∈ [Es, Es], where Es the maximum

energy level of the storage device. Clearly, S = 0 leads to a non-convex H, therefore,

for the application of the MP it is assumed that S ≡ 1 and Pf (Pm) and Ps(Pm, Es)

are (approximated) such that H is convex. The situation S = 0 is implemented in the

real-time control on a heuristic basis in Section 4.3.

Applying the non-smooth MP as in Theorem 9.3.1 of (Vinter, 2000, p. 375), the control

is optimal if there exists a nontrivial continuous multiplier function:

p(t) 6= 0, (4.10)

such that the following necessary conditions are satisfied:

• the multiplier function obeys the differential equation:

−p(t) = ∂EsH(t, Es, p, Pm), (4.11)

where ∂EsH denotes the generalized subdifferential of H. The generalized subdif-

ferential is the set of all subgradients in U . Since Pm = 0 if Ps = 0 it follows that

(4.5) is smooth with respect to the state Es, such that (4.11) reduces to:

−p(t) =∂H

∂Es= p(t)

∂Ps∂Es

, (4.12)

i.e., the non-smoothness associated with Ps, (4.2), does not affect the differentia-

bility of p. Often it is assumed that ∂Ps

∂Es= 0 such that p = 0.

• the Hamiltonian has a global minimum with respect to Pm:

P ∗m(t) = arg minPm

H(t, E∗s , Pm, p∗), (4.13)

where ∗ indicates an optimal trajectory.

4.3 Real-time implementable energy management strategies 63

The necessary conditions of optimality lead to the following observations:

• the multiplier function, p(t) is a scalar continuously differentiable function, and

when the state dependent losses are ignored, the multiplier function p becomes

constant,

• given the optimal multiplier function p∗, the optimal control is obtained by mini-

mizing the Hamiltonian locally at each time instant.

So, indeed the optimal solution for the power split control problem, also for the non-

smoothness associated with charging/discharging of the storage device, can be described

with one (constant) multiplier function. This generalizes the observations obtained with

the Pontryagin MP for smooth systems to systems with non-smooth characteristics of

the type used here. These observations are used for the design of a real-time imple-

mentable strategy which is discussed next.

4.3 Real-time implementable energy managementstrategies

Input to the real-time EMS are actual vehicle signals, such as accelerator pedal posi-

tion, vehicle velocity, operating conditions of the power converters and state-of-energy.

Additional demands on the real-time implementable EMS are imposed by the limited

computational power and storage capacity of the ECU. To cope with these difficulties,

several approaches are suggested in literature which are discussed in this section.

EMS that take advantage of the observations obtained by the MP, essentially boil down

to: i) find a suitable estimation of the multiplier function by using real-time available

information only, and ii) solve the local minimization of the Hamiltonian with the limited

available computation power.

4.3.1 Real-time estimation of the multiplier function

Real-time estimation of the multiplier p based upon available vehicle signals is the key

factor for the success of optimal control based EMS.

It is convenient to define the state-of-energy as the actually stored energy normalized

for the total storage capacity:

SOE(t) =Es(t)

Ec, (4.14)

where, Ec is the theoretical battery capacity. The battery limitations SOE > 0 and

SOE < 1 are used.


Since p is closely related to the state-of-energy, a too large multiplier results in over-

charging of the storage device while a too small multiplier leads to depletion. Delprat

et al. (2002) and Kleimaier and Schroder (2002) suggest to apply proportional feedback

on the battery state-of-energy to estimate the multiplier. Paganelli et al. (2001); Pisu

and Rizzoni (2007) and Ambuhl et al. (2010) suggest to apply nonlinear feedback on

the state-of-energy, to reduce control actions for a state-of-energy around the reference.

A polynomial description of the feedback becomes:

p(t) = p0(t) +n∑i=1

K2i−1(SOEr(t)− SOE(t))2i−1, (4.15)

here, p is the estimated multiplier, p0 an “initial” guess, K2i−1 the (2i− 1)th feedback

gain, SOEr the reference state-of-energy, and SOE the actual state-of-energy. Fig-

ure 4.3 depicts the relation between SOE and p for n = 1 (solid) and n = 2 (dashed).

Typically, the nonlinear feedback controllers allow for more freedom around the refer-

ence state-of-energy, while having a “higher” feedback gain close to the state-of-energy

bounds. Disadvantage of nonlinear feedback algorithms is that the tuning requires more

effort.

p(t)

SOE(t)

p0

SOErSOE

SOE

Figure 4.3 / Linear (solid) vs nonlinear (dashed) feedback on the state-of-energy toestimate the multiplier function p(t).

The feedback approaches discussed so far use a constant SOEr. Since energy recovery

forms the major part of the fuel savings due to hybridization, it was observed (Rodatz

et al., 2005) that the vehicle itself is a reversible energy storage system. Therefore,

Van Keulen et al. (2008) reasoned that the state-of-energy reference set point can be

corrected for expected future energy recoveries, by accounting for current kinetic and

potential energy of the vehicle:

SOEr(t) = SOE0r (t)−KEm

12v2(t)−Khmga(h(t)− h0), (4.16)

where SOE0r is the preferred reference at zero kinetic and potential energy, KE a tuning

parameter that estimates the, driver dependent, kinetic energy recovery efficiency, m the

estimated vehicle mass, v the actual vehicle longitudinal velocity, Kh a tuning parameter

estimating the recovery efficiency of potential energy, ga the gravitational constant, h the

estimated elevation, and h0 the reference elevation. Real-time estimation of m based

4.3 Real-time implementable energy management strategies 65

on actual vehicle parameters is possible (Vahidi et al., 2005). An adaptive state-of-

energy reference leads to more freedom in the storage device operation since deviations

of state-of-energy are only penalized if they are different from the expected kinetic and

potential energy recovery.

For charge-sustaining hybrids, the control set point SOE0r can be constant and predic-

tive information is not necessarily required, while, for plug-in hybrids, or green-zone

entering, a time (or distance) dependent set point is required which involves, at least,

information about the distance to the next green-zone or plug-in location; e.g.,

SOE0r (t) = SOE0 +

SOEf − SOE0

strips(t), (4.17)

where SOEf is the endpoint constraint state-of-energy, SOE0 the initial state-of-energy,

strip the distance to the next plug-in station or green-zone, and s(t) the distance traveled

at time t.

Finally, EMS based on (4.15) are suited to include predictive information. If geographic

information of the route ahead is available, optimization methods can be used to esti-

mate p0 and optimal state-of-energy reference trajectory SOEr (see, e.g., Kessels and

Van den Bosch, 2008; Van Keulen et al., 2010a). This contribution limits itself to EMS

that has access to in-vehicle real-time information only, however.

4.3.2 Real-time minimization of the Hamiltonian

Real-time optimization of (4.13), with the limited available computational power forms

the second challenge, after estimation of the multiplier function, for the successful im-

plementation of optimal control based EMS. In the literature several approaches are

suggested.

Koot et al. (2005a) suggests to approximate the Hamiltonian, locally for Pm ∈ U(t),

given the actual operating conditions of the power converters, as a quadratic function,

which allows for an analytical solution, see Fig. 4.4a. Solving a quadratic programming

problem requires little computational effort, and, thus, forms a good candidate for real-

time implementation.

A similar approach is used in Ambuhl et al. (2010) where clutch opening is included,

and H is approximated as convex piecewise quadratic, allowing for non-smoothness in

the approximation but still having an analytical solution, see Fig. 4.4b. The fuel cost

associated with clutch opening and stop-start, where the power request is completely

covered by the secondary power converter, Pm = Pr, is calculated by Pf,i+ p(t)Ps(t, Pr),

where Pf,i describes the fuel cost of the prime mover after clutch opening and is com-

pared with the analytical solution of the two quadratic functions presented in Ambuhl

et al. (2010).


H

PmPmPm

Pm = Pr

Pm = 0

↑p

a) b) c)

Figure 4.4 / Minimization of a local H with in dark solid the original data, the darkdisk indicates the minimum of the approximation, with a) a quadraticapproximation (gray), b) a piecewise quadratic approximation (gray),the circle shows the cost associated with clutch opening (S=0) wherethe power request is covered by the secondary power converter Pm=Pr,and c) the grid approximation.

Note that by including non-convex behavior, the existence of a unique minimizer is not

guaranteed anymore by the MP. However, stop-start of the prime mover can lead to

lower fuel consumption under certain conditions. Therefore, it is useful to evaluate the

cost of stop-start, while compromising on the guarantee of a global optimal solution.

Technically speaking, the resulting EMS is somewhat heuristic, rather than fully optimal

control based.

In case one would like to include also the non-convex behavior of the components, a

numerical solution using a grid is possible, see Van Mullem et al. (2010) and Fig. 4.4c.

This involves calculation of the cost at each grid point and finding the minimum.

The suggested optimization problems could be solved in real-time. If the computational

load is too high, in Van Mullem et al. (2010) it is suggested to solve the optimization

off-line for a range of multiplier values and relevant component characteristics, and store

the solution in a look-up table. Disadvantage of look-up tables is that the size of the

table grows exponentially with the number of dimensions. Storage capacity of the ECU

can become an issue, if one likes to include more component characteristics, such as

temperature influences or ageing.

The influence of multiplier p can be seen in Fig. 4.4a, an increasing p “rotates” the

Hamiltonian anticlockwise around Pm = 0. For a convex H this implies that a contin-

uous monotonically decreasing relation between p and the optimal Pm can be found,

see Fig. 4.5. A minimum can be found at Pm = 0 for a range of multiplier values, see

Fig. 4.4b, due to the non-smoothness for charging/discharging. The non-smoothness

creates a dead-zone in the relation between p and Pm.

If the underlying data is non-convex, for instance, due to introduction of stop-start, the

relation between p and Pm is not continuous anymore and a jump in Pm may occur.

4.4 Component characteristics and implemented strategy 67

Pm

p(t)

Pm

Pm

0p

p

(discharging)

(charging)

Figure 4.5 / The continuous monotonically decreasing relation (solid) between mul-tiplier p and optimal secondary power converter output power Pm. Incase H is non-convex, the relation is discontinuous (dashed).

4.4 Component characteristics and implemented strat-egy

This section describes the component characteristics of a test vehicle, and the imple-

mentation of an optimal control based strategy on a standard ECU.

A parallel hybrid electric truck is used as carrier, see Fig. 4.6 for a schematic overview

of the drive train. The prime mover is a 142 kW diesel engine, and the secondary power

converter is a 44 kW electric machine which is connected to a 5.2 Ah lithium-ion battery

pack. Power electronics match the electric machine power request to the battery open-

circuit voltage which ranges between 310 and 360 V. Both power converters are situated

in front of a six speed Automated Manual Transmission (AMT). The gearshift strategy

is implemented separate from the power split control and is not further discussed.

engineelectric

clutch

gearbox final

power battery wheels

service

machinedrive

electronics

brakes

Figure 4.6 / Schematic overview of the drive train components in a parallel hybridelectric truck.

The vehicle can operate engine-only, where all tractive power is delivered by the diesel

engine, full-electric where the vehicle is powered solely by the electric machine, or the

tractive power can be split over the two power converters. The clutch can be opened to

disconnect the diesel engine from the drive train. Engine stop-start is not implemented

since several engine auxiliaries (power steering and pneumatic pump) required the engine

to operate continuously. To brake the vehicle the electric machine can operate as a

generator and assist the conventional service brakes.


4.4.1 Component characteristics

The steady-state component characteristics of the diesel engine are depicted in Fig. 4.7.

The characteristics of the electric machine and power electronics combined are depicted

in Figs. 4.8 and 4.9. The battery characteristics are depicted in Figs. 4.10 and 4.11. The

influence, of rotational velocity and battery state-of-energy, on the power conversion

of the two power converters is included. Other influences as component dynamics,

temperature, or ageing, are not included since no information was available.

The conversion efficiency of the diesel engine, of fuel power to output power at the

crank shaft, is measured under steady-state conditions at different rotational velocities

ωp, see Fig. 4.7. Around 1500 rpm and T > 400 Nm the fuel conversion is relatively

efficient compared to operating points below 1500 rpm. This results in non-convexity

of the Hamiltonian.

Engine speed [rpm]

Tor

que

[Nm

]

800 1000 1200 1400 1600 1800 2000 2200−200

−100

0

100

200

300

400

500

600

700

800

0

100

200

300

400

500

Figure 4.7 / Contour plot of the engine fuel consumption Pf [kW] as a functionof torque Tp and rotational velocity ωp. The dark line indicates themaximum engine torque T p. The gray line indicates the engine dragtorque T p.

The conversion of electric machine mechanical power Pm to electrical power Pb, and

visa versa, is also measured under steady-state conditions, see Fig. 4.8. Since the

rotational velocity of the electric machine ωm is prescribed by the vehicle velocity and

shift strategy, the rotational velocity dependent steady-state losses can be excluded

from the component description, since these losses are independent of control Pm. If

the clutch is closed (S = 1) the rotational velocity of the engine and electric machine is

equal ωp = ωm. To indicate that a non-smoothness is present, the gradient ∂Pb

∂Pm, at one


rotational velocity, is also depicted in Fig. 4.9.

Rotational velocity [rpm]

Tor

que

[Nm

]

200 600 1000 1400 1800 2200

−400

−300

−200

−100

0

100

200

300

400

−60

−40

−20

0

20

40

60

Figure 4.8 / Electric power Pb [kW] as a function of electric machine torque Tm androtational velocity ωm. The upper dark line indicates the maximumelectric machine torque Tm. The lower dark line indicates the maximumgenerator torque Tm.

−50 −40 −30 −20 −10 0 10 20 30 40 500

0.5

1

1.5

2

Electric machine mechanical power Pm

[kW]Sto

rage

pow

er g

radi

ent ∂ P b/∂

Pm

[−]

Figure 4.9 / Storage power gradient ∂Pb∂Pm

[kW] as a function of electric machine me-chanical power Pm [kW] at rotational velocity ωm = 1000 [rpm].

The conversion characteristics of the lithium-ion battery can be described with an in-

ternal resistance model, see Pop et al. (2008), this leads to the relation:

Ps(t, Pm) = Is(t)Uoc(t) = I2s (t)R(t) + Pb(t, Pm), (4.18)

where Is is the battery current, Uoc the open-circuit voltage, and R the internal resis-


tance. Solving (4.18) for Is gives:

Is(t, Pm) =Uoc(t)−

√U2oc(t)− 4R(t)Pb(t, Pm)

2R(t), (4.19)

The battery open-circuit voltage Uoc is estimated by discharging the battery with the

lowest possible current request (where the vehicle drives at a footpace), see Fig. 4.10.

Besides, under the assumption that the internal resistance R is equal for charging and

discharging (Pop et al., 2008), the measured terminal voltage Ut during charging and

discharging is averaged to estimate Uoc. The measured voltage is quantized with 2 V.

Figure 4.10 shows that a fully charged battery has a higher voltage then a depleted

20 30 40 50 60 70 80310

330

350

370

State−of−energy SOE [%]

Bat

tery

vol

tage

U oc[V

]

Figure 4.10 / Estimated battery open-circuit voltage Uoc as a function of the state-of-energy SOE.

battery. Therefore, a charged battery requires a lower current to deliver a certain power

request, and the internal losses are also smaller.

Given the estimated open-circuit voltage Uoc, the measured terminal voltage Ut and

current Is, the battery internal resistance is estimated with:

R =Uoc − Ut

Is. (4.20)

The computed internal resistance for one charge/discharge cycle is depicted in Fig. 4.11.

4.4.2 Implemented strategy

As discussed in Section 4.3.1 and 4.3.2, the implemented strategy boils down to estima-

tion of the multiplier function with real-time available information, and approximation

and optimization of the Hamiltonian with the, limited, available storage capacity and

computation power.

The multiplier p is estimated with linear feedback on the state-of-energy SOE, so n = 1

in (4.15). This results in tuning parameters p0, K1 and SOE0r , which were added to the


20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

State−of−energy SOE [%]

Res

ista

nce

R [Ω

]

chargedischarge

Figure 4.11 / Measured battery resistance R as a function of the state-of-energySOE.

vehicle CANbus such that they are real-time adjustable. The implemented strategy is

evaluated with a constant state-of-energy reference SOE0r = 0.5, as well as an adaptive

SOEr(t) by (4.16).

To exploit the non-convex behavior of the diesel engine and electric machine, the Hamil-

tonian is approximated with a grid rather than a (piecewise) quadratic approximation,

as discussed in Section 4.3.2. Clutch opening and idling of the engine is also considered.

The idle velocity is lower than the normal operating range of the engine, therefore,

clutch opening can provide fuel benefits.

The ECU manufacturer indicated that storage capacity of the ECU is more abundant

than computational capacity, thus, the optimal solution is derived off-line and stored

in a look-up table. For a total number of 8704 operating points (16 different rotational

velocities, 32 torque requests, and 17 different multiplier values), the Hamiltonian was

constructed locally and the optimal control calculated. In the implemented strategy,

the battery open-circuit voltage Uoc and resistance R are approximated with a constant.

The dependency of Uoc and R on the state-of-energy SOE could be incorporated in the

look-up table with an extra dimension taking more memory.

The resulting look-up table is depicted in Fig. 4.12. Four subfigures are depicted,

indicating the solution for four of the seventeen different multiplier values p. Each

subfigure depicts the optimal electric machine torque T ∗m (in color) as a function of the

electric machine rotational velocity ωm, and the torque request Tr, with discharging

(T ∗m > 0) in yellow/red, (T ∗m = 0) in green, and charging (T ∗m < 0) in blue. Clutch

opening (S = 0) is indicated by the horizontal lines in Fig. 4.12 where T ∗m = Tr.

At the upper left subfigure, with p = 1.8, electric driving is favored because T ∗m = Trfor Tr < Tm, almost everywhere, and T ∗m > 0 for Tr > 0 almost always, especially at

engine operating points where the engine efficiency is relatively low. Looking at the

second subfigure, with p = 2.3, it can be seen that discharging is preferred less, in fact,

the non-smoothness in Pm = 0 leads to engine-only fields (in green) for large parts of

the operating range. In the lower two subfigures, it can be seen that for increasing


Rotational velocity ωm

[rpm]

Tor

que

requ

est

T r [Nm

]

p = 1.8

800 1000 1200 1400 1600 1800 2000 2200

−400

−200

0

200

400

600

800


[rpm]

Tor

que

requ

est

T r [Nm

]

p = 2.3

800 1000 1200 1400 1600 1800 2000 2200

−400

−200

0

200

400

600

800


[rpm]

Tor

que

requ

est

T r [Nm

]

p = 2.8

800 1000 1200 1400 1600 1800 2000 2200

−400

−200

0

200

400

600

800


[rpm]

Tor

que

requ

est

T r [Nm

]

p = 3.3

800 1000 1200 1400 1600 1800 2000 2200

−400

−200

0

200

400

600

800

−420

Charge

0

Discharge

420

Figure 4.12 / Iso contour plot of electric machine torque T ∗m [Nm] for various valuesof p, ωm and Tr. The dark solid line indicates torque limitations, andthe dark dashed line indicates the electric machine torque limitationwith opened clutch.

multiplier p, charging the battery is used more and more.

An overview of the implemented power split control is shown in Fig. 4.13. The gas

pedal position ϑ determines the torque (power) request Tr. The brake pedal operates

the service brakes only. The percentage gas pedal position is scaled, between rotational

velocity dependent bounds to obtain Tm(ωm) for ϑ = 0 and T p(ωm) for ϑ = 100%.

The optimal electric machine torque T ∗m, resulting from the look-up table, is corrected

afterwards to protect the battery from over or undercharging with:

Tm r = min

(max

[T ∗m,

1

ωmPm (P s)

],

1

ωmPm(P s

)), (4.21)

where Tm r is the resulting electric machine torque request which is communicated to

the lower level controller operating on the same ECU, Pm(Ps) the functional inverse of

(4.2) and Fig. 4.8, and the control bounds are given by:

P s =UR(t, SOE)

RUoc, (4.22)

4.5 Experimental results 73

look-uptable

Fig. (4.12)

controlbounds

Eq. (4.21)

T ∗m Tp r

S

torquerequestscaling

inputs

ωm

SOE

Tr

multiplierestimation Eqs.

(4.15)-(4.17)

ϑ

SOE0r ,

p0, K1

p SOE,temp. limits,etc.

outputs−

Tm r

Figure 4.13 / Overview of implemented power split controls.

and,

P s =UR(t, SOE)

RUoc, (4.23)

in which, UR is the state-of-energy dependent allowable discharge over-potential which

becomes zero at SOE, UR the state-of-energy dependent allowable charge over-potential

which becomes zero at SOE. For simplicity, the temperature related bounds are not

discussed here. The torque request to the engine becomes:

Tp r = Tr − Tm r, (4.24)

where Tp r is the resulting torque request for the engine which is communicated via the

CANbus to another ECU governing the engine power. If Tp r = 0 the clutch command

S = 0 is communicated to the engine ECU, otherwise S = 1.

4.5 Experimental results

In this section, the experimental results, obtained with the power split control algorithm

discussed in Section 4.4.2, are presented. Firstly, the experimental setup is introduced.

Secondly, tuning rules for the feedback algorithm (4.15), for estimation of the multiplier

function, are presented. Thirdly, the performance of the optimal control based power

split with i) a constant state-of-energy set point, ii) an adaptive state-of-energy set

point is evaluated, using a non-hybrid baseline and heuristic power split algorithm as

reference.

4.5.1 Experimental setup

The experiments are carried out on the Eindhoven University of Technology heavy-duty

chassis dynamometer, with a hybrid electric truck. A schematic overview of the setup

is depicted in Fig. 4.14. Advantage of such a setup, compared to measurements on the

road, is the good reproducibility of the road load forces, the forces a vehicle normally


experiences on the road. The influence of the driver, traffic, and weather conditions is

excluded. The road load forces can be described with the following function:

Frl = c0mga cosα(t) +mga sinα(t) + c1v(t) + c2v2(t), (4.25)

here, c0 is the coefficient for rolling resistance, α the road slope, m the vehicle mass, c1

the resistance proportional to the vehicle velocity v and c2 the aerodynamic coefficient.

The inertial force Fa of the vehicle is also simulated:

Fa = medv

dt, (4.26)

in which me is the effective vehicle mass including rotating parts of the vehicle.

dyno controls

Td Cv

α

pedalpositions

evvr-Id s

s

τ

dt

vd

vehicle controls

vd

observerTset

Cd

-

FF

+

rd

Figure 4.14 / Schematic overview of the test setup with a hybrid truck on the dy-namometer.

In the dynamo test setup, the wheels of the driven axle are placed on drums. The

driver is replaced by the vehicle controls where the covered distance s is calculated by

integration of the measured velocity vd of the drums. Predefined distance-velocity and

distance-elevation trajectories, stored in the control system, provide the velocity error

signal and send it to the velocity controller Cv, which essentially is a PID controller with

anti-windup. A feed forward controller FF is added to reduce the tracking error. Output

of the velocity controller is an accelerator and brake pedal position. The accelerator

pedal input is an analog signal replacing the potentiometer of the pedal. Since the

braking system is not controlled “by-wire”, the brake pedal is actuated mechanically

with an electric piston, see Fig. 4.15.

The drums are connected to a 220 kW electric machine which simulates the road load

and inertial forces Frl + Fa of the truck. The dyno controls observer generates a torque

set point:

Tset = (Frl(t, v) + Fa(t, v))rd + Tc, (4.27)


Figure 4.15 / Brake pedal actuator.

using the measured velocity of the drums vd and by estimating dvd

dtwith a non-linear (Lu-

enberger) observer and using the road grade α provided by the corresponding distance-

elevation profile. Here, rd is the drum radius and Tc compensates for the drum dynamics.

A feedback algorithm Cd, using the measured torque Td, controls the dynamo electric

machine output power with control signal Id.

The engine cooling water is cooled with an external heat exchanger. The air tempera-

ture, pressure and humidity of the test setup room is not controlled. All measurements

where carried out in summer, where the ambient temperature was between 20 and 30 oC.

Before every measurement sequence, a warm-up procedure was carried out, while be-

tween two velocity trajectories the setup was cooled. The temperature of engine cooling

water, hybrid system cooling water, final drive oil and gearbox oil was hereby used as

an indicator.

Several signals are measured. A torque flange measures Td, the velocity of the drums vdis measured with an encoder. The fuel consumption is measured using a dedicated fuel

flow meter. The battery voltage, electric machine input current, and rotational velocity

of electric machine and engine, are obtained using the standard vehicle sensors via the

CANbus interface, as well as the signals SOE, p, Tr, etc.

4.5.2 Tuning of the multiplier estimation

In this section, the implemented power split control, which is motivated in Section 4.4.2,

is evaluated with experiments.

A proportional feedback n = 1 on the state-of-energy is used to estimate multiplier p.

The resulting power split control strategy has two tuning parameters, see (4.15):

• feedback gain K1,


• initial multiplier guess p0.

The evaluation of K1 and p0 is performed on the basis of dynamometer tests with a

medium-duty truck, the test setup is discussed in Section 4.5.1.

The Urban Dynamometer Driving Schedule (UDDS) is used as reference velocity tra-

jectory for the velocity controller, see the dark line in Fig. 4.16. The UDDS is explicitly

developed for chassis dynamometer testing of heavy-duty vehicles, see SAE Standard

J2711 (2002).

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

Distance s [km]

Vel

ocity

v [k

ph]

UDDSFTP−75

Figure 4.16 / Reference velocity trajectories.

The main aim of the evaluation is to compare the relative fuel consumption of different

power split control strategies and tuning parameters, therefore, reproducibility of the

measurements is of primary interest, while the exact tracking of the velocity set point

is of secondary importance. The dynamo mechanical energy consumption trajectory

Ed(t) =∫

Tdvd

rddt is used to indicate the reproducibility of the velocity tracking and road

load simulation.

The feedback gain K1 is varied between 0 and 3 while initial guess p0 is varied between

2.5 and 3.0. The experimental results for the UDDS velocity trajectory are depicted

in Fig. 4.17. The figure displays 9 subfigures, where the first row depicts the energy

consumption trajectories Ed(t), the second row depicts the state-of-energy trajectories

SOE(t), and the third row depicts the multiplier estimation p trajectories. The three

columns indicate the three different settings of K1. Each subfigure shows the results of

6 different settings of p0. The measurements are repeated three times for each setting.

In the top row of Fig. 4.17, it can be seen that the reproducibility of the velocity and

dynamometer control is good. The standard deviation of Ed(t1), for the 18 different


0

4

8

12

16

20

UDDS trajectory with K1=1

Ene

rgy

E d [MJ]

0.25

0.35

0.45

0.55

0.65

0.75

Sta

te−

of−

ener

gy S

OE

[−]

0 2 4 6 8 102.4

2.5

2.6

2.7

2.8

2.9

3

Distance s [km]

Mul

tiplie

r p

[−]

p0=2.5

p0=2.6

p0=2.7

p0=2.8

p0=2.9

p0=3.0


0 2 4 6 8 10Distance s [km]


0 2 4 6 8 10Distance s [km]

Figure 4.17 / Influence of tuning variables p0 and K1 on the UDDS velocity trajec-tory.

control settings, is 0.017 MJ, which is a standard deviation of 0.1%. To evaluate the

reproducibility of fuel consumption, six measurements with one control setting where

carried out. The standard deviation in fuel consumption was less than 1.0%.

The resulting state-of-energy trajectories are depicted on the second row of Fig. 4.17.

All measurements are started with an initial battery SOE of approximately 0.5. An

estimation of the state-of-charge is provided by the battery management system. The

exact algorithm for state-of-charge estimation is unknown, but relies mainly on Coulomb

counting. The open-circuit voltage of Fig. 4.10 is used to compute the state-of-energy.

As expected the battery is depleted for multiplier values, p0, that are estimated too

small, and charged for multiplier values that are estimated too large. For a small feed-

back gain, K1 = 1, the state-of-energy bounds are reached for p0 = 2.5, and p0 = 3.0.

The bandwidth of the (linear) feedback is a trade-off between avoiding under- or over-

charging and minimizing the control actions on short term state-of-energy fluctuations.

It can be seen that with K1 = 2, that the feedback allows enough “freedom” for fluctu-

ations without reaching the battery bounds for a large range of p0.

The bottom row of Fig. 4.17 depicts the estimated multiplier. With sufficient feedback,

the estimated multiplier converges to the same trajectory. A steady-state difference in


state-of-energy is a consequence.

The measurements for each setting on the UDDS cycle where repeated three times, the

mean value of the fuel consumption is depicted in Fig. 4.18. The figure indicates the

percentage increase in equivalent fuel consumption, the fuel consumption corrected for

state-of-energy deviations, as a function of tuning parameters p0 and K1. The lowest

fuel consumption is used as reference fuel consumption.

A minimum fuel consumption is found with p0 = 2.8 and K1 = 1. At low feedback

gain the fuel consumption increases with several percents for increasing or decreasing

p0. For larger feedback gains this increase in fuel consumption is smaller and remains

within 1.5% of the minimum.

12

3

2.42.52.62.72.82.930

0.5

1

1.5

2

2.5

3

3.5

4

Feedback gain K1 [−]Multiplier p0 [−]

Equ

ival

ent f

uel c

onsu

mpt

ion

[%]

Figure 4.18 / Percentage increase of fuel consumption corrected for state-of-energydeviation, as a function of tuning parameters K1 and p0.

Open loop measurements (K1 = 0) where also performed, providing a near charge-

sustaining trajectory with p0 = 2.75. According to the MP this provides the global

optimal solution (assuming that the non-convexity, associated with the clutch opening,

does not alter optimality and battery state-of-energy dependent losses can be ignored).

The value p0 = 2.75 is also used for the calculation of the equivalence factor for fuel

consumption correction

Eeq(t) = Ef (t) + p0 (Es(t)− Es(t0)) . (4.28)

The mean fuel consumption of three repeated open loop measurements with p0 = 2.75

is 0.65% higher than the minimum in Fig. 4.18. This difference is not significant.

Based upon the experimental results it is concluded that:


• tuning parameter p0 is relatively easy to tune, although p0 depends on the char-

acteristics of the components, since the influence of different settings on the fuel

consumption is small as long as K1 > 1,

• the parameter setting K1 = 2 and p0 = 2.7, provides good performance, (the

settings are chosen after evaluating also other velocity trajectories and, therefore,

differ from the optimum found in Fig. 4.18),

• fuel consumption results within 1.5% of the global optimum can be expected on

routes where the state-of-energy bounds are not reached. The optimum is estab-

lished with open loop charge sustaining measurements using a constant multiplier

which, according to optimal control theory, leads to the global optimum.

4.5.3 Strategy comparison

In Section 4.5.2, it is shown that fuel consumption results close to optimal can be

achieved for power request trajectories at which the state-of-energy bounds are not

reached. A bigger challenge, for the real-time power split control, is to obtain good

performance on routes where the state-of-energy bounds are reached. In general this

happens on routes with considerable elevation differences, where the recoverable poten-

tial energy is large compared to the capacity of the storage device. Four different power

split control strategies are evaluated:

• the optimal control algorithm, with the settings K1 = 2, p0 = 2.7, using SOEr =

0.5,

• the optimal control algorithm, with the settings K1 = 2, p0 = 2.7, using an

adaptive SOEr, by (4.16), with KE = Kh = 0.2, and SOE0r = 0.5,

• a heuristic algorithm, tuned by the manufacturer of the hybrid system hardware,

• a non-hybrid baseline strategy, using the same test vehicle in engine-only mode.

To evaluate the performance of the three power split algorithms in situations where the

state-of-energy bounds are reached, a road grade trajectory, see Fig. 4.19, is added to

the FTP-75 velocity trajectory depicted in gray in Fig. 4.16. The road grade is scaled

so α varies between 0 and 0.05 rad, in steps of 0.005 rad. This leads to 11 different

velocity-power trajectories which where used to evaluate the four strategies.

The results are depicted in Fig. 4.20, where 3 columns display three of the eleven

evaluated road grades. The first row depicts the cumulative energy produced by the

dynamometer Ed. Parameter Ed is used to indicate a fair comparison between the

strategies. Notice that Ed of the baseline strategy, on all trajectories, is lower than


0 2 4 6 8 10 12 14 16 18−40−30−20−10

010203040

α

Distance s [km]

Ele

vatio

n h

[m]

Figure 4.19 / Elevation-distance trajectory h for α = 0.01 rad.

the other strategies. This is caused by an increase in gearshift time since in the hybrid

strategies the electric machine is also used to alter the rotational velocity of the gearbox

primary spindle during shifting. The increased gearshift time decreased the average

velocity and, thus, the magnitude of the simulated road load losses. Also the other

strategies have some deviations, therefore, the fuel consumption results are corrected

for deviations of Ed(t) with the average Ed(t) of the three hybrid strategies.

At routes with α ≤ 0.02 rad recovery of the potential energy is not possible. At routes

with α ≥ 0.025 rad, dEd

dt< 0 for some time intervals, and recovery of the potential

energy is possible. Note also that the cumulative energy delivered by the dynamometer

decreases for routes with increasing grades. During the uphill trajectories the maximum

power of the prime mover is reached, such that the average velocity of the cycle is

reduced, and the road load decreases due to lower aerodynamic losses.

The second row depicts the state-of-energy trajectories of the three hybrid strategies

(solid) and the reference trajectory of the adaptive strategy (dashed). The heuristic

strategy (dark) is to deplete the battery right from the start of the trajectory. For

routes with mild grades, the state-of-energy remains on the lower boundary of the

battery. This is suboptimal as shown by the fuel consumption results discussed below.

However, on routes with a large recovery potential, the heuristic strategy is performing

well. Since the time in between the two descents is relatively small, immediate depletion

is successful here.

The optimal control is depicted in gray, and the adaptive optimal control in light gray.

In routes with low-recovery potential the optimal control strategies are both charge

sustaining. In routes with high-recovery potential (α > 0.035 rad) the upper bound

of the storage device is reached by the optimal control with constant set point. The

optimal control with adaptive reference is more successful in discharging the battery in

between the two descents. Note that an integration term in the feedback algorithm for

the multiplier estimation (4.15) would be undesirable here, since this limits freedom in

deviation from SOEr even further.


0

5

10

15

20

25

30

Ene

rgy

E d [MJ]

α=0.025 [rad]

baseline

heuristic

optimal control

adaptive optimal control

α=0.035 [rad] α=0.045 [rad]

20

30

40

50

60

70

80

Sta

te−

of−

ener

gy S

OE

[%]

heuristic

optimal control


adaptive reference

0 4 8 12 16 202.2

2.4

2.6

2.8

3

3.2

3.4

Distance s [km]

Mul

tiplie

r es

timat

ion

p [−

]

optimal control


0 4 8 12 16 20Distance s [km]

0 4 8 12 16 20Distance s [km]

Figure 4.20 / Test results at FTP-75 velocity trajectory with elevations.

The third row shows the estimated multiplier function. Recall from Section 4.2.2, that

the optimal multiplier function, on trajectories where the state-of-energy boundaries

are not reached, and neglecting the state-of-energy dependent losses, is described with a

constant. The multiplier estimation, based upon the reference state-of-energy adaptive

for kinetic and potential energy, shows a much smaller variation, than the constant

state-of-energy reference, indicating that the multiplier estimation is improved.

The fuel consumption results, as a function of road grades α, of the four strategies,

are depicted in Fig. 4.21. Unfortunately, the results of the adaptive strategy at α =

0.04 where erroneous and are omitted. The fuel consumption of the baseline vehicle,

corrected for the smaller dynamo energy input, is increasing for routes with larger

grades. The optimal control strategies attain a better fuel consumption, up to 4%,

than the heuristic controller on trajectories where recoverable energy is smaller than

the battery capacity. In fact, the fuel consumption of the optimal control with adaptive

reference is superior to the heuristic strategy at all trajectories, except at α = 0.045

rad.

From the evaluation of four strategies the following can be concluded:

• the optimal control with adaptive state-of-energy reference leads to a good estima-


0 0.01 0.02 0.03 0.04 0.054

4.25

4.5

4.75

5

5.25

5.5

Road grade α [rad]

Equ

ival

ent f

uel c

onsu

mpt

ion

[L]

BaselineHeuristic strategyOptimal control with SOE

r=0.5

Optimal control with adaptive SOEr

Figure 4.21 / Fuel consumption results of various strategies.

tion of the multiplier function, is superior to the other strategies, and is a useful

extension on the optimal control algorithm at routes where the recovery potential

is large,

• the measured fuel consumption reduction potential of hybridizing a medium-duty

truck on the FTP velocity trajectory, with a vehicle mass and component size and

characteristics as presented in Section 4.4, ranged from 7% on velocity and power

trajectories with low-recovery potential (small slopes) up to 16% on velocity and

power trajectories with high-recovery potential (large slopes),

• the adaptive optimal control strategy achieves up to 45% larger savings than the

heuristic controller.

4.6 Conclusions

In this research, the design, implementation, and evaluation of optimal power split

control in hybrid vehicles is discussed.

To obtain a good real-time implementable power split control, a design based upon

optimal control theory is useful, also for systems where charging and discharging are

described with a non-smooth characteristic. Using a non-smooth Maximum Principle,

it is shown that the design of real-time implementable power split strategies boils down

to:

• estimating the multiplier based on real-time available vehicle parameters,

4.6 Conclusions 83

• minimizing a local approximation of the Hamiltonian, with the limited ECU ca-

pacity.

A real-time strategy is implemented in a hybrid truck using linear feedback on the

state-of-energy to estimate the optimal multiplier function. The storage capacity and

computational power, of standard Electronic Computational Units, form no obstacle for

the implementation of optimal control based power split strategies.

The proposed strategy is evaluated with chassis dynamo measurements on 12 differ-

ent velocity and grade trajectories. The results indicate that linear feedback on the

state-of-energy is a useful strategy for multiplier estimation and is easy to tune. Fuel

consumption results within 1.5% of the global optimum can be expected in velocity and

grade trajectories where the capacity bounds of the storage device are not reached.

The performance can be enhanced, by using a state-of-energy reference trajectory, adap-

tive for kinetic and potential energy of the vehicle, which still requires only information

of the actual vehicle status. Compared to the non-hybrid baseline strategy, the op-

timal power split obtained fuel consumption reductions between 7% and 16%. The

optimal controller performed up to 45% better than the heuristic controller tuned by

the manufacturer.

A power split control design for hybrid electric vehicles, based on optimal control and

requiring only information available in the vehicle, is easy to tune, can be implemented

on standard hardware, and offers a fuel consumption very close to the optimal one for

a wide variety of operating conditions. Therefore, predictive information, for example

coming from a navigation system, can offer only marginal fuel consumption benefits

in charge sustaining hybrid vehicles, and for the control of plug-in hybrids, only the

distance to the next plug-in station or inner city is required.

Further research is required to improve the state-of-energy reference trajectory calcu-

lation of the adaptive strategy, e.g., to i) include the maximum available power of the

secondary power converter in the estimation, ii) use only large road grades, that lead

to energy recovery downhill, in the determination of useful potential energy.

Chapter five

Conclusions and recommendations

Abstract / The main ideas and methods are recapitulated. The conclusions for this thesis arepresented. Recommendations for future work are given.

5.1 Conclusions

This thesis deals with the fuel optimal control of vehicles with a hybrid drive train. The

main results obtained in this thesis are:

• a novel algorithm for the computation of the optimal velocity trajectory, for ve-

hicles with energy recovery options, based on the non-smooth maximum principle

using elevation-distance and velocity limitations, e.g., coming from a geographical

information system, as input,

• a novel numerical solution of state constrained optimal control problems with a

scalar state. Application of the novel numerical solution for state constrained

problems for the power split control problem for a known power and velocity

trajectory, resulting in an efficient search for optimal hybridization factors,

• an original methodology for the design and implementation of a real-time optimal

control based power split control, adaptive for vehicle mass and elevations, on

standard hardware. An experimental evaluation indicates that a performance close

to the global optimum can be expected for a wide variety of operating conditions.

These results followed from the research question posed in the introduction:

Derive a methodology for the fuel optimal design and operation of a hybrid vehi-

cle, i.e. by application of predictive and adaptive control strategies. Implement

85

86 5 Conclusions and recommendations

a real-time strategy on standard industry hardware and evaluate the strategy with

experiments.

The main results in relation to this research question are now further discussed.

5.1.1 Optimal trajectories for vehicles with energy recovery options

Chapter 2 presented a modeling framework to compute the optimal gear ratio, power

split, and velocity trajectories for vehicles with energy recovery options. The resulting

optimal trajectories can be used as (predictive) set point for the lower-level controllers.

A novel cost function description is proposed in which the control of gear shift, power

split and combined power output of the power converters, is approximated with a scalar

convex piecewise affine function. This approach reduces the complexity of the problem

considerably as only one control signal is used instead of three. Using a piecewise affine

description of the cost function has the advantage that the control appears linearly in

the Hamiltonian. Optimal control theory can then effectively be used to derive a limited

set of optimal control sub-arcs fulfilling the necessary conditions of optimality.

It is shown that the optimal control solution can be structured to use only a few se-

quences of control sub-arcs. Each sequence can be analytically described with the veloc-

ities at the junction points of the sub-arcs. Hence, the original problem is rewritten as a

standard nonlinear programming problem in terms of the length of the sub-arcs which

can be solved with sequential quadratic programming like algorithms. The feasibility

of this approach is indicated with a numerical example.

5.1.2 Optimal power split control for predefined trajectories

Chapter 3 presented a novel numerical procedure for solving state constrained non-

smooth optimal control problems with a single state. Optimality of the solution is

guaranteed. The proposed method is computationally more efficient than other numer-

ical solutions (in the sense of fulfilling the necessary conditions of optimality) applied

for such problems, e.g., dynamic programming.

This novel method is applied to the optimal control of power split in hybrid vehicles for a

predefined power and velocity trajectory. The resulting optimal power split trajectory

can be used to bench mark real-time implemented controllers, to provide predictive

information based on predicted power trajectories, and to determine the optimal degree

of hybridization for vehicles operating on a certain duty cycle.

The novel numerical method is compared with a dynamic programming algorithm. The

method offers a similar accuracy as this often applied approach, however, up to 250

5.1 Conclusions 87

times faster.

In a case study, the effect of hybridization is determined for a 40 ton heavy-duty vehicle

operating on a high way trajectory in a hilly environment with a length of over 500 km.

The fuel consumption benefit for this type of vehicles on the driving cycle specifically

used is limited. A reduction of 6.2% is found for a hybridization factor of 0.4.

5.1.3 Real-time power split control in hybrid vehicles

Chapter 4 discussed the design, real-time implementation and evaluation of optimal

power split control in hybrid vehicles. Based upon the necessary conditions of optimality

it is stated that an optimal control based design boils down to:

• estimation of a Lagrange multiplier adjoining the fuel cost function to the energy

conversion in the hybrid system, based on real-time available information,

• solving an optimization procedure with the limited computational power and stor-

age capacity available on-board.

It is shown that a good approximation of the multiplier is obtained by using feedback

on the current state-of-energy of the storage device. Moreover, an adaptive reference

state-of-energy set point that accounts for kinetic and potential energy of the vehicle,

and, so, deals with varying vehicle mass and elevation, without the use of predictive

information, provides a satisfactory approximation of the multiplier and is easy to tune.

The proposed strategy is implemented in a 9 ton distribution truck on standard hard-

ware and evaluated with chassis dynamo meter experiments on 12 different velocity

and elevation trajectories. The test results indicate that the adaptive, optimal control

attains a fuel consumption close to the global optimal on velocity and grade trajectories

where the capacity bounds of the storage device are not reached.

It is, therefore, concluded that application of predictive information, for instance coming

from geographical information systems, offers only limited fuel consumption benefits for

the power split control. In some applications, e.g., plug-in vehicles or when temporary

electric driving in an inner city is desired, or in case capacity limits are approached, e.g.,

when the battery is chosen smaller, predictive information is useful compared to only

real-time available data. Predictive formation can be communicated via the standard

CANbus interface using a predefined multiplier and optimal state-of-energy reference

trajectory as carrier.


5.2 Recommendations

Here, directions that can be pursued in future work are outlined.

Over the last 15-20 years considerable efforts have been made to the power split con-

trol in hybrid vehicles by several research groups around the world, see Sciarretta and

Guzzella (2007). As real-time implemented controllers achieve fuel consumption re-

sults close to optimal, it could be concluded that this research area is maturing. Some

extensions are possible, however. New battery technologies might lead to new require-

ments on the power split control problem. One example is boost charging of which the

possibilities are further discussed in Section 5.2.1.

A straightforward extension of the power split control discussed in Chapter 4 could be

to include gear shifting in a similar way as the clutch opening is implemented. An

experimental implementation should point out whether describing a gear shift with

instantaneous cost leads to satisfying results. Besides, other forms of energy storage in

the auxiliaries such as the air pressure of the service brake system in a truck, the air

conditioning temperature in the cabin, and state-of-energy of the low voltage battery

system can be included and might offer a fuel benefit.

New trends in power train control are arising. The stringent emission legislation, e.g.,

Euro VI in 2013, have a strong impact on engine design. It is expected that complex

systems for exhaust gas after treatment are required to meet the new legislation. Ex-

tending the fuel optimal control described in this thesis with constraints on the after

treatment is not trivial. Extensions of the real-time power split control to the combina-

tion of fuel consumption and engine emission control is referred to as integrated power

train control; some comments are given in Section 5.2.2.

Another trend is to alleviate the driver from more and more tasks. One of the examples

is cruise control which is now moving towards (cooperative) adaptive cruise control

(Naus et al., 2010). Ultimately, this trend can result in autonomous driving. Predefined

optimal velocity trajectories, presented in Chapter 2, are then even compulsory rather

than optional. This is further elaborated on in Section 5.2.3.

Finally, in this thesis, some theoretical advances are made in the numerical solution of

state constrained optimal control problems with a single state. It might be possible to

extend these results to more general state constrained optimal control problems. Some

directions are indicated in Section 5.2.4.

5.2.1 Advanced battery modeling

Batteries are critical components in HEV drive trains, firstly, due to their high cost,

and secondly, since the process of ageing (capacity decrease and impedance increase)

5.2 Recommendations 89

is not fully understood (Vetter et al., 2005). It is known, however, that limiting the

battery terminal voltage and the maximum C-rate ((dis)charge current proportional to

the battery capacity) can temper the process of ageing, see Notten et al. (2005) and

Vetter et al. (2005).

In general, this results in a battery capacity that is unnecessary large compared to the

energy that can be recuperated during normal operation of the vehicle. This holds

especially for vehicles driving in city traffic where recovery of potential energy due to

elevations is small and vehicle decelerations are harsh such that the tractive powers are

high, however, recuperated (kinetic) energy is small.

To enhance the power/capacity ratio of the battery, without compromising (much) on

the cycle life, it is possible to boost-charge the battery. Boost charging is proposed as a

new, fast recharging policy for Li-ion batteries (Notten et al., 2005). Characteristic of

boost charging is that close-to-fully discharged batteries can be recharged, up to one-

third of its rated capacity, with high currents (power), without having a (large) negative

effect on the cycle life of the battery.

The boost charging current is limited by the maximum terminal voltage level of the

battery. Adjusting the battery current depending on the state-of-energy of the battery

to meet voltage limitations leads to a class of constraints in optimal control theory

referred to as mixed state-control constraints (Hartl et al., 1995; Gollmann et al., 2009).

Normally, battery modeling for HEV control only considers pure state and pure control

constraints. So, including boost charging in the real-time power split control is not a

trivial extension.

5.2.2 Towards integrated power train control

The results presented in this thesis indicate that a fuel optimal control choice by selecting

the best power split action based on the instantaneous cost, where this instantaneous

cost includes (a convex approximation of) static maps describing the conversion cost of

electric machine and engine, provides good results.

Optimizing fuel consumption without accounting for emissions could lead to misleading

conclusions since a decrease in fuel consumption might lead to an increase in emissions.

In literature, attempts are made to include engine emissions, such as NOx and Partic-

ulate Matter (PM), with static maps (Koot et al., 2005b). This might not work for

modern engines, however.

To meet the Euro VI legislation, diesel engines are equipped with a complex after treat-

ment system involving, e.g., an urea-based selective catalytic reduction (SCR) technol-

ogy, exhaust gas recirculation (EGR), and a diesel particulate filter (DPF), see Willems

and Cloudt (2010). The performance of such after treatment systems relies on the dy-


namic behavior, e.g., the temperature of the catalyst and pressure of the turbocharger.

For instance, the dosing of the SCR depends on the temperature and the DPF requires

a certain light-off temperature. The catalyst extracts heat from the exhaust gasses, so if

the prime mover delivers more power this results in an increased catalyst temperature.

Using the hybrid system temporarily to reduce the engine load could, therefore, result

in a decrease of the after treatment performance. On the other hand, idle-stop leads to

less heat loss than engine idle (Foster et al., 2008). Since temperature increase/decrease

is dynamic, the instantaneous framework using static maps might not lead to satisfying

results.

5.2.3 Towards autonomous driving

The optimal velocity trajectories derived in Chapter 2 can be used as reference input

for a cruise control system. Ideally this trajectory is exactly followed, however, in real

traffic situations it might not be possible to track the optimal reference. In that case,

a switching from velocity tracking to car following could be made (Van Keulen et al.,

2009). An experimental validation should point out the practical relevance.

In other applications than road vehicles such as train and metro services as well as Au-

tonomous Guided Vehicles (AGV) for industries, exact following of velocity trajectories

is more easy to implement, since these vehicles already have a prescribed distance and

time schedule without interference of other “road” users. A more dedicated evaluation

for these applications is required.

5.2.4 Numerical solutions for state constrained optimal controlproblems

The unconstrained solution (without state constraints) of a convex optimal control

problem can be found with relatively low computational effort by solving a two point

boundary value problem. Solving a state constrained problem is more involved especially

if the structure of the solution (number and sequence of contact times at the boundary)

is unknown a priori.

In Chapter 3, a novel algorithm is proposed to solve a scalar state constrained optimal

control problem by evaluating the unconstrained solution. The conditions under which

optimality is proven suit the applications described in this thesis, however, could be

restrictive for other applications. Further research is required to indicate whether this

technique can be generalized to problems with more than one state.

Appendix A

Singular solution1

singular solution

Consider the cost function

Peq(t) = σ(t)Pr(t),

with

σ(t) =

γp,1 in int [P p, P p],

γp,1

γ+m,1γ

−m,1

in int [P q, P p],(A.1)

and dynamic equations

v(t) = a1Pr(t)

v(t)− a2(t, s)− a3v(t)− a4v

2(t), (A.2)

s(t) = v(t), (A.3)

which leads to a Hamiltonian of the form

H(t) = g(t) + h(t)Pr(t) + λ1(t)(v(t)− v) + λ2(t)(v − v(t))

where Pr is the scalar control signal, g and h are independent of Pr, λ1 and λ2 are

piecewise constant multiplier functions for the state constraint functions. The multiplier

conditions are:

p1(t) = −∂L∂v

= p1(t)

(a1Pr(t)

v2(t)+ a3 + 2a4v(t)

)− p2(t)− λ1(t) + λ2(t), (A.4)

p2(t) = −∂L∂s

= p1(t)∂a2(t, s)

∂s. (A.5)

1Derivation of higher order necessary conditions of optimality for the singular solution presented inChapter 2.

91

92 A Singular solution

Then a singular solution occurs if h ≡ 0 and higher order necessary conditions of

optimality are required, i.e., h = 0, h = 0, and so on.

The following necessary condition for optimality is obtained:

h(t) =a1p1(t)

v(t)+ σ ≡ 0,

which leads to a condition on multiplier p1:

p1(t) ≡ −σv(t)

a1

. (A.6)

The condition on the first derivative of h becomes:

h(t) = p1(t)a1

v(t)− vp1(t)

a1

v2(t)≡ 0.

Using the derivative of (A.6) and (A.2) this becomes

h(t) =

[p1(t)

(a1Pr(t)

v2(t)+ a3 + 2a4v(t)

)− p2(t)− λ1(t) + λ2(t)

]a1

v(t)

− p1(t)

[a1Pr(t)

v(t)− a2(t, s)− a3v(t)− a4v

2(t)

]a1

v2(t)≡ 0

which can be simplified to

h(t) = p1(t)a1

(a2(t, s)

v2(t)+

2a3

v(t)+ 3a4

)− λ1(t)a1

v(t)+λ2(t)a1

v(t)− p2(t)a1

v(t)≡ 0.

A condition on multiplier p2 is obtained:

p2(t) = p1(t)

(a2(t, s)

v(t)+ 2a3 + 3a4v(t)

)− λ1(t) + λ2(t). (A.7)

The condition on the second derivative of h becomes:

h(t) = p1(t)

(a1a2(t, s)

v2(t)+

2a1a3

v(t)+ 3a1a4

)+ v(t)

(−2p1(t)a1a2(t, s)

v3(t)− 2p1(t)a1a3

v2(t)+p2(t)a1

v2(t)+λ1(t)a1

v2(t)− λ2(t)a1

v2(t)

)− λ1(t)

a1

v(t)+ λ2(t)

a1

v(t)− p2(t)

a1

v(t)+ a2(t, s)

p1(t)a1

v2(t)≡ 0

substituting p1 with (A.4), implementing the condition on multiplier p2, (A.7), and

93

using (A.5) the following equation is obtained:

h(t) =[p1(t)

(a1Pr(t)

v2(t)+ a3 + 2a4v(t)

)−[p1(t)

(a2(t, s)

v(t)+ 2a3 + 3a4v(t)

)− λ1(t) + λ2(t)

]−λ1(t) + λ2(t)] ·

(a1a2(t, s)

v2(t)+

2a1a3

v(t)+ 3a1a4

)+ v(t)

(−2p1(t)a1a2(t, s)

v3(t)− 2p1(t)a1a3

v2(t)+

(p1(t)

(a2(t, s)

v(t)+ 2a3 + 3a4v(t)

)−λ1(t) + λ2(t))

a1

v2(t)+λ1(t)a1

v2(t)− λ2(t)a1

v2(t)

)− λ1(t)

a1

v(t)+ λ2(t)

a1

v(t)− p1(t)

∂a2(t, s)

∂s

a1

v(t)+∂a2(t, s)

∂t

p1(t)a1

v2(t)≡ 0

which can be simplified to

h(t) =p1(t)

v(t)

(a1Pr(t)

v2(t)− a2(t, s)− a3v(t)− a4v

2(t)

)︸︷︷︸

v

(a1a2(t, s)

v2(t)+

2a1a3

v(t)+ 3a1a4

)

+ v(t)

(−p1(t)a1a2(t, s)

v3(t)+

3p1(t)a1a4

v(t)

)≡ 0,

this results in the condition

h(t) = p1(t)a1v

(2a3

v2(t)+

6a4

v(t)

)≡ 0.

This condition is satisfied if v ≡ 0, by (A.2) it can be seen that the control appears

explicitly in this necessary condition for optimality.

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Samenvatting

Hybride voertuigen hebben, tenminste, twee vermogensomzetters. Normaal gesproken

een primaire vermogensomzetter, die een aandrijvend vermogen levert, gebruikmakend

van een brandstof middels een onomkeerbaar proces, en een secundaire vermogensom-

zetter, die een aandrijvend vermogen, omkeerbaar, omzet in een grootheid geschikt voor

de accumulator, of andersom. De brandstof optimale regeling van hybride voertuigen

omvat de regeling van voertuigsnelheid, van de aandrijflijn overbrengsverhouding, van

de vermogensdeling tussen de primaire vermogensomzetter en de secundaire vermogens-

omzetter(s), en van stop-start van de primaire vermogensomzetter. Het potentieel van

hybride voertuigen is niet geheel benut door een gebrek aan regelstrategieen die om

kunnen gaan met de onbekende toekomstige vermogensvragen, die kunnen worden in-

gebed in standaard apparatuur gebruikt in de industrie, en die een brandstofverbruik

behalen dat dichtbij het globale optimum ligt.

De doelstelling voor de regelstrategie is het voertuig naar de volgende bestemming te

rijden met een minimum aan brandstof terwijl de reistijd begrensd is. De gecombineerde

optimalisatie regelstrategie van voertuigsnelheid, de aandrijflijn overbrengsverhouding

en vermogensdeling is benaderd met een stuksgewijs continu scalair regelsignaal -de

gecombineerde vermogensvraag- en geoptimaliseerd met een niet-gladde optimale regel-

theorie. De stop-start van de primaire vermogensomzetter en capaciteitsgrenzen van

de accumulator zijn hierbij verwaarloosd. Gebruikmakend van een navigatiesysteem

aan boord van het voertuig, dat informatie geeft over, bijvoorbeeld, de bochtstraal, de

weghelling en de snelheidslimieten, kan de optimale vermogensvraag, voertuigsnelheid,

aandrijflijn overbrengsverhouding en vermogensverdeling worden berekend. Het optima-

le snelheid en aandrijflijn overbrengverhoudingspad kan worden gebruikt als referentie

voor de snelheidsregeling (cruise control) en schakelstrategie, ook voor niet hybride

voertuigen.

Het optimale vermogensvraag- en snelheidspad kan worden gebruikt door meer gecom-

pliceerde optimalisatie methodes die de optimale vermogensdeling berekenen, inclusief

stop-start van de primaire vermogensomzetter, en met in achtneming van de capaciteits-

grenzen van de accumulator. In het geval dat de kostenfunctie van de vermogensdeling

kan worden benaderd met een convexe functie en er een monotoon stijgende relatie

101

102 Samenvatting

is tussen het opslagvermogen en het mechanisch vermogen van de secudaire vermo-

gensomzetter, kan een nieuwe numerieke methode worden gebruikt die is ontworpen

met behulp van het -in optimale regeltheorie bekende- Pontryagin Maximum Principe.

De resulterende optimale vermogensdeling kan worden gebruikt als referentie voor de

geımplementeerde regelaar welke een robuustheid geeft tegen fouten in het voorspelde

pad. De berekende optimale paden kunnen ook gebruikt worden voor het ontwerp van,

en als maat staaf voor, de geımplementeerde vermogensverdeling regelstrategie, of om

de optimale technologie, topologie en component grote in het ontwerp van hybride voer-

tuigen vast te stellen. In dit proefschrift is de optimale hybridisatie factor voor lange

afstand vrachtauto’s vastgesteld voor een 513 km lange route.

Het ontwerp van een geımplementeerde regelstrategie haalt voordeel uit de resulta-

ten behaald met de benodigde condities voor optimaliteit van het eerder genoemde

Maximum Principe, en komt neer op: i) schatten van een multiplicator functie, die

de opgeslagen energie in de accumulator relateert aan de brandstofkosten, gebruikma-

kend van voorhanden zijnde actuele informatie in het voertuig, en ii) optimalisatie van

een lokaal benaderde Hamiltonian functie, gegeven de begrensde rekenkracht van de

standaard apparatuur gebruikt in de industrie. De op optimale regeltheorie gebaseerde

geımplementeerde vermogensdeling strategie schat de multiplicator functie met pro-

portionele terugkoppeling op een adaptieve referentie welke is gebaseerd op de actueel

opgeslagen energie in de accumulator en de actuele kinetische en potentiele energy van

het voertuig. De regelstrategie is geımplementeerd in een hybride elektrische vrachtauto

op door de industrie gebruikte standaard apparatuur. De regellus is geevalueerd met

experimenten op een rollenbank. De regeling is gemakkelijk te verfijnen en behaald een

brandstofverbruik, zonder gebruik te maken van voorspelde vermogensvragen, binnen

1.5% van het globale optimum op routes waar de capaciteitsgrenzen van de accumulator

niet worden geraakt. In het geval dat de capaciteitsgrenzen van de accumulator wel wor-

den geraakt kunnen optimale vermogensdelingpaden berekend op basis van informatie

komend van navigatiesystemen die de werking dichtbij het optimum brengen.

De berekening van optimale paden, gebaseerd op informatie komend van een navigatie

systeem, de nieuwe numerieke methode voor scalaire optimale regelproblemen met toe-

stand afhankelijke grenzen, en de implementatie van een vermogensdeling strategie die

adaptief is voor voertuigmassa, voertuigsnelheid en hoogte verschillen, samen met de

opmerking wanneer voorspellende informatie voordelen biedt, kan worden gezien als de

belangrijkste resultaten van dit onderzoek.

Dankwoord

Dit proefschrift had nooit tot stand kunnen komen zonder de hulp van meerdere perso-

nen.

Allereerst wil ik mijn promotor Maarten Steinbuch en co-promotor Bram de Jager

bedanken voor de mogelijkheid om dit promotietraject te doorlopen. Maarten, jouw

vertrouwen en gedrevenheid hebben erg motiverend gewerkt. Als promovendus in jouw

groep hoef je je over niks anders druk te maken dan je eigen onderzoek en deze unieke

werksfeer komt de kwaliteit zeker ten goede. Bram, van je heldere analytische bena-

dering heb ik veel geleerd, je vasthoudendheid om de onderste steen boven te krijgen

werkt erg inspirerend en heeft een grote invloed gehad op het eindresultaat. Verder heb

ik het als erg prettig ervaren dat het vrijwel altijd mogelijk was om binnen te lopen

voor een korte of lange discussie.

Dit onderzoek is gefinancierd door TNO Automotive. Naast de financiele steun, wil ik

alle mensen die betrokken zijn geweest bij mijn onderzoek bedanken voor de inspannin-

gen. In het bijzonder Olaf op den Camp en Gertjan Koornneef voor het vertrouwen en

de vrijheid waarmee ik dit onderzoek richting kon geven en John Kessels voor de vele

inhoudelijke discussies. Ook wil ik alle medewerkers bij DAF Trucks die betrokken zijn

geweest bij dit onderzoek bedanken voor de technische ondersteuning, informatie voor-

ziening en waardevolle suggesties. In het bijzonder ben ik Mark Nievelstein erkentelijk

voor de altijd snelle afhandeling van allerhande vragen en Jack Martens voor de nuttige

tips bij de projectvergaderingen. Daarnaast wil ik Alex Serrarens bedanken voor zijn

rol aan het begin van mijn promotie.

De kerncommissieleden Henk Nijmeijer, Thierry Marie Guerra en Lars Nielsen bedank

ik voor het kritisch doorlezen en becommentarieren van dit proefschrift.

Tijdens mijn promotie heb ik het geluk gehad dat een aantal uitstekende afstudeerders

aan dit onderzoek hebben meegewerkt, ik bedank Loek Marquenie, Dominique van

Mullem en Jan Gillot voor de prettige samenwerking en hun waardevolle bijdragen.

Een experimentele validatie is alleen mogelijk met de hulp van “echte” techneuten;

graag wil ik daarom Ruud van de Bogaert, Erwin Meinders en Toon van Gils bedanken

voor de technische ondersteuning in het “automotive lab”.

103

104 Dankwoord

Vier jaar onderzoek is niet vol te houden zonder de nodige afleiding. Ik wil daarom alle

(oud)collega’s van de “DCT” groepen bedanken voor de fantastische werksfeer, tijdens,

en na, het werk. Dankzij jullie heeft het mij geen enkele moeite gekost om in Eindhoven

in te burgeren. In het bijzonder wil ik mijn kamergenoot Tijs bedanken voor vier jaar

prettig samenwerken. Verder wil ik mijn vrienden in het “zeeuwse”: Lennard, Edwin en

Rens bedanken voor de jarenlange vriendschap. Ondanks dat we elkaar niet elke week

meer zien, blijft het toch altijd gezellig.

Als zoon in een vervoerdersfamilie was ik al vroeg betrokken bij het reilen en zijlen in de

transportwereld. De kennis op het gebied van autotechniek, transport en logistiek, die

ik, vanaf “het moment dat ik een stuur kan vasthouden”, heb opgedaan heeft in grote

mate bijgedragen aan de inhoud van dit proefschrift. Graag wil ik mijn opa, vader en

ooms Wim en Kees bedanken voor de mooie leerschool.

Tenslotte, maar zeker niet het minste, wil ik mijn familie bedanken; papa, mama,

Maarten en Astrid, voor de onvoorwaardelijke liefde en steun.

Thijs van Keulen

April 2011

Curriculum Vitae

Thijs van Keulen was born on April 27th, 1982 in Biggekerke, gemeente Valkenisse, the

Netherlands. He graduated from secondary school at the Stedelijke Scholen Gemeen-

schap Middelburg, Middelburg, the Netherlands, in 1999. He received the B.eng. degree

in Mechanical Engineering from the Hogeschool Zeeland, Vlissingen, the Netherlands,

in 2003. He started his study Mechanical Engineering at the Delft University of Techno-

logy in Delft, the Netherlands. His M.Sc. project ’Feasibility of a novel ABS controller’

was carried out at Bridgestone Technical Center Europe, Rome, Italy. He received his

M.Sc. degree from the Delft University of Technology, Delft, the Netherlands, in May

2007.

In June 2007, he started as a Ph.D. candidate in the Control Systems Technology group

at the Eindhoven University of Technology. The Ph.D. project focused on the optimal

energy management of hybrid vehicles.

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