Fuel optimal control of hybrid vehicles

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Citation for published version (APA):

Keulen, van, T. A. C. (2011). Fuel optimal control of hybrid vehicles. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR712657

DOI:

10.6100/IR712657

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

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extensive project in the development of advanced energy management for urban distribution trucks which has been made possible by TNO Automotive.

A catalogue record is available from the Eindhoven University of Technology Library

Fuel optimal control of hybrid vehicles / by Thijs van Keulen. – Eindhoven : Technische Universiteit Eindhoven, 2011

Proefschrift. – ISBN: 978-90-386-2502-7 NUR 978

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Fuel Optimal Control of Hybrid Vehicles

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

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prof.dr.ir. M. Steinbuch

Copromotor:

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Summary

Fuel Optimal Control of Hybrid Vehicles

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

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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.

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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

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Roman uppercase

Symbol Description Unit

Cd dynamo road load control –

Cv velocity controller –

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

E_s0 initial energy in storage device J

Ef

s 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

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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 –

a₂ polynomial coefficient –

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Symbol Description Unit (continued ) a4 polynomial coefficient – 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 –

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Nomenclature xiii

Greek

α road slope rad

γ conversion characteristic –

γm,0 electric motor conversion characteristic –

γp,0 prime mover conversion characteristic –

γ_p,1 prime mover conversion characteristic –

γw well-to-tank conversion characteristic –

1 allowed error of end point constraint –

k 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 estimate

0 _{initial value}

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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) pRe(a)2_{+ Im(a)}2

int(a) interior point a of a set _X a_{⊂ X}

¯ a averaged value _N1 PN 1 ak ∂a(x,t) ∂x partial derivative ∂(bx+ct) ∂x = b

˙a time derivative ∂(bx+ct)_∂t = c + b ˙x

∂x Clarke’s generalized sub-differential: a vector v_{∈ R}n _{is said to be a}

sub-gradient of a convex continuous function f at a point x if the inequalityf (z)_{≥ f(x) + v}T_(z_{− x) holds for all z ∈ R}n_{. 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)

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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 the research 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

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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 50 100 150 0 5 10 15 20 25 30 35 40 45 50 Diesel Ethanol Gasoline

Hydrogen Gas (700 bar) Hydrydrogen liquid Natural Gas liquid

LPG Propane LPG 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.

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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 = mv ˙v |{z} (1) + mvgasin α | {z } (2) + c0mvgacos α | {z } (3) + c2v3 |{z} (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, gathe 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:

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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 Pm

into 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:

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• 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.

prime mover Pr Pf Pp Ef Es Pm generator Ps prime mover storage device Pr Pf Pp Ef Es Pm S motor-generator Ps gearbox motor-generator storage device rg Pw Pw Pk Pk

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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).

engine

electric clutch

gearbox final

power battery _wheels

service machine

drive 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 =R Pfdt. Control problems that deal with optimization as

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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= P_Pm_r = 1− _PPp_r, with Pr= Pp+ Pm.

The control of the tractive power at the wheels Pwto 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,

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• 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 inin-formation system or signals available in the vehicle. This is further discussed in Subsection 1.1.3.

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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

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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 tratra-jectory 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.

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1.3 Contributions and outline 11 Chapter 2 Chapter 3 Chapter 4 ˆp, ˆEs ˆ Pr, ˆv α, v, ˆm Pm, Pp, Pk, rg, S power request and

velocity trajectory

multiplier and state-of-energy requested travel time route data and vehicle parameters

real-time control

trajectory optimization

power split for

predefined trajectories real-time power split cruise control topology design

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, v

the vehicle velocity, Ef the fuel consumption, p a Lagrange multiplier

denoting the equivalent cost of stored energy in fuel, Es the

state-of-energy of the storage device, Pm the power output of the secondary

power converter, Pp the power output of the prime mover, Pk the power

of 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.

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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 inforinfor-mation, 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.

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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, HyHy-brid 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.

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Chapter two

Optimal Trajectories for Vehicles with

Energy Recovery Options

1

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 of brake 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 is involved 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 energy recovery options is rewritten like a nonlinear programming problem, and numerical solutions are 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

1_{This 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.

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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 brak(includ-ing capabilities have been made (Monastyrsky and Golownykh, 1993; Ko et al., 2004; Hellstr¨om et al., 2008; Va˜sak et al., 2009). In Hell-str¨om 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.

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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 ≤ ω max(Tp(ω))

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

min

ω

Z

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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 5₈Pp the

power reserve is linearly build off to become zero at Pp. 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 < Pp with P_p(Pv) = −γ p,0(Pv) γp,1(Pv)

the engine drag power and Pp 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 P r [kW] P eq [kW]

HEV e−line with P_v > 0 [kW] HEV e−line with P

v > 40 [kW]

HEV approximation at P

v>0

EV approximation

Figure 2.1 /Cost function, Peq is the equivalent fuel power, and Pr the tractive

power. 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.

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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(γm,1+ Pm, Pm/γm,1− )

or visa versa

Pm = min(γm,1− Ps, 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

min

Pm

Z

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 < Pp, 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 > Pp. Because the incremental cost

γp,1 and discharge/motor factor γm,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 _{≤ P}r ≤ Pp. The amount of fuel saved

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

γm,1+ γ

− m,1

Pm, with

Pm = Pr− Pp ≤ 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

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machine overload capability is not accounted for, the maximum tractive power is limited to the maximum engine output power Pr ≤ Pp, 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 < Pq, 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 γ_p,1(Pr(t)− Pp), γp,1(Pr(t)− Pp) γ_m,1+ γ_m,1− , γp,1(Pq− Pp) γ_m,1+ γ_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 Rt1 t0 Peq(t, Pr)dt subject to: dx_dt = 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) = a1

Pr(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 = _m1_e > 0 the reciprocal

effective vehicle mass (including rotating parts of the drive train), a2 = _mm_ec0gacos α + m

megasin α 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 = _mc1_e > 0 a loss parameter proportional to velocity, a4 = _mc2_e > 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)

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2.3 Necessary conditions for optimality 21

The control variable Pr is bounded to the set:

U = [Pd, Pp]. (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

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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)− Pp), γp,1(Pr(t)− Pp) γ_m,1+ γ_m,1− , γp,1(Pq− Pp) γ_m,1+ γ_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≡     0 0 0 0     (2.14)

such that the following necessary conditions are satisfied:

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

subdifferential2of 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)

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• 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 Z ta ˙ p1(t)dt + Z (ta,tb] dξ1(t)− Z (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 _{∈ [t}0, t1] where ξ1 and ξ2 are not differentiable.

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

P_r∗ = arg min

Pr

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:

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Switching function h is of first order with respect to the control Pr, and described by: h(t) =        p1(t)_v(t)a1 + γp,1 in int [Pp, Pp], p1(t)_v(t)a1 + _γ+γp,1 m,1γ − m,1 in int [P_q, P_p], p1(t)_v(t)a1 in int [Pd, Pq]. (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 [Pp, Pp], γp,1 γ_m,1+ γ_m,1− in int [Pq, Pp], 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) a₁˙v(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

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

¨ h(t) = p1(t)a1˙v(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

∗_{+ a}

3v∗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 _{≡ −λ}1 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)_∈                            Pp for p1 < −γp,1v a1 , [P_p, Pp] for p1 ≡ −γ_ap,1₁ v, P_p for −γp,1v a1 < p1 < −γp,1v γ+_m,1γ−_m,1a1, [P_q, P_p] for p₁ _≡ −γp,1v γ_m,1+ γ_m,1− a1, 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)

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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 p2is 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 ∈ [Pp, Pp]. The same reasoning holds for regenerative braking

if Pm ∈ [Pq, Pp]. Therefore, the road slopes can be aggregated if power requests are

known to remain in the intervals [P_p, Pp] and [Pq, Pp].

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, Pp] and the above mentioned assumptions are satisfied. It is acknowledged that

Fuel optimal control of hybrid vehicles

Fuel Optimal Control of Hybrid Vehicles

PROEFSCHRIFT

Thijs Adriaan Cornelis van Keulen

Contents

Summary

Fuel Optimal Control of Hybrid Vehicles

Nomenclature

Acronyms

Roman uppercase

Roman lowercase

Greek

Superscripts

Symbols and Operations

Chapter one

Introduction

1.1 General introduction

1.1.1 What is a hybrid vehicle

1.1.2 Control applied in hybrid vehicles

1.1.3 Route information and estimation of the vehicle conditions

1.2 Problem statement and objectives

1.3 Contributions and outline

1.4 Publications

Chapter two

Optimal Trajectories for Vehicles with

Energy Recovery Options

1

2.1 Introduction

2.2 Derivation of the cost function

2.3 Necessary conditions for optimality

2.4 Numerical solution

2.4.1 Structure of the solution