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공학박사학위논문공학박사학위논문공학박사학위논문공학박사학위논문

Effective Optimization of Power Management for Fuel Cell Hybrid

Vehicles Based on Pontryagin’s Minimum Principle

폰트리아진 최적 원리 기반의 연료전지 하이브리드

차량의 효과적인 동력관리 최적화

2012201220122012년년년년 8888월월월월

서울대학교서울대학교서울대학교서울대학교 대학원대학원대학원대학원

기계항공공학부기계항공공학부기계항공공학부기계항공공학부

Chunhua Zheng

Effective Optimization of Power Management for

Fuel Cell Hybrid Vehicles Based on Pontryagin’s

Minimum Principle

A DISSERTATION SUBMITTED TO THE SCHOOL OF

MECHANICAL AND AEROSPACE ENGINEERING OF

SEOUL NATIONAL UNIVERSITY IN PARTIAL

FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

By

Chunhua Zheng

August 2012

i

Abstract

Effective Optimization of Power Management for

Fuel Cell Hybrid Vehicles Based on Pontryagin’s

Minimum Principle

Chunhua Zheng

School of Mechanical and Aerospace Engineering

The Graduate School

Seoul National University

A considerable amount of research on power management strategies of

hybrid vehicles has been conducted during last few decades in order to

improve fuel economy and performance of hybrid vehicles. This dissertation

introduces a Pontryagin’s Minimum Principle (PMP)-based power

management strategy for fuel cell hybrid vehicles (FCHVs) and extends this

strategy mathematically for considering three important factors in FCHVs.

These factors include limitations on the battery state of charge (SOC) usage,

the fuel cell system (FCS) lifetime, and the effects of battery thermal

management on the fuel economy. The PMP-based power management

strategy is implemented in a computer simulation for each case.

ii

The limitation problem on the battery SOC usage is solved by introducing

a new cost function other than the fuel consumption rate to the PMP-based

optimal control problem. The limitation requirements on the battery SOC are

satisfied while minimizing the fuel consumption by this solution. In order to

take into account the lifetime of an FCS while considering its fuel

consumption minimization, a second cost function is defined and added to

the PMP-based optimal control problem. The second cost function is related

to the power changing rate of the FCS. Simulation results show that the

lifetime of the FCS can be prolonged by the reformulation of the PMP-based

optimal control problem. However, there is a tradeoff between the FCS

lifetime and the fuel consumption because of the added cost function. The

effect of battery thermal management on the total fuel consumption is

considered by designating the battery temperature as an extra state variable

other than the battery SOC in the PMP-based optimal control problem. The

relationship among the final battery SOC, the final battery temperature, and

the total fuel consumption is illustrated by simulation results. This

relationship can be expressed by a surface, which is composed of two

intersecting half-planes with similar gradients. This surface is defined as an

optimal surface in this dissertation, which indicates the optimal solutions, as

it is derived from the PMP-based power management strategy. Fuel economy

potential gains attributed to the battery thermal management are determined

using the optimal surface. The battery thermal management can improve the

iii

fuel economy of an FCHV up to 4.77% depending on the driving cycle. A

discussion on the combined case is carried out to consider the three factors

together.

For the three extended cases, global optimality of the PMP-based power

management strategies is discussed. Simulation results of the PMP-based

strategy are also compared to those of Dynamic Programming (DP) approach

for the three cases. The PMP-based power management strategy saves much

time compared to DP approach while it guarantees global optimality under

battery assumptions. The time-saving effect of the PMP-based strategy is

outstanding especially when there are more than two state variables.

Key words: Fuel cell hybrid vehicle, Pontryagin’s Minimum Principle,

Power management strategy, Mathematical extension, Time-

saving effect

Student Number: 2007-31059

iv

Contents

Abstract ....................................................................................................................... i

Contents .................................................................................................................... iv

List of Figures ......................................................................................................... vi

List of Tables ......................................................................................................... xv

Chapter 1 Introduction .......................................................................................... 1

1.1 Background ................................................................................... 1

1.2 Contributions ................................................................................ 6

1.3 Outline of this dissertation ............................................................ 9

Chapter 2 Vehicle Model ................................................................................... 11

2.1 Configuration of an FCHV ......................................................... 11

2.2 FCS model .................................................................................. 13

2.2.1 Fuel cell stack model ......................................................................................... 13

2.2.2 Compressor model .............................................................................................. 15

2.2.3 Air cooler and humidifier models ............................................................ 18

2.2.4 FCS characteristics ............................................................................................. 18

2.3 Battery model .............................................................................. 23

2.3.1 Battery internal resistance model ............................................................. 23

2.3.2 Battery thermal model ...................................................................................... 24

2.4 Components sizing ...................................................................... 25

2.4.1 Traction motor power design ...................................................................... 25

2.4.2 FCS power design ............................................................................................... 27

2.4.3 Battery power design and energy capacity design ....................... 29

Chapter 3 PMP-based power management strategy for FCHVs ............. 31

3.1 Theoretical study ......................................................................... 32

v

3.2 Optimal lines ............................................................................... 36

3.3 Fuel economy evaluation based on optimal lines ....................... 44

3.4 Comparison between PMP-based power management strategy

and DP approach ............................................................................... 48

Chapter 4 Extended PMP-based power management strategy for

FCHVs ..................................................................................................................... 52

4.1 PMP-based power management strategy considering battery

SOC constraint .................................................................................. 52

4.2 PMP-based power management strategy considering FCS

lifetime .............................................................................................. 60

4.3 PMP-based power management strategy considering battery

thermal management ......................................................................... 69

4.3.1 PMP-based power management strategy without considering

battery thermal management ..................................................................................... 71

4.3.2 PMP-based power management strategy considering battery

thermal management ....................................................................................................... 72

4.3.3 Global optimality of the two-state variable PMP-based

power management strategy ...................................................................................... 81

4.3.4 Control parameters of the PMP-based power management

strategy ..................................................................................................................................... 88

4.4 Discussions on the combined case .............................................. 93

Chapter 5 Concluding remarks ......................................................................... 95

5.1 Conclusion .................................................................................. 95

5.2 Future work ................................................................................. 98

References ............................................................................................................. 100

Abstract (korean) ................................................................................................ 107

Acknowledgement (korean) ............................................................................ 110

vi

List of Figures

Fig. 1.1 Improvement of the distribution pattern of the FCS operating points

by hybridization: (a) fuel cell vehicle and (b) fuel cell hybrid vehicle

Fig. 2.1 Configuration and energy flows of an FCHV

Fig. 2.2 Configuration of an FCS

Fig. 2.3 Simulation results derived from the compressor model

Fig. 2.4 OCV, activation loss, ohmic loss, concentration loss, and fuel cell

voltage of one single cell versus the stack current for the FCS used in this

dissertation

Fig. 2.5 Stack-provided power, auxiliary power, and FCS net power of the

FCS used in this dissertation

Fig. 2.6 Relationship between the FCS net power and the fuel consumption

rate of the FCS used in this dissertation

Fig. 2.7 FCS efficiency versus the FCS net power for the FCS used in this

dissertation

Fig. 2.8 Electrical schematic of the battery internal resistance model

Fig. 2.9 Relationship between the acceleration time and the required motor

power when the vehicle fully accelerates

Fig. 2.10 Relationship between the vehicle speed and the required FCS net

power when the vehicle drives on a flat road at a constant speed and it is

powered only by the FCS

vii

Fig. 2.11 Relationship between the vehicle speed and the required FCS net

power when the vehicle drives on a 5% grade road at a constant speed and it

is powered only by the FCS

Fig. 2.12 Relationship between the FCS net power and the battery power

when the selected motor power and FCS net power are 75 kW and 62 kW,

and the average motor efficiency is 85 %

Fig. 3.1 OCV and internal resistance of the battery used in this dissertation:

(a) OCV, (b) internal resistance

Fig. 3.2 Fuel consumption rate 2hm

•

, time derivative of the battery SOC

SOC•

, and Hamiltonian H for the whole range of the FCS net power

Fig. 3.3 Simulation results comparison between a constant costate and a

variable costate on the FTP75 urban driving cycle


variable costate on the NEDC 2000


variable costate on the Japan 1015 driving cycle

Fig. 3.6 Simulation results of the final battery SOC and the total fuel

consumption while changing both constant costates and variable costates on

the FTP75 urban driving cycle



the NEDC 2000

viii



the Japan 1015 driving cycle

Fig. 3.9 Optimal lines for the FTP75 urban driving cycle

Fig. 3.10 Optimal lines for the NEDC 2000

Fig. 3.11 Optimal lines for the Japan 1015 driving cycle

Fig. 3.12 A rule-based power management strategy

Fig. 3.13 Fuel economy evaluation of the rule-based strategy on the FTP75

urban driving cycle

Fig. 3.14 Fuel economy evaluation of the rule-based strategy on the NEDC

2000

Fig. 3.15 Fuel economy evaluation of the rule-based strategy on the Japan

1015 driving cycle

Fig. 3.16 Distribution patterns of the FCS operating points: (a) Rule-based

power management strategy, (b) PMP-based power management strategy

Fig. 3.17 Simulation results comparison between PMP-based power

management strategy and DP approach on the FTP75 urban driving cycle: (a)

FCS net power trajectories, (b) battery power trajectories, (c) SOC

trajectories


management strategy and DP approach on the NEDC 2000: (a) FCS net

power trajectories, (b) battery power trajectories, (c) SOC trajectories

ix


management strategy and DP approach on the Japan 1015 driving cycle: (a)

FCS net power trajectories, (b) battery power trajectories, (c) SOC

trajectories

Fig. 4.1 Comparison of the optimal battery SOC trajectories for the case

without battery SOC boundary and the case with battery SOC boundary

when the cost function C is used

Fig. 4.2 Comparison of the optimal costate trajectories for the case without

battery SOC boundary and the case with battery SOC boundary when the

cost function C is used

Fig. 4.3 Comparison of the optimal power trajectories for the case without


cost function C is used

Fig. 4.4 Comparison of the optimal battery SOC trajectories for the case

without battery SOC boundary and the case with battery SOC boundary

when the cost function S is used

Fig. 4.5 Comparison of the optimal costate trajectories for the case without


cost function S is used

Fig. 4.6 Comparison of the optimal power trajectories for the case without


cost function S is used

x

Fig. 4.7 Comparison between DP approach and PMP-based power

management strategy when the battery SOC constraint is considered: (a) FCS

net power trajectories, (b) battery power trajectories, (c) SOC trajectories

Fig. 4.8 Optimal trajectories for the cases when the FCS lifetime is

considered and is not considered on the FTP75 urban driving cycle


considered and is not considered on the NEDC 2000


considered and is not considered on the Japan 1015 driving cycle

Fig. 4.11 Optimal lines for the cases when the FCS lifetime is considered and

is not considered on the FTP75 urban driving cycle


is not considered on the NEDC 2000


is not considered on the Japan 1015 driving cycle


management strategy when the FCS lifetime is considered: (a) FCS net

power trajectories, (b) battery power trajectories, (c) SOC trajectories

Fig. 4.15 Battery characteristics: (a) OCV, (b) charging resistance, (c)

discharging resistance

Fig. 4.16 Optimal surface on the FTP75 urban driving cycle for the case

when the initial battery SOC is 0.6 and the initial battery temperature is

xi

25 °C

Fig. 4.17 Optimal surface on the NEDC 2000 for the case when the initial

battery SOC is 0.6 and the initial battery temperature is 25 °C

Fig. 4.18 Optimal surface on the Japan 1015 driving cycle for the case when

the initial battery SOC is 0.6 and the initial battery temperature is 25 °C

Fig. 4.19 Optimal surface on the FTP75 urban driving cycle for the case

when the initial battery SOC is 0.6 and the initial battery temperature is 5 °C

Fig. 4.20 Optimal trajectories for different initial battery temperature

conditions: (a) battery SOC trajectories, (b) battery temperature trajectories,

(c) battery power trajectories, (d) FCS net power trajectories

Fig. 4.21 Effects of battery thermal management on the optimal trajectories

on the FTP75 urban driving cycle: (a) battery SOC, (b) battery temperature,

(c) battery power, (d) FCS net power


on the NEDC 2000: (a) battery SOC, (b) battery temperature, (c) battery

power, (d) FCS net power


on the Japan1015 driving cycle: (a) battery SOC, (b) battery temperature, (c)

battery power, (d) FCS net power

Fig. 4.24 The effect of the battery thermal management on the fuel economy

over the FTP75 urban driving cycle


xii

over the NEDC 2000


over the Japan 1015 driving cycle

Fig. 4.27 Concavity and convexity of the state equations based on the

characteristics of the battery used in this dissertation: (a) state equation F

versus battery SOC and battery temperature for different battery power, (b)

state equation F versus battery temperature and battery power for different

battery SOC, (c) state equation F versus battery SOC and battery power

for different battery temperature, (d) state equation f versus battery

temperature and battery SOC for different battery power, (e) state equation

f versus battery temperature and battery power for different battery SOC,

(f) state equation f versus battery SOC and battery power for different

battery temperature

Fig. 4.28 Simulation results derived from DP approach for the two-state

variable case (driving cycle 1): (a) simulation results of the two-state variable,

(b) simulation result of the battery SOC, and (c) simulation result of the

battery temperature

Fig. 4.29 Driving cycle 1 used in comparison between PMP-based strategy

and DP approach


management strategy when the effect of battery thermal management is

considered (driving cycle 1): (a) battery SOC, (b) battery temperature, (c)

xiii

battery power, and (d) FCS net power

Fig. 4.31 Simulation results derived from DP approach for the two-state

variable case (driving cycle 2): (a) simulation results of the two-state variable,

(b) simulation result of the battery SOC, and (c) simulation result of the

battery temperature

Fig. 4.32 Driving cycle 2 used in comparison between PMP-based strategy

and DP approach


management strategy when the effect of battery thermal management is

considered (driving cycle 2): (a) battery SOC, (b) battery temperature, (c)

battery power, and (d) FCS net power


•

, time derivative of the battery SOC

SOC•

, time derivative of the battery temperature T•

, and Hamiltonian H

for the whole range of the FCS net power

Fig. 4.35 Simulation results of two costates for different initial values of

them over the FTP75 urban driving cycle

Fig. 4.36 Relationship between initial value of 1p and final battery SOC

and relationship between initial value of 2p and final battery temperature

over three typical driving cycles: (a) final battery SOC versus initial value of

1p on the FTP75 urban driving cycle, (b) final battery SOC versus initial

value of 1p on the NEDC 2000, (c) final battery SOC versus initial value of

1p on the Japan 1015 driving cycle, (d) final battery temperature versus

xiv

initial vale of 2p on the FTP75 urban driving cycle, (e) final battery

temperature versus initial vale of 2p on the NEDC 2000, (f) final battery

temperature versus initial vale of 2p on the Japan 1015 driving cycle

xv

List of Tables

Table 2.1 Parameters of the vehicle

Table 2.2 Parameters used to the fuel cell voltage calculation

Table 2.3 Parameters related to the FCS used in this dissertation

Table 2.4 Driving performance requirements on the FCHV

Table 2.5 Selected power source components

Table 3.1 Fuel economy evaluation results of the rule-based power

management strategy on three driving cycles

Table 3.2 Fuel economy comparison between the PMP-based power

management strategy and DP approach on three driving cycles

Table 4.1 Fuel consumption comparison of three PMP-based power

management strategies on the FTP75 urban driving cycle

Table 4.2 Comparison of the PMP-based power management strategies for

the cases when the FCS lifetime is considered and is not considered on three

driving cycles

Table 4.3 Influence of the tuning parameter

Table 4.4 Effects of the battery thermal management on the total fuel

consumption on three driving cycles

1

Chapter 1 Introduction

1.1 Background

Hybrid vehicles use two or more than two kinds of power sources, and they

have become a major area of interest in academia and in the automotive

industry recently owing to the energy supply problem and environmental

problems. The power management strategy of hybrid vehicles is one of the

most important and popular research topics in this area, as it determines the

power split between power sources and because it is related to the fuel

economy of the vehicles. Several types of power management strategies for

hybrid vehicles have been developed during last few decades. These power

management strategies can be divided into two major groups: those based on

the heuristic concept and those based on the optimal control theory. The

former mainly includes rule-based algorithms and fuzzy logic algorithms [1-3].

Earlier in the development of hybrid vehicles, power management strategies

were dominated by these types of strategies owing to their simplicity when

actually realizing them. These types of strategies, however, cannot guarantee

the optimal power distribution and the optimal fuel economy as well. In

addition, the rules and fuzzy logic need expert knowledge. To remedy this

problem, the optimal control theory was introduced as part of the power

management strategy of hybrid vehicles, including both Dynamic

2

Programming (DP) as developed by R. E. Bellman [4-6] and Pontryagin’s

Minimum Principle (PMP) [7-10]. The DP approach examines all admissible

control inputs at every state, thus guaranteeing global optimality if the driving

cycle information is given in advance [4-6]. However, the DP approach cannot

be used directly for the real-time control of hybrid vehicles due to the

backward-looking calculation process and the long calculation time. Being

confronted with the drawbacks of the DP, some researchers have proposed

stochastic dynamic programming [11-13] to overcome these problems. The

PMP-based power management strategy optimizes the power distribution

between power sources and minimizes the performance measure by

instantaneously providing the necessary optimality conditions. One of the

major advantages of the PMP-based strategy is that there is only one parameter

to be tuned in this strategy in order to obtain optimal results over a specific

driving cycle [9]. Moreover, the core of this strategy is implementable in a

real-time controller, even if the driving cycle information is not known in

advance [10]. Furthermore, previous research [8] proved from a mathematical

point of view that the PMP-based power management strategy can serve as a

global optimal solution (DP) under the assumption that the open-circuit

voltage (OCV) and the internal resistance of a battery are independent of the

battery state of charge (SOC). This assumption is reasonable for charge-

sustaining types of hybrid vehicles, especially for those which use lithium-ion

batteries. There is also a power management strategy known as the equivalent

3

consumption minimization strategy (ECMS), which is similar to the PMP-

based strategy. The ECMS is originally based on the heuristic concept holding

that the electric energy usage can be transformed to the equivalent fuel

consumption [14-17]. Although the ECMS is based on this heuristic idea, it

also works in conjunction with the optimal concept and can be applied for use

with a real-time control scheme. However, in the PMP-based strategy, control

parameters and their relationship can be explained physically and

mathematically from an optimal control viewpoint, given that this strategy

stems from the optimal control theory. This is notable especially for the case

where there are two state variables and this is the main difference between the

ECMS and the PMP-based strategy.

A fuel cell hybrid vehicle (FCHV) uses a fuel cell system (FCS) as its

primary power source. The FCS converts hydrogen and oxygen into electric

energy with water and heat as the by-products [18]. Therefore, FCHVs are

considered as one of the most promising candidates for future transportation.

As a power source, an FCS has relatively slow power response and cannot

recover the braking energy. Thus, the size and cost of the FCS will be

increased if the FCS is the only power source in a vehicle. The secondary

power source which has relatively fast power response and can recuperate the

braking energy is needed. A battery could be one of the candidates for the

secondary power source. An FCHV can provide sufficient power during its

acceleration and can recuperate the kinetic or potential energy of the vehicle

4

during braking by hybridization of an FCS and a battery. Our research has

shown that the fuel economy can be improved around 20% by hybridization

when the FCHV and the vehicle which is powered only by an FCS (fuel cell

vehicle) have similar mass. Fig. 1.1 illustrates the distribution patterns of the

FCS operating points in the two vehicles. It can be observed that the operating

points of the FCS are shifted to high efficiency region by hybridization in the

FCHV, while the FCS operation does not have an option in the fuel cell

vehicle (FCV), as the FCS is the only power source of the FCV. FCHVs have

many outstanding advantages, such as higher energy efficiency and lower

emissions compared to internal combustion engine vehicles.

(a) (b)

Fig. 1.1 Improvement of the distribution pattern of the FCS operating points by hybridization:

(a) fuel cell vehicle, (b) fuel cell hybrid vehicle

The PMP-based power management strategy is applied to an FCHV in this

dissertation. Some researchers have studied this power management strategy

for engine/battery powered hybrid vehicles and for plug-in hybrid vehicles as

well [18-23]. In the optimal control problem formulation of earlier research,

the performance measure to be minimized is the total fuel consumption, the

state variable of the control system is the battery SOC, and the control variable

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

(%

)

0 10 20 30 40 50 600

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

/Fre

qu

ency

(%

)

5

of the control system is the battery power or the engine power. Some

researchers have extended the basic form of the optimal control problem to

achieve some specific goals [19, 22, 23]. In the research [19], the limitation

problem on the battery SOC usage is considered by defining a cost function

regarding the battery SOC and adding it to the PMP-based optimal control

problem in an engine/battery powered hybrid vehicle. In the research [22],

engine oil temperature is added to the PMP-based control problem as an extra

state variable other than the battery SOC in order to assess the effects of

engine thermal management on the fuel consumption in an engine/battery

powered hybrid vehicle. In the research [23], the battery aging factor is

defined by a parameter, and this parameter is added to the PMP-based control

problem as a second cost function in order to take into account the battery

lifetime together with the total fuel consumption in an engine/battery powered

hybrid vehicle.

In spite of the previous research on the PMP-based power management

strategy, there are still some important factors which are ignored or which

need to be improved when applying this strategy to FCHVs. In this dissertation,

these factors are considered mathematically in the formulation of the PMP-

based optimal control problem. These factors include the battery SOC

constraint, the FCS lifetime, and the effect of battery thermal management on

the fuel economy. These factors are considered by adding a new cost function

or a new state variable to the optimal control problem. In the previous research

6

[8], the global optimality of the PMP-based strategy is proved for the case

where there are one state variable and one cost function. In this dissertation,

global optimality is discussed when the new factors are considered in the

PMP-based strategy, and simulation results of the PMP-based strategy are

compared to those of DP approach which gives global optimal solution. The

comparison result shows that the PMP-based power management strategy still

guarantees global optimality when the new factors are considered. Time-

saving effect of the PMP-based power management strategy is outstanding

considering that the DP approach needs much more time to obtain results,

especially for two-state variable cases. Also, the control parameters and their

relationship in the PMP-based strategy are explained physically and

mathematically in this dissertation.

1.2 Contributions

As introduced in 1.1, a great deal of research on power management

strategies of hybrid vehicles has been conducted during last decades. The

research is still in progress aiming to reduce fuel consumption and improve

vehicle performance. This dissertation introduces a PMP-based power

management strategy for FCHVs, and extends this strategy mathematically in

order to consider some important factors in FCHVs. There are two main

contributions that this dissertation has made compared to the previous research

introduced in 1.1.

7

The first contribution is that the basic formulation of the PMP-based optimal

control problem is extended mathematically in order to take into account some

important factors in FCHVs. In the previous research on the PMP-based power

management strategy, some important factors are ignored in FCHVs. This will

cause some practical problems, such as lifetime shortening of power sources

and increase of fuel consumption. These factors include upper limit and lower

limit of the battery SOC, the FCS lifetime, and the influence of battery thermal

management on the fuel economy. These extensions are achieved by defining

a new state variable or a new cost function and mathematically adding it to the

optimal control problem formulation. Previously, the battery SOC limitation

problem is considered by defining a cost function regarding the battery SOC

and adding it to the PMP-based optimal control problem [19]. This solution,

however, makes the optimal battery SOC trajectory fluctuating. In order to

overcome this drawback, a new cost function regarding the FCS net power is

defined and added to the PMP-based optimal control problem in this

dissertation, and the drawback has been disappeared by this method. In order

to prolong the FCS lifetime, a wavelet transform method is used to decompose

the required power signal into low frequency component and high frequency

component in the literature [3]. In this literature, a fuzzy logic controller is

applied to distribute the low frequency and high frequency components to the

FCS and the battery, respectively. This strategy, however, cannot guarantee

the optimal solution, as the fuzzy logic controller is used. In this dissertation, a

8

new cost function is defined and added to the PMP-based optimal control

problem in order to consider the FCS lifetime. By this extension, the FCS

lifetime is prolonged while the fuel consumption is minimized. Few

researchers have focused on the effect of battery thermal management on the

fuel economy in FCHVs so far. In this dissertation, the battery thermal model

is applied and the effect of battery thermal management on the total fuel

consumption is assessed by designating the battery temperature as a second

state variable in the PMP-based optimal control problem. By this extension,

the effect is assessed and an optimal surface is defined, which expresses the

relationship among the final battery SOC, the final battery temperature, and

the total fuel consumption.

The second contribution of this dissertation is that it provides an effective

method of power management for hybrid vehicles. The PMP-based power

management strategy guarantees global optimality under some reasonable

assumptions while it saves much time compared to the DP approach. The time-

saving effect is outstanding especially for those systems where there are more

than two state variables. Few researchers have provided elapsed time

comparison result of the PMP-based strategy and the DP approach for two-

state variable cases so far. This dissertation compares the simulation time

consumed in the PMP-based strategy and the DP approach for two-state

variable cases so that the effectiveness of the PMP-based power management

strategy is proved.

9

1.3 Outline of this dissertation

There are five chapters in this dissertation. Chapter 1 introduces the research

background on the power management strategies of hybrid vehicles. The

contributions that this dissertation has made compared to the previous research

and outline of this dissertation are also included in the chapter 1. Chapter 2

presents the control-oriented powertrain component models used in this

dissertation. It mainly covers the FCS model and the battery model. The sizing

process of the power source components of an FCHV is also covered in

chapter 2. Chapter 3 mathematically introduces the PMP-based power

management strategy for FCHVs. Formulation of the optimal control problem

here is aimed at a system in which there are one state variable and one cost

function. The PMP-based strategy introduced in chapter 3 is mathematically

extended according to three important factors in chapter 4. These extensions

are achieved by defining a new state variable or a new cost function and

mathematically adding it to the optimal control problem formulation. The

global optimality of the extended PMP-based strategies is discussed and the

simulation results of the extended PMP-based strategy are compared to those

of DP approach. Chapter 4 also compares the elapsed time of the PMP-based

strategy and the DP approach so that the effectiveness of the PMP-based

strategy is emphasized. Chapter 4 is the main part of this dissertation. Chapter

10

5 concludes the whole contents of this dissertation and gives some concluding

remarks and proposes the future work.

11

Chapter 2 Vehicle Model

The complexity of a vehicle model depends on its application. When the

objective is to develop and evaluate a power management strategy or to

estimate the fuel economy, the quasi-static vehicle model is enough. When the

objective is to evaluate the drivability of the vehicle, such as jerks and surges,

the detailed dynamic vehicle model is required. In this dissertation, the

primary objective is to evaluate power management strategies in FCHVs, and

thus a quasi-static vehicle model is used. A quasi-static vehicle model is

sufficient to calculate energy flows in the powertrain and is appropriate for

performance optimization problems [24].

2.1 Configuration of an FCHV

Fig. 2.1 illustrates the configuration of an FCHV and the energy flows in an

FCHV. The architecture of an FCHV is similar to that of a series hybrid

electric vehicle, considering that the electric motor is the only powertrain

component that is directly connected to the wheels. The FCS and the battery

are the power sources and they are connected to the wheels through the

traction motor. The motor receives power from both the FCS and the battery

through the DC-DC converter and the DC-AC inverter. The motor can be

controlled to operate as a generator to convert the kinetic or potential energy

of the vehicle into electrical energy and store it in the battery. The arrows in

12

Fig. 2.1 indicate the energy flow directions. The motor uses a map to express

its efficiency, and the converters are assumed to be ideal converters with a

constant efficiency of 95%. The final drive gear efficiency is considered to be

a constant. The vehicle parameters used in this dissertation are shown in Table

2.1. Parts of these data are sourced from available literature [25].

Fig. 2.1 Configuration and energy flows of an FCHV

Table 2.1 Parameters of the vehicle

Item Value

Vehicle total mass (kg) 1700

Mass factor 1.1

Final drive gear efficiency (%) 95

Tire radius (m) 0.29

Aerodynamic drag coefficient 0.37

Vehicle frontal area (m2) 2.59

Air density (kg/m3) 1.21

Rolling resistance coefficient 0.014

13

2.2 FCS model

The FCS is the primary power source of FCHVs. Fig. 2.2 illustrates the

configuration of an FCS. It also shows the information of gas flows and of

control signals. In this subsection, the components marked in red are

introduced and modeled, which are the fuel cell stack, the compressor, the air

cooler, and the humidifier. We consulted the literature [26-29] for these

models. Here, the fuel cell stack is the main device and others are auxiliary

devices. The characteristics of the FCS used in this dissertation are also

presented in this subsection.

Fig. 2.2 Configuration of an FCS

2.2.1 Fuel cell stack model

A fuel cell stack converts chemical energy of reactants into electrical

energy and provides power to the vehicle and to its auxiliary devices. A fuel

14

cell stack is composed of many single cells connected in series. Here, these

cells are assumed to be identical in performance. The voltage of a single cell

fcv is calculated as follows [26-29]:

fc act ohm concv E v v v= − − −

(2.1)

Here, E is the open circuit voltage (OCV). actv , ohmv , and concv represent

activation loss, ohmic loss, and concentration loss. These losses are

considered by physical and empirical equations here.

The OCV here is calculated from the energy balance between the reactants

and products, and the Faraday Constant [26, 27, 30], as follows:

( ) ( ) ( )( )2 2

4 51.229 8.5 10 298.15 4.3085 10 ln 0.5 lnfc fc h oE T T P P− −= − × − + × ⋅ + (2.2)

Here, fcT represents the fuel cell stack temperature, 2hP and

2oP represent

the reactant partial pressures.

The activation loss, which is caused by the need to move electrons and to

break and form chemical bonds in the anode and cathode, is dominated by the

cathode reaction conditions [27, 31]. This loss can be expressed by the Tafel

equation [26, 27, 32] as follows,

( )10 1 c i

act av v v e− ⋅= + ⋅ − (2.3)

where 0v is the voltage drop at zero current and av depends on the oxygen

partial pressure and the stack temperature. i represents the current density,

and 1c is a constant.

15

The ohmic loss is attributed to the electrical resistance of the electrodes and

the electrolyte, and the voltage drop caused by this loss is proportional to the

current density, as follows:

ohm ohmv i R= ⋅ (2.4)

Here, the internal electrical resistance ohmR depends on the membrane

thickness mt and the membrane conductivity mσ , as follows,

mohm

m

tR

σ= (2.5)

where, the membrane conductivity strongly depends on the membrane water

content and the cell temperature.

The concentration of the reactants in the fuel cell decreases along with the

electrochemical reaction. It results in the fuel cell voltage drop. Especially in

the high current density region, it leads to a rapid voltage drop. This is the

concentration loss, which can be expressed as follows [26, 27, 33]:

3

2max

c

conc

iv i c

i

= ⋅

(2.6)

Here, 2c , 3c , and maxi are constants and depend on the temperature and

reactant partial pressure and can be determined empirically [26].

2.2.2 Compressor model

An air compressor is the main auxiliary device of the fuel cell stack, and it

consumes the greatest amount of auxiliary power from the fuel cell stack. An

air compressor is needed to provide the air to the cathode side with a certain

16

pressure and a certain air flow rate. The performance of the air compressor

affects the overall efficiency of the fuel cell stack, as the reaction rate in the

membrane is influenced by the air pressure. Here, the compressor model is

divided into two parts. The first part is a static compressor map, which is

derived based on the Jensen & Kristensen method [34], and the second part is

the compressor and motor inertia.

The dynamic behavior of the air compressor is expressed by a dynamic

equation, as follows:

cpcp m cp

dJ

dt

ωτ τ= − (2.7)

Here, cpJ represents the total inertia of the compressor and the motor, cpω

represents the compressor speed, mτ is the torque provided by the motor,

and cpτ is the torque required to drive the compressor. Here, mτ is

calculated by a static equation, as follows [27]:

( )tm m m v cp

m

kv k

Rτ η ω= − (2.8)

Here, mη represents the motor mechanical efficiency, and mv represents the

motor input voltage. tk , mR , and vk are motor constants. cpτ is calculated

using a thermodynamic equation, as follows [27, 35]:

1

1p atm smcp cp

cp cp atm

C T PW

P

γγ

τω η

− = −

(2.9)

Here, pC is the constant-pressure specific heat capacity of air, atmT is the

atmospheric temperature, cpη is the compressor efficiency, γ is the ratio of

17

the specific heats of air, and cpW is the air flow rate of the compressor. smP

and atmP represent the air pressure in the supply manifold (compressor output)

and the atmospheric pressure, respectively. The only dynamic state in this

compressor model is the compressor speed. Here, we consulted the literature

[26] for the related data, and obtained our own maps on the relationship

between the stack current and the compressor output pressure, the relationship

between the stack current and the compressor air flow rate, and the

relationship between the stack current and the compressor power consumption.

Fig. 2.3 shows the simulation results derived from the compressor model.

Fig. 2.3 Simulation results derived from the compressor model

0 10 20 30 40 50 600

200

400

Cu

rren

t (A

)

0 10 20 30 40 50 600

100

200300

Mo

tor

inpu

t (V

)

0 10 20 30 40 50 601.5

2

2.5

Pre

ssu

re r

atio

0 10 20 30 40 50 600.5

1

1.5x 10

4

Co

mp

ress

or

spee

d (

rpm

)

0 10 20 30 40 50 600

0.1

0.2

Co

mp

ress

or

flow

(kg

/s)

0 10 20 30 40 50 600

2

4

Oxy

gen

exc

ess

ratio

0 10 20 30 40 50 600

10

20

Co

mp

ress

or

pow

er (

kW)

Time (s)

18

2.2.3 Air cooler and humidifier models

An air cooler is needed for the fuel cell stack to reduce the temperature of

the air entering the stack, considering that the pressurized air leaving the

compressor is at a higher temperature. In the air cooler model, the vapor

saturation map and thermodynamic properties are used to calculate the vapor

pressure, dry air pressure, vapor mass flow, and dry air mass flow. A

humidifier is also needed for the fuel cell stack to prevent dehydration of the

membrane. The humidifier model here calculates the amount of water which

is required to be injected. The water injected is assumed to be in the form of

vapor [26]. In addition, the humidifier model also gives the total flow rate

change and pressure change caused by the added water.

2.2.4 FCS characteristics

Fig. 2.4 illustrates the OCV, the activation loss, the ohmic loss, the

concentration loss, and the fuel cell voltage of one single cell versus the stack

current for the FCS used in this dissertation. The data used here are listed in

Table 2.2.

19

Fig. 2.4 OCV, activation loss, ohmic loss, concentration loss, and fuel cell voltage of one single

cell versus the stack current for the FCS used in this dissertation

Table 2.2 Parameters used to the fuel cell voltage calculation

Item Value

Maximum stack current (A) 400

Anode pressure (Hydrogen) (atm) 2

Cathode pressure (air) (atm) 1-1.5

Stack temperature (°C) 80

Active area (cm2/cell) 280

Membrane thickness (cm) 0.01275

0 50 100 150 200 250 300 350 4001.175

1.18

1.185

1.19

Ope

n c

ircu

it vo

ltage

(V

)

0 50 100 150 200 250 300 350 4000.2

0.3

0.4

Act

ive

loss

(V

)

0 50 100 150 200 250 300 350 4000

0.2

0.4

Ohm

ic lo

ss (

V)

0 50 100 150 200 250 300 350 4000

0.02

0.04

0.06

Co

ncen

tra

tion

loss

(V

)

0 50 100 150 200 250 300 350 4000.4

0.6

0.8

1

Cel

l vol

tag

e (V

)

Time (s)

20

The stack-provided power stackP is related to the stack current stackI and

cell voltage fcv as follows:

stack cell fc stackP N v I= ⋅ ⋅ (2.10)

Here, cellN represents the cell number of the stack. The stack-provided power

is partially used to maintain the auxiliary devices of the FCS, such as the air

compressor. The part of power used to propel the vehicle is called FCS net

power. The relationship between the FCS net power fcsP and the stack-

provided power stackP is as follows:

fcs stack auxP P P= − (2.11)

Here, auxP represents the power consumption of the auxiliary components.

Fig. 2.5 illustrates the stack-provided power, auxiliary power, and FCS net

power of the FCS used in this dissertation. The parameters related to the FCS

are listed in Table 2.3.

Fig. 2.5 Stack-provided power, auxiliary power, and FCS net power of the FCS used in this

dissertation

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

Stack current (A)

Po

wer

(kW

)

Stack powerFCS net powerCompressor powerOther auxiliary power

21

Table 2.3 Parameters regarding the FCS

Item Value

Maximum stack power (kW) 77

Maximum net power (kW) 62

Cell number 350

Compressor efficiency (%) 80

For a fuel cell stack, the fuel consumption rate 2hm

• is related to the stack

current according to the following equation:

2

2

cell hh stack

N Mm I

n Fλ

• ⋅= ⋅ ⋅

⋅ (2.12)

In equation (2.12) [19, 36, 37], 2hM represents the molar mass of hydrogen,

n represents the number of electrons acting in the reaction, F is the

Faraday constant, and λ is the hydrogen excess ratio.

The FCS net power and the fuel consumption rate have a specific

relationship, as both of them are related to the fuel cell stack current according

to Fig. 2.5 and equation (2.12). Fig. 2.6 illustrates the relationship between the

FCS net power and the fuel consumption rate of the FCS used in this

dissertation.

22

Fig. 2.6 Relationship between the FCS net power and the fuel consumption rate of the FCS

used in this dissertation

In an FCS, its efficiency is defined as

2

fcsfcs

h

P

m LHVη •=

⋅ (2.13)

In equation (2.13) [38], LHV=120000 kJ/kg is the lower heating value of

hydrogen. Fig. 2.7 illustrates the FCS efficiency versus the FCS net power for

the FCS used in this dissertation.

Fig. 2.7 FCS efficiency versus the FCS net power for the FCS used in this dissertation

0 10 20 30 40 50 600

0.5

1

1.5

2

FCS net power (kW)

Hyd

rog

en c

on

sum

ptio

n r

ate

(g/s

)

0 10 20 30 40 50 600

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

(%

)

23

2.3 Battery model

2.3.1 Battery internal resistance model

An internal resistance battery model [39] is used in this dissertation. This

battery model consists of a voltage source (OCV) and an internal resistance

component [40-42]. The effects of the battery temperature on both the internal

resistance R and the OCV, V , were typically neglected in previous

research [43-46]. In this dissertation, the effects are taken into account and the

parameters of the battery model are related according to the following

equation:

2( , ) ( , ) 4 ( , )

2 ( , )

bat

bat

ISOC

Q

V SOC T V SOC T R SOC T PI

R SOC T

•= −

− − ⋅=

(2.14)

Here, batQ is the battery charge capacity, I is the battery current, T is the

battery temperature, and batP is the battery power at the battery terminals. Fig.

2.8 illustrates the electrical schematic of the battery internal resistance model.

Fig. 2.8 Electrical schematic of the battery internal resistance model

24

2.3.2 Battery thermal model

In this dissertation, a lumped capacitance thermal model [47] is used to

estimate the battery temperature changes. The temperature change in the

battery is calculated in accordance with the energy balance between the

battery heat generation and heat lost, the thermal mass of the battery, and the

duration of the battery use, as follows [47]:

_ _

,

bat gen bat case

bat p bat

Q QT

m C

−=

⋅ɺ (2.15)

Here, _bat genQ represents the battery heat generation caused by

electrochemical reactions and resistive heating, _bat caseQ represents heat loss

from the battery, batm is the battery mass, and ,p batC is the battery heat

capacity.

In this battery thermal model, the parallel airflow approach is used to cool

the battery. Thus, _bat caseQ is a combination of conduction and convection

loss from the battery to the surrounding air; it can be expressed as follows:

_

1

,

,

air

eff

bat caseamb

eff

T TT a

RQ

T TT a

R

− ≥= − <

(2.16)

In this equation, airT represents the temperature of the air surrounding the

battery, ambT represents the ambient temperature. a is a set point of the

battery temperature, which is the starting point of the air cooling. effR is the

effective thermal resistance for the case when the battery temperature is above

25

a and the air cooling system operates, and 1effR is that for the case when the

battery temperature is below a and the air cooling system does not operate.

The exit air temperature airT can be expressed as follows [47]:

_

,

0.5 bat caseair amb

air p air

QT T

m C= +

⋅ɺ (2.17)

Here, airmɺ is the airflow rate, and ,p airC is the heat capacity of the air. In this

thermal model, it is assumed that 50% of the battery heat loss is used to warm

the air.

2.4 Components sizing

Before the power management strategy, the sizing process on power source

components of an FCHV needs to be done in order to meet the driving

performance requirements, which are bounds or constraints on the component

sizing. The components sizing process of an FCHV includes the power design

of the traction motor, the FCS power design, the battery power design, and the

energy capacity design of the battery. The driving performance requirements

include acceleration requirement and maximum speed requirement.

2.4.1 Traction motor power design

As stated in subsection 2.1, the configuration of an FCHV is similar to that

of a series type of hybrid vehicle, considering that the traction motor is the

only powertrain component that is directly connected to the wheel side. The

motor transforms the electrical energy of the power sources into the

26

mechanical energy to propel the vehicle. Therefore, the traction motor power

is required to meet the acceleration demand, the maximum speed demand, and

the gradeability demand of the vehicle. Out of these demands, the acceleration

demand requires the greatest amount of power, and thus the power of a

traction motor is usually determined by this factor. The acceleration ability of

a vehicle is usually evaluated by its acceleration time for accelerating it from

zero to a certain high speed. The total traction power p for accelerating the

vehicle from zero to a certain speed fV in the time t can be expressed as

follows [48]:

2 2 32 1( )

2 3 5f b r f a D f f

Mp V V Mgf V C A V

t

δ ρ= + + + (2.18)

Here, δ is the mass factor that equivalently converts rotational inertias of

rotating components into translational mass [49], M is the total mass of the

vehicle, bV is the vehicle base speed which is dependent on the motor base

speed, g is the gravity acceleration, rf is the rolling resistance coefficient,

aρ is the air density, DC is the aerodynamic drag coefficient, and fA is the

vehicle frontal area. The values of these parameters are listed in Table 2.1. In

this dissertation, the maximum rotational speed of the motor is 5000 rpm, and

the speed ratio of the motor, which is defined as the ratio of maximum speed

to base speed of the motor [48], is 4. The final gear ratio is 4, and fV is 100

km/h. Under this condition, the relationship between the acceleration time and

the required motor power is illustrated in Fig. 2.9. It can be observed that if

27

the vehicle is required to accelerate itself from 0 to 100 km/h in 13 s, the

required motor power is around 75 kW.

Fig. 2.9 Relationship between the acceleration time and the required motor power

2.4.2 FCS power design

An FCS is the primary power source in an FCHV, and thus it is required to

provide enough power to the vehicle when the vehicle drives at a high constant

speed on a flat road or a grade road without help of the secondary power

source. When the vehicle drives at a constant speed V , the required traction

power is as follow [48]:

31cos sin

2r a D fP Mgf V C A V MgVα ρ α= + + (2.19)

Here, α represents the gradient of the road. Fig. 2.10 illustrates the

relationship between the vehicle speed and the required FCS net power when

the vehicle drives on a flat road at a constant speed and it is powered only by

the FCS.

10 11 12 13 14 15 1660

65

70

75

80

85

90

95

Acceleration time (s)

Req

uire

d m

oto

r p

ow

er (

kW)

28

Fig. 2.10 Relationship between the vehicle speed and the required FCS net power when the

vehicle drives on a flat road at a constant speed and it is powered only by the FCS

Fig. 2.11 shows the same relationship for the case where the gradient of the

road is 5 %. The average motor efficiency is assumed to be 85 % here. It can

be observed that if the FCS net power is 62 kW, the maximum constant speed

is around 148 km/h on a flat road and 111 km/h on a 5 % grade road.

Fig. 2.11 Relationship between the vehicle speed and the required FCS net power when the

vehicle drives on a 5% grade road at a constant speed and it is powered only by the FCS

110 120 130 140 150 160 17020

30

40

50

60

70

80

90

Vehicle speed (km/h)

Req

uire

d F

CS

net

po

wer

(kW

)

80 85 90 95 100 105 110 115 12035

40

45

50

55

60

65

70

75

Vehicle speed (km/h)

Req

uire

d F

CS

net

po

wer

(kW

)

29

2.4.3 Battery power design and energy capacity design

The battery power is decided based on the motor power and the FCS net

power, which are determined by the preceding process, and the average motor

efficiency. Fig. 2.12 illustrates the relationship between the FCS net power and

the battery power when the selected motor power and FCS net power are 75

kW and 62 kW, and the average motor efficiency is 85 %. It can be seen that

the minimum battery power is about 26 kW in this case.

Fig. 2.12 Relationship between the FCS net power and the battery power when the selected

motor power and FCS net power are 75 kW and 62 kW, and the average motor efficiency is 85 %

As a second power source, the battery provides power to the vehicle and

also recovers energy from the vehicle or from the FCS. The energy change

changeE in the battery during driving can be expressed as follows:

traction recoveringchange out inE P dt P dt= −∫ ∫ (2.20)

Here, outP is the battery power corresponding to the case when the battery

provides power to the vehicle, inP is the battery power corresponding to the

case when the battery recovers energy from the regenerative braking or from

45 50 55 60 65 70 75 80 850

5

10

15

20

25

30

35

40

FCS net power (kW)

Bat

tery

po

wer

(kW

) Admissible area

30

the FCS. Considering the available battery SOC usage, the battery energy

capacity can be determined by

changeb

EE

K= (2.21)

where, K is the allowed percentage of the battery SOC usage. changeE

strongly depends on the power management strategy and the driving cycle [48].

Thus, the battery energy capacity is also dependent on specific conditions.

The driving performance requirements on the FCHV in this dissertation are

listed in Table 2.4, and the powertrain components of the FCHV are selected

based on these requirements which are listed in Table 2.5. Here, two types of

batteries are selected for chapter 3 and chapter 4, respectively.

Table 2.4 Driving performance requirements on the FCHV

Requirement Value

Acceleration time from 0 to 100 km/h (s) 13

Maximum speed (flat road) (km/h) 148

Maximum speed (5% grade road) (km/h) 111

Table 2.5 Selected power source components

Component Value

Motor power (kW) 75

FCS net power (kW) 62

Battery capacity (kWh) 1.5

Battery capacity (kWh) 1.9 at 25°C

31

Chapter 3 PMP-based power management strategy

for FCHVs

PMP stems from the optimal control theory; it is a general case of the

fundamental theorem of the Calculus of Variations [7]. PMP instantaneously

provides the necessary conditions to optimal control problems to let them find

optimal control laws. In this chapter, the basic formulation of the PMP-based

optimal control problem, in which there are one state variable and one cost

function, is presented. The optimal lines are defined based on the simulation

results of the PMP-based power management strategy, and they are used to

the fuel economy evaluation of a rule-based power management strategy. The

simulation results of the PMP-based power management strategy are also

compared to those of DP approach (global optimal solution). The powertrain

components listed in Table 2.5 are used in this chapter. Here, the first battery

is used. Fig. 3.1 illustrates the characteristics of the first battery. The

characteristics of the FCS were described in subsection 2.2.

(a) (b)

Fig. 3.1 OCV and internal resistance of the battery: (a) OCV, (b) internal resistance

0 0.2 0.4 0.6 0.8 1270

280

290

300

310

320

330

Battery SOC

OC

V (

V)

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

Battery SOC

Inte

rnal

res

ista

nce

(O

hm

)

dischargingcharging

32

3.1 Theoretical study

The motor power comes from both the FCS and the battery in an FCHV, as

shown in Fig. 2.1. Thus, the fuel economy of an FCHV depends on how the

power required for the vehicle is distributed between the FCS and the battery.

The objective of the optimal control problem of an FCHV is to find an

optimal power split trajectory which minimizes the fuel consumption when

the vehicle is being driven. We solve this problem by finding the optimal

trajectory of the FCS net power, which is the control variable of the optimal

control problem. The battery SOC is the state variable of the optimal control

problem. In this chapter, the battery temperature is not taken into account in

the fuel consumption minimization.

The state equation of the system, which describes the dynamics of the state

variable, is given in (2.14). Considering that the battery temperature is not

taken into account here, and the internal resistance and OCV of the battery are

functions of the battery SOC, equation (2.14) can be simplified using a

function f as follows:

( )( ), ( )( ) batSOC f SOC t P tt•

= (3.1)

The power required for the motor reqP , the FCS net power fcsP , and the

battery power batP have the following relationship:

( ) ( ) ( )bat req fcsP t P t P t= − (3.2)

33

As the power required for the motor can be derived when selecting a driving

cycle, we can transform the state equation (3.1) into

( ) ( )( ), ( )fcsSOC t F SOC t P t•

= (3.3)

using a different function F .

The performance measure to be minimized here is the total fuel

consumption when the FCHV drives over a specified driving cycle from time

0t to time ft . Given that the FCS net power and the fuel consumption rate

are related to each other as shown in Fig. 2.6, the performance measure J is

expressed as follows:

( ) ( )2

0

( ) ( )ft

fcs h fcstJ P t m P t dt

•= ∫ (3.4)

Considering the state equation (3.3), which is a constraint of the optimal

control problem, together with the cost function, the performance measure is

to be

( ) ( ) ( ) ( ) ( )2

0

( ) ( ) ( ), ( )ft

fcs h fcs fcstJ P t m P t p t F SOC t P t SOC t dt

• • = + ⋅ −

∫ (3.5)

where, p is the Lagrange multiplier, which is also called the costate in the

PMP-based control.

The objective of the optimal control problem here is to minimize the total

fuel consumption while the dynamic state equation (3.3) is satisfied. Thus, the

necessary conditions of the optimal control problem are given when the

variation of the performance measure Jδ from equation (3.5) is zero [7, 8].

If we introduce a Hamiltonian H [7, 8], which is defined as

34

( ) ( )( ) ( ) ( ) ( )2

, ( ), ( ) ( ), ( )fcs h fcs fcsH SOC t P t p t m P t p t F SOC t P t•

= + ⋅ (3.6)

then the necessary conditions that derive the optimal trajectories are as

follows:

0fcs

HSOC

p

Hp

SOCH

P

•

•

∂ =∂∂ = −

∂∂ =

∂

(3.7)

The necessary conditions in (3.7) should be satisfied all the time in order to

obtain the optimal results. In the definition of the Hamiltonian (3.6), the first

term is about the fuel usage and the second term can be considered as the

electric usage. The costate p can also be considered as an equivalent

parameter between the fuel usage and the electric usage [8, 43, 44]. The first

necessary condition in (3.7) is actually the state equation (3.3), which is a

constraint of the optimal control problem. The second necessary condition is

called the costate equation that determines the optimal trajectory of the costate

p when the initial value of the costate is given. The third necessary condition

determines the optimal trajectory of the control variable fcsP by minimizing

the Hamiltonian H .

PMP is a general case of the Euler-Lagrange equation of the Calculus of

Variation [8], in which the third necessary condition in (3.7) is expressed as

follows:

( ) ( ) ( )( ) ( ) ( ) ( )( )* * * * *, ,, ,fcs fcsH SOC t P t p t H SOC t P t p t≤ (3.8)

35

The advantage of form (3.8) is that it can be applied to a non-linear, non-

differentiable, or a non-convex function [8]. In the computer calculation of the

PMP-based optimal control, the optimal fcsP is obtained at every calculation

time step by finding out the fcsP among all admissible FCS net power values,

which minimizes the Hamiltonian H . Now, the necessary conditions of the

PMP-based optimal control can be written in a specific form, as follows:

( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )( )

* * * * * *

* * * * * * *

* * * * *

, , ,

, , ,

, , , ,

fcs fcs

fcs fcs

fcs fcs

HSOC t SOC t P t p t F SOC t P t

p

H Fp t SOC t P t p t p t SOC t P t

SOC SOC

SOC t P t p t SOC t P t p tH H

•

•

∂= =∂∂ ∂= − = − ⋅

∂ ∂≤

(3.9)

Boundary conditions also need to be satisfied other than the necessary

conditions, which are as follows [7]:

( ) ( ) ( ) ( )( )* * * *, , 0T

f f f fcs f f fp t SOC H SOC t P t p t tδ δ − + = (3.10)

Here, ft represents the final time.

Fig. 3.2 illustrates the fuel consumption rate 2hm

•

, the time derivative of the

battery SOC SOC•

, and the Hamiltonian H for the whole range of the

FCS net power when the power required for the motor reqP is 30 kW, the

battery SOC is 0.6, and the costate p is set to -90. The convexity of the

Hamiltonian indicates that the optimal fcsP , which minimizes the Hamiltonian,

can be determined for this calculation time step. In this example, the FCS net

power ranges from 0 to 62 kW. Admissible FCS net power range for each

calculation time step is decided by considering the maximum and minimum

36

power of the battery. It can be seen from Fig. 3.2 that the costate p should

be a negative value all the time in the PMP-based optimal control problem.

Otherwise zero will be always selected for the FCS net power and there is no

the concept of the optimization.


•, time derivative of the battery SOC SOC

•, and

Hamiltonian H for the whole range of the FCS net power

3.2 Optimal lines

Previous research proved from a mathematical point of view that the PMP-

based power management strategy can work as a global optimal solution (DP)

under the assumption that the internal resistance and OCV of a battery do not

depend on the battery SOC [8]. This assumption is reasonable for FCHVs,

0 10 20 30 40 50 600

1

2

Fuel c

onsu

mptio

n rate

(g/s

)

0 10 20 30 40 50 60-0.01

0

0.01

Tim

e d

eriv

ativ

e o

f S

OC

(1/s

)

0 10 20 30 40 50 600.5

1

1.5

Ham

iltonia

n (g/s

)

FCS net power (kW)

37

considering that FCHVs are charge-sustaining types of hybrid vehicles. This

assumption can lead to the following equation because the function F does

not depend on the battery SOC.

( ) ( ) ( )( )* * * 0fcs

Fp t p t P t

SOC

• ∂= − ⋅ =∂

(3.11)

Equation (3.11) indicates that the costate is a constant value. From equation

(3.11), previous research [8] proved that the PMP-based power management

strategy can work as a global optimal solution (DP).

A variable costate, derived from the second necessary condition in (3.7),

can be replaced with a constant costate when the above assumption is satisfied

[43]. In order to validate this fact, a simulation comparison is carried out here.

Fig. 3.3 illustrates the comparison results for the FTP75 urban driving cycle.

This figure shows that the optimal trajectories of the fuel consumption for

both a constant costate and a variable costate are very similar to each other.

The optimal trajectories of the battery SOC for both cases are also very

similar to each other. Fig. 3.4 and Fig. 3.5 illustrate the comparison results of

the NEDC 2000 and the Japan 1015 driving cycle, respectively. The two

figures also validate the same fact that a variable costate, derived from the

necessary condition of the PMP, can be replaced with a constant costate. The

second result in each figure pertains to the costate information. For each

driving cycle, the constant costate selected is close to the average value of the

variable costate. In fact, the optimal trajectories of the fuel consumption for

both a constant costate and a variable costate can be identical by adjusting the

38

value of the constant costate and the initial value of the variable costate. The

optimal trajectories of the battery SOC can also be the same by adjusting the

value of the costates.

Fig. 3.3 Simulation results comparison between a constant costate and a variable costate on the

FTP75 urban driving cycle

0 200 400 600 800 1000 1200 14000

50

100

Veh

icle

spe

ed (

km/s

)

0 200 400 600 800 1000 1200 1400-87

-86.5

-86

Cos

tate

(g

)

Variable costateConstant costate

0 200 400 600 800 1000 1200 14000.4

0.6

0.8

Bat

tery

SO

C


0 200 400 600 800 1000 1200 14000

50

100

150

Time (s)

Fu

el c

ons

umpt

ion

(g

)


39


NEDC 2000


Japan 1015 driving cycle

0 200 400 600 800 1000 12000

50

100

150

Veh

icle

sp

eed

(km

/h)

0 200 400 600 800 1000 1200-89

-88

-87C

ost

ate

(g)


0 200 400 600 800 1000 12000.4

0.6

0.8

Bat

tery

SO

C


0 200 400 600 800 1000 12000

50

100

150

Time (s)

Fu

el c

on

sum

ptio

n (

g)


0 100 200 300 400 500 600 7000

50

100

Veh

icle

sp

eed

(km

/h)

0 100 200 300 400 500 600 700-86

-85.5

-85

Cos

tate

(g

)


0 100 200 300 400 500 600 7000.55

0.6

0.65

Bat

tery

SO

C


0 100 200 300 400 500 600 7000

20

40

60

Fu

le c

onsu

mpt

ion

(g)

Time (s)


40

The trajectories of the battery SOC and the fuel consumption depend on the

costate value. Thus, the final battery SOC and the total fuel consumption will

change if a different costate is used in the simulation. We assessed the

simulation results of the final battery SOC and the total fuel consumption

while changing both constant costates and variable costates. Fig. 3.6, Fig. 3.7,

and Fig. 3.8 illustrate the assessed simulation results for the FTP75 urban

driving cycle, the NEDC 2000, and the Japan 1015 driving cycle, respectively.

Asterisks correspond to constant costates and circles correspond to variable

costates. The initial battery SOC is set to 0.6 here. The three figures show the

strong similarity in the relationship between the final battery SOC and the

total fuel consumption when using both constant costates and variable costates.

Fig. 3.6 Simulation results of the final battery SOC and the total fuel consumption while

changing both constant costates and variable costates on the FTP75 urban driving cycle

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)

Variable costatesConstant costates

41


changing both constant costates and variable costates on the NEDC 2000


changing both constant costates and variable costates on the Japan 1015 driving cycle

Fig. 3.9 shows the simulation results of the relationship between the final

battery SOC and the total fuel consumption for the FTP75 urban driving cycle

when the vehicle mass is increasing. Here, only constant costates are used. It

can be observed that the simulation results form an approximately straight line

for each vehicle mass. These lines are defined as optimal lines, as it is proved

that the PMP-based power management strategy can serve as a global optimal

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

Fu

el c

on

sum

ptio

n (

g)

Final battery SOC


0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.830

40

50

60

70

80

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)


42

solution [8]. The initial battery SOC is set to 0.6 here. It is clear that the

gradient of the optimal lines is the same when the vehicle mass is increasing

and the optimal line moves up along with increasing the vehicle mass. Also,

the distance between nearby optimal lines is the same. Fig. 3.10 and Fig. 3.11

illustrate the cases of the NEDC 2000 and the Japan 1015 driving cycle,

respectively. It can be seen from the three figures that the distance between

nearby optimal lines is different for each driving cycle.

Fig. 3.9 Optimal lines for the FTP75 urban driving cycle

Fig. 3.10 Optimal lines for the NEDC 2000

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

180

190

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)

Vehicle mass=1700 kgVehicle mass=1900 kgVehicle mass=2100 kg

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

180

190

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)


43

Fig. 3.11 Optimal lines for the Japan 1015 driving cycle

In an optimal line, each simulation result point corresponds to a different

costate value. A lower final battery SOC value and a lower total fuel

consumption value correspond to a higher costate value, and a higher final

battery SOC value and a higher total fuel consumption value correspond to a

lower costate value. As stated before, the costate can be considered as an

equivalent parameter between the fuel usage and the electric usage. The

physical meaning of the costate here can be explained as the amount of fuel

that can replace the battery electrical energy usage. Thus the unit of the

costate is g . A higher costate value indicates that more fuel is needed to

replace the same amount of battery electrical energy and that the fuel is more

valuable. Thus, the PMP-based power management strategy attempts to use

the electrical energy more during the power distribution. On the contrary, a

lower costate value implies that less fuel is needed to replace the same amount

of battery electrical energy and the electrical energy is more valuable.

Therefore, the strategy tries to use the fuel more during power distribution.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.830

40

50

60

70

80

90

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)


44

The costate is the only control parameter in the PMP-based power

management strategy introduced in this chapter. The physical meaning of the

costae helps us to predict it based on the future driving information in the

realization of the PMP-based strategy.

3.3 Fuel economy evaluation based on optimal lines

As the optimal lines are derived from the PMP-based power management

strategy, they can be used to evaluate other power management strategies.

Any other simulation result points derived from other power management

strategies will always locate above optimal lines. In this subsection, a rule-

based power management strategy for the FCHV is introduced and its fuel

economy is evaluated based on the optimal lines. Fig. 3.12 illustrates the rule-

based power management strategy which is based on the efficiency

characteristics of the FCS. For the FCS used in this dissertation, it is clear that

the FCS efficiency is very low when the FCS net power is less than 7 kW;

hence, the battery mode is used in this region as much as possible. On the

other hand, the FCS efficiency is high when the FCS net power is greater than

7 kW and less than 25 kW; therefore, the FCS mode is used in this region.

When the required power is greater than 25 kW, the hybrid mode or battery

charging mode is used. The FCS provides constant power in this region. The

battery charging mode is selected if the constant power is greater than the

45

required power, and the hybrid mode is used if the constant power is less than

the required power.

Fig. 3.12 A rule-based power management strategy

In Fig. 3.13, Fig. 3.14, and Fig. 3.15, the solid lines are optimal lines

derived on the FTP75 urban driving cycle, the NEDC 2000, and the Japan

1015 driving cycle, respectively. The process of a fuel economy evaluation

based on the optimal line is as follows: (1) Obtain the simulation results of the

final battery SOC and fuel consumption when the rule-based power

management strategy illustrated in Fig. 3.12 is applied to the FCHV and the

initial battery SOC is set to 0.6 for the three driving cycles. (2) Plot the point,

which corresponds to the simulation results of each driving cycle, to each

figure to compare with each optimal line. (3) Add a straight line parallel to the

optimal line that intersects the point in each figure. (4) Check the fuel

consumption value which corresponds to the final battery SOC of 0.6 on the

line from step (3) in each figure. In Fig. 3.13, Fig. 3.14, and Fig. 3.15, the

asterisks are derived from step (2) and the dashed lines are obtained from step

0 10 20 30 40 50 600

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

(%

)

0.4<SOC<0.8

Hybrid modePfcs=28 kW

FCS mode

Battery mode

46

(3). In this subsection, the fixed-step size (sample time) is 0.1 s for both the

PMP-based strategy and rule-based strategy, while it is 1 s in other sections.

Fig. 3.13 Fuel economy evaluation of the rule-based strategy on the FTP75 urban driving cycle

Fig. 3.14 Fuel economy evaluation of the rule-based strategy on the NEDC 2000

Fig. 3.15 Fuel economy evaluation of the rule-based strategy on the Japan 1015 driving cycle

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

120

130

140

150

160

170

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

120

130

140

150

160

170

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.825

30

35

40

45

50

55

60

65

70

Final battery SOC

Fu

el c

on

sum

ptio

n (

g)

47

Table 3.1 shows the fuel economy evaluation result of the rule-based power

management strategy on the three driving cycles. The discrepancy between

the rule-based strategy and the PMP-based strategy is 3.46%, 4.44%, and 4.23%

on the three driving cycles.

Table 3.1 Fuel economy evaluation result of the rule-based power management strategy on

three driving cycles

Driving cycle

Fuel consumption (kg/100 km)

Discrepancy (%) PMP-based power

management strategy

Rule-base power

management strategy

FTP75 urban 1.127 1.166 3.46

NEDC 2000 1.269 1.325 4.44

Japan 1015 1.119 1.166 4.23

Fig. 3.16 illustrates the distribution patterns of the FCS operating points for

the rule-based power management strategy and the PMP-based strategy. The

FTP75 urban driving cycle is used here, and the final battery SOC is lower

than the initial battery SOC for the rule-based strategy while they are the same

for the PMP-based strategy. It can be observed that more FCS operating

points are located in the high efficiency region for the PMP-based power

management strategy.

48

(a) (b)

Fig. 3.16 Distribution patterns of the FCS operating points for the rule-based power

management strategy and the PMP-based strategy: (a) rule-based strategy, (b) PMP-based

strategy

3.4 Comparison between PMP-based power management

strategy and DP approach

Previous research [7, 50] theoretically investigated the relationship between

the Hamilton-Jacobi-Bellman (HJB) equation, which is the recurrence relation

of the DP, and the PMP, and concluded that the PMP can be derived from the

HJB equation under certain conditions. Previous research [8] also proved from

a mathematical point of view that the PMP-based power management strategy

can be a global optimal solution (DP) under the assumption that the internal

resistance and OCV of a battery do not depend on the battery SOC. In this

subsection, simulation results of the two power management strategies are

compared to each other. Here, the control variable is the FCS net power when

the PMP is applied to the power management strategy of the FCHV while the

control variable is the battery power when the DP approach is applied.

0 10 20 30 40 50 600

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

/Fre

qu

ency

(%

)

0 10 20 30 40 50 600

10

20

30

40

50

60

FCS net power (kW)

FC

S e

ffici

ency

/Fre

qu

ency

(%

)

49

Fig. 3.17 illustrates the optimal trajectories solved by the PMP and DP on

the FTP75 urban driving cycle. The results include the FCS net power, the

battery power, and the battery SOC. Here, the optimal battery power

trajectory of the PMP is obtained by considering the optimal FCS net power

trajectory and the power required for the vehicle, which is given when

selecting a driving cycle. As we can see, the two trajectories of each term

nearly overlap each other most of the time. Fig. 3.18 and Fig. 3.19 are for the

cases of the NEDC 2000 and the Japan 1015 driving cycle.

(a)

(b) (c)

Fig. 3.17 Comparisons of PMP and DP on the FTP75 urban driving cycle: (a) FCS net power

trajectories, (b) battery power trajectories, (c) SOC trajectories

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

35

Time (s)

FC

S n

et p

ow

er (kW

)

DPPMP

0 200 400 600 800 1000 1200 1400-25

-20

-15

-10

-5

0

5

10

15

Time (s)

Bat

tery

pow

er (kW

)

DPPMP

0 200 400 600 800 1000 1200 14000.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

Time (s)

Batte

ry S

OC

DPPMP

50

(a)

(b) (c)

Fig. 3.18 Comparisons of PMP and DP on the NEDC 2000: (a) FCS net power trajectories, (b)

battery power trajectories, (c) SOC trajectories

(a)

(b) (c)

Fig. 3.19 Comparisons of PMP and DP on the Japan 1015 driving cycle: (a) FCS net power


0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40

Time (s)

FC

S n

et p

ow

er (k

W)

DPPMP

0 200 400 600 800 1000 1200-30

-20

-10

0

10

20

Time (s)

Bat

tery

pow

er (k

W)

DPPMP

0 200 400 600 800 1000 12000.5

0.55

0.6

0.65

0.7

0.75

Time (s)

Bat

tery

SO

C

DPPMP

0 100 200 300 400 500 600 7000

5

10

15

20

25

Time (s)

FC

S n

et p

ow

er (kW

)

DPPMP

0 100 200 300 400 500 600 700-10

-5

0

5

10

Time (s)

Bat

tery

pow

er (k

W)

DPPMP

0 100 200 300 400 500 600 700

0.58

0.59

0.6

0.61

0.62

0.63

0.64

0.65

Time (s)

Bat

tery

SO

C

DPPMP

51

Table 3.2 shows the discrepancy of the two strategies on the fuel economy.

It can be seen that the discrepancy over the three driving cycles is within

plus/minus 0.4%. However, the elapsed time discrepancy between the two

strategies is very large. The PMP-based power management strategy can save

much time compared to the DP approach while it guarantees the global

optimality. This is one of the strong advantages of the PMP-based strategy.

Table 3.2 Fuel economy comparison between the PMP-based power management strategy and

DP approach

Driving cycle Fuel consumption (kg/100 km)

Discrepancy (%) PMP-based strategy DP approach

FTP75 urban 1.154 1.149 0.44

NEDC 2000 1.282 1.279 0.23

Japan 1015 1.137 1.133 0.35

52

Chapter 4 Extended PMP-based power management

strategy for FCHVs

The basic formulation of the PMP-based optimal control problem of

FCHVs is carried out in chapter 3, in which there are only one state variable

and one cost function and the state variable is not constrained. In this chapter,

the basic formulation is mathematically extended based on three important

factors which are limitations on the battery SOC usage, the FCS lifetime, and

the effect of the battery thermal management on the fuel economy.

4.1 PMP-based power management strategy considering

battery SOC constraint

In chapter 3, we assumed that the state variable is not bounded. But in fact,

most batteries operate in a certain SOC range in the charge-sustaining types of

hybrid vehicles. Thus the battery SOC should be constrained in the optimal

control problem. In this subsection, the constraints on the battery SOC usage

are considered by defining a new cost function and adding the cost function to

the PMP-based optimal control problem. The first battery in Table 2.5 is used

in this subsection. The objective of the optimization problem here is to

minimize the fuel consumption while the battery SOC usage boundary is

satisfied. The formulation of the optimal control problem is introduced below.

53

The state equation is the same with equation (3.3), as the state variable is

also the battery SOC here. In the literature [19], a new cost function is defined

in order to consider the battery SOC limitation factor in the fuel consumption

optimization problem. Here, we first introduce a similar cost function with the

literature [19]. The cost function C is defined as follows:

( )( )

( ) ( ) ( )

( ) ( ) ( )

2

minmin

max

2

maxmax

max

0

SOC t SOCa SOC t SOC

SOC

SOC t SOCC SOC t SOC t SOC

SOC

Otherwise

b

δδ

δδ

− + ⋅ ≤ +

− − = ≥ −

⋅ (4.1)

Here, minSOC and maxSOC represent the upper limit and the lower limit of

the battery SOC, respectively. a and b are tuning parameters, and δ is a

constant. The performance measure to be minimized is then

( ) ( ) ( )( ) ( ) ( ) ( )2

0

( ) ( ) ( ), ( )ft

fcs h fcs fcstJ P t m P t C SOC t p t F SOC t P t SOC t dt

• • = + + ⋅ −

∫ (4.2)

The necessary conditions of the PMP-based optimal control here are as

follows,

( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )( )

* * * * * *

* * * *

* * * *

* * * * *

, , ,

, ,

,

, , , ,

fcs fcs

fcs

fcs

fcs fcs


p

Hp t SOC t P t p t

SOCF dC

p t SOC t P t SOC tSOC dSOC

H SOC t P t p t H SOC t P t p t

•

•

∂= =∂∂= −

∂∂= − ⋅ −

∂≤

(4.3)

when Hamiltonian is defined as

( ) ( )( ) ( ) ( )( ) ( ) ( )2

, ( ), ( ) ( ), ( )fcs h fcs fcsH SOC t P t p t m P t C SOC t p t F SOC t P t•

= + + ⋅ (4.4)

54

It can be observed from (4.3) that there is a new term in the second

necessary condition compared to (3.9). This new term is originated from the

added cost function and affects the optimal trajectory of the costate when the

battery SOC is about to reach its limits. Fig. 4.1, Fig. 4.2, and Fig. 4.3

illustrate comparison results of the PMP-based strategies introduced in

chapter 3 and here. The upper limit and lower limit of the battery SOC are

0.6936 and 0.5073, and the FTP75 urban driving cycle is used here.

Fig. 4.1 Comparison of the optimal battery SOC trajectories for the case without battery SOC

boundary and the case with battery SOC boundary by cost function C

Fig. 4.2 Comparison of the optimal costate trajectories for the case without battery SOC


0 200 400 600 800 1000 1200 14000.45

0.5

0.55

0.6

0.65

0.7

Time (s)

Bat

tery

SO

C

Not constrainedConstrained

0 200 400 600 800 1000 1200 1400-95

-90

-85

-80

-75

Time (s)

Co

stat

e (g

)


55

Fig. 4.3 Comparison of the optimal power trajectories for the case without battery SOC


It can be seen that there are fluctuations in the trajectories because of the

changed costate value for the case with the battery SOC boundary, although

the battery SOC is constrained in this case. This is attributed to the

characteristics of the cost function C . The fluctuations of the battery SOC

may shorten the battery lifetime. In order to remedy the drawback of the cost

function (4.1), a cost function S is newly defined regarding the battery SOC

limitation factor. Here, S is a function of the FCS net power, whereas C is

a function of the battery SOC.

The definition of the new cost function S is as follows:

( )( )( ) ( )( ) ( )

min

max

0

fcs

fcs fcs

P t SOC t SOC

S P t P t SOC t SOC

Otherwise

αβ

⋅ ≤= ⋅ ≥

(4.5)

Here, α and β are tuning parameters. The performance measure using S

is then

0 200 400 600 800 1000 1200 14000

10

20

30

40

FC

S n

et pow

er (k

W)

Not constrained

Constrained

0 200 400 600 800 1000 1200 1400-40

-20

0

20

Time (s)

Batte

ry p

ow

er (k

W)

Not constrained

Constrained

56

( ) ( ) ( )( ) ( ) ( ) ( )2

0

( ) ( ) ( ), ( )ft

fcs h fcs fcs fcstJ P t m P t S P t p t F SOC t P t SOC t dt

• • = + + ⋅ −

∫ (4.6)


follows,

( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )( )

* * * * * *

* * * * * * *

* * * * *

, , ,

, , ,

, , , ,

fcs fcs

fcs fcs

fcs fcs


p


SOC SOC


•

•

∂= =∂∂ ∂= − = − ⋅

∂ ∂≤

(4.7)

when Hamiltonian is defined as

( ) ( )( ) ( ) ( )( ) ( ) ( )2

, ( ), ( ) ( ), ( )fcs h fcs fcs fcsH SOC t P t p t m P t S P t p t F SOC t P t•

= + + ⋅ (4.8)

Necessary conditions in (4.7) look the same with those in (3.9), but the

third necessary condition expresses different contents, given that the new cost

function S is added to the Hamiltonian here. The influence of the cost

function S on the optimal trajectories is different from that of C . C

affects the optimal trajectory of the costate when the battery SOC is about to

reach its limits, and this consequently affects other optimal trajectories

including the optimal battery SOC trajectory. On the other hand, S has the

function of shifting the optimal value of the FCS net power when the battery

SOC reaches its boundaries. This accordingly influences the optimal battery

SOC trajectory. S affects the shape of Hamiltonian illustrated in Fig. 3.2, as

S is a function of the FCS net power. A lower value of the FCS net power

will be selected as the optimal solution when the battery SOC reaches its

upper limit, and a greater value of the FCS net power will be chosen as the

optimal solution when the battery SOC reaches its lower limit. Fig. 4.4, Fig.

57

4.5, and Fig. 4.6 show comparison results of the PMP-based power

management strategies introduced in chapter 3 and here using the cost

function S . The upper limit and lower limit of the battery SOC are also

0.6936 and 0.5073, and the FTP75 urban driving cycle is used here. It can be

seen that the tendency of the battery SOC trajectories is similar while the

battery SOC is constrained in the case where the cost function S is used.

The FCS net power trajectories overlap each other most of the time and the

battery power trajectories as well. This is the main advantage of using the cost

function S .

Fig. 4.4 Comparison of the optimal battery SOC trajectories for the case without battery SOC

boundary and the case with battery SOC boundary by cost function S

0 200 400 600 800 1000 1200 14000.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

Time (s)

Bat

tery

SO

C


58

Fig. 4.5 Comparison of the optimal costate trajectories for the case without battery SOC


Fig. 4.6 Comparison of the optimal power trajectories for the case without battery SOC


Table 4.1 shows the fuel consumption comparison of the three power

management strategies introduced in chapter 3 and in this subsection. Here,

the initial battery SOC and the final battery SOC are both 0.6, and the FTP75

urban driving cycle is used. The table indicates that there is a tradeoff between

the limitation requirement of the battery SOC usage and the fuel consumption.

The vehicle consumes slightly more fuel when the battery SOC is constrained,

0 200 400 600 800 1000 1200 1400-87

-86.5

-86

-85.5

Time (s)

Co

stat

e (g

)


0 200 400 600 800 1000 1200 14000

20

40

60

FC

S p

ow

er

(kW

)

Not constrained

Constrained

0 200 400 600 800 1000 1200 1400-40

-20

0

20

Time (s)

Ba

ttery

po

we

r (k

W)

Not constrained

Constrained

59

and the fuel economy of the case where the cost function S is used is better

than that of the case where the cost function C is used.

Table 4.1 Fuel consumption comparison of three PMP-based power management strategies on

the FTP75 urban driving cycle

Power management strategy Fuel consumption

(kg/100 km)

PMP-based strategy without battery

SOC boundary 1.154

PMP-based strategy with battery

SOC constraint

(cost function C )

1.166

PMP-based strategy with battery

SOC constraint

(cost function S )

1.159

Equation (3.11) is still true when the new cost function S is added if the

battery OCV and internal resistance are not dependent on the battery SOC.

Because S is only related to fcsP as shown in equation (4.5) and there is no

state variable other than the battery SOC. Thus, the PMP-based power

management strategy with the cost function S can serve as a global optimal

solution. Fig. 4.7 illustrates the comparison result between DP approach and

the PMP-based power management strategy when the battery SOC constraint

is considered. The FTP75 urban driving cycle is used here. The optimal

trajectories overlap each other most of the time and simulation results show

60

that the fuel consumption discrepancy is within 0.5%. It can be concluded that

the PMP-based power management strategy still guarantees the global

optimality when the battery SOC constraint is considered by the cost function

S . Meanwhile, the PMP-based power management strategy can save much

time compared to the DP approach.

(a)

(b) (c)

Fig. 4.7 Comparison between DP approach and PMP-based power management

strategy when the battery SOC constraint is considered: (a) FCS net power


0 200 400 600 800 1000 1200 14000

10

20

30

40

50

Time (s)

FC

S n

et p

ow

er (

kW)

DPPMP

0 200 400 600 800 1000 1200 1400-25

-20

-15

-10

-5

0

5

10

15

Time (s)

Bat

tery

po

wer

(kW

)

DPPMP

0 200 400 600 800 1000 1200 14000.5

0.52

0.54

0.56

0.58

0.6

0.62

Time (s)

Bat

tery

SO

C

DPPMP

61

4.2 PMP-based power management strategy considering FCS

lifetime

In chapter 3, it was assumed that there are no limits on the FCS power

changing rate. However, the power changing rate of an FCS is actually limited

because of the slow dynamic of its air circuit [25, 51]. Besides, frequent and

rapid changes of the dynamic load shorten the FCS lifetime. Hence, these

changes of the load should be avoided in order to improve the FCS durability

and prolong the FCS lifetime. In this subsection, a new cost function is

defined for the FCS lifetime factor and is introduced to the PMP-based

optimal control problem. The first battery in the Table 2.5 is also used in this

subsection. The objective of the optimal control problem here is to minimize

the fuel consumption while considering the FCS lifetime.

The state equation is also the same with equation (3.3) here, as there is also

one state variable in this optimal control problem. The new cost function L

which is related to the objective of prolonging the FCS lifetime is defined as

follows:

( ) ( )2( ) ( ) ( )fcs fcs fcsL P t c P t P t d= ⋅ − − (4.9)

Here, c is a tuning parameter, t represents a time step, and t d−

represents its previous time step. d is the duration of one time step.

Considering the cost functions and the state equation, which is a constraint on

the optimization problem, the performance measure to be minimized is

62

( ) ( ) ( ) ( ) ( ) ( )2

0

( ) ( ) ( ) ( ), ( )ft

fcs h fcs fcs fcstJ P t m P t L P t p t F SOC t P t SOC t dt

• • = + + ⋅ −

∫ (4.10)


follows:

( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )( )

* * * * * *

* * * * * * *

* * * * *

, , ,

, , ,

, , , ,

fcs fcs

fcs fcs

fcs fcs


p


SOC SOC


•

•

∂= =∂∂ ∂= − = − ⋅

∂ ∂≤

(4.11)

It looks the same with (3.9) and (4.7), but the third necessary condition is

different for each case, as Hamiltonian here is defined as,

( ) ( ) ( )( ) ( ) ( ) ( ) ( )2

, , ( ) ( ) ( ), ( )fcs h fcs fcs fcsH SOC t P t p t m P t L P t p t F SOC t P t•

= + ⋅+ (4.12)

where, a new term is added compared to the definition (3.6). This new term

makes the optimal trajectory of the FCS net power smooth in order to prolong

the FCS lifetime.

Fig. 4.8, Fig. 4.9, and Fig. 4.10 illustrate the simulation results on the

FTP75 urban driving cycle, NEDC 2000, and Japan 1015 driving cycle,

respectively. These figures indicate that the optimal trajectory of the FCS net

power becomes smooth through the reformulation of the PMP-based optimal

control problem introduced in this subsection. The power changing rate of the

FCS also becomes smaller for the case of reformulation. The initial battery

SOC and the final SOC are both 0.6 here.

63

Fig. 4.8 Optimal trajectories when the FCS lifetime is considered and is not considered on the

FTP75 urban driving cycle

0 200 400 600 800 1000 1200 14000

50

100

Veh

icle

sp

eed

(km

/h)

0 200 400 600 800 1000 1200 14000

20

40F

CS

net

po

wer

(kW

)

0 200 400 600 800 1000 1200 1400-50

0

50

FC

S p

ow

er c

han

ge

rate

(kW

/s)

0 200 400 600 800 1000 1200 1400-50

0

50

Bat

tery

pow

er (

kW)

0 200 400 600 800 1000 1200 14000.4

0.6

0.8

Time (s)

Bat

tery

SO

C

FCS lifetime is not consideredConsidered




64


NEDC 2000

0 200 400 600 800 1000 12000

50

100

150

Veh

icle

spe

ed (

km/h

)

0 200 400 600 800 1000 12000

20

40

60F

CS

net

pow

er (

kW)

0 200 400 600 800 1000 1200-40

-20

0

20

FC

S p

ower

cha

nge

rate

(kW

/s)

0 200 400 600 800 1000 1200-50

0

50

Bat

tery

pow

er (

kW)

0 200 400 600 800 1000 12000.4

0.6

0.8

Bat

tery

SO

C

Time (s)





65


Japan 1015 driving cycle

Table 4.2 shows comparisons of the simulation results illustrated in the

three figures. It can be observed that the mean power changing rate of the

FCS is reduced through the reformulation. Thus, the FCS lifetime can be

increased. However, the reformulation increases the fuel consumption instead

of prolonging the FCS lifetime.

0 100 200 300 400 500 600 7000

50

100

Veh

icle

spe

ed (

km/h

)

0 100 200 300 400 500 600 7000

10

20

30F

CS

net

pow

er (

kW)

0 100 200 300 400 500 600 700-20

-10

0

10

FC

S p

ower

cha

nge

rate

(kW

/s)

0 100 200 300 400 500 600 700-20

0

20

Bat

tery

pow

er (

kW)

0 100 200 300 400 500 600 7000.55

0.6

0.65

Time (s)

Bat

tery

SO

C





66

Table 4.2 Comparison of the PMP-based power management strategies

Driving cycle

Fuel consumption (kg/100 km) Mean power changing rate of

FCS (kW/s)

Basic PMP-

based strategy

PMP-based

strategy

considering FCS

lifetime

Basic PMP-

based strategy

PMP-based

strategy

considering FCS

lifetime

FTP75 urban 1.154 1.204 2.974 1.009

NEDC 2000 1.282 1.310 2.396 0.696

Japan 1015 1.137 1.170 1.681 0.841

Fig. 4.11 illustrates the optimal lines of the PMP-based power management

strategies introduced in chapter 3 and here over the FTP75 urban driving cycle.

It can be observed that the optimal line moves up for the strategy introduced

here. The gap between the two optimal lines indicates the amount of fuel

usage attributed to the new cost function L . Fig. 4.12 and Fig. 4.13 show the

comparisons of optimal lines for the NEDC 2000 and the Japan 1015 driving

cycle, respectively. Table 4.2, Fig. 4.11, Fig. 4.12, and Fig. 4.13 indicate that

there is a tradeoff between the FCS lifetime and the fuel economy. However,

prolonging the FCS lifetime will be significant from an economic viewpoint

even though there is a fuel consumption loss because the price of the fuel cell

stack is very high currently.

67

Fig. 4.11 Optimal lines for the basic case and for the case where the FCS lifetime is considered

on the FTP75 urban driving cycle


on the NEDC 2000


on the Japan 1015 driving cycle

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

Final battery SOC

Fuel

consu

mptio

n (g)

Basic optimal controlFCS lifetime is considered

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8120

130

140

150

160

170

Final battery SOC

Fuel

consu

mptio

n (g)


0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.830

35

40

45

50

55

60

65

70

Final battery SOC

Fuel

consu

mptio

n (g)


68

Table 4.3 shows the influence of the tuning parameter c on the fuel

consumption and on the mean power changing rate of the FCS. The FTP75

urban driving cycle is used here. It can be seen that a higher value of c

results in higher fuel consumption and lower mean power changing rate of the

FCS.

Table 4.3 Influence of the tuning parameter

Tuning parameter

Fuel consumption

(kg/100 km)

Mean power

changing rate of FCS

(kW/s)

0.001c = 1.187 1.307

0.002c = 1.204 1.009

0.003c = 1.216 0.836

Equation (3.11) is still true in this subsection under the assumption that the

battery OCV and internal resistance are not dependent on the battery SOC, as

the new cost function L is only dependent on the FCS net power and there is

no other state variable except the battery SOC. Thus, the PMP-based power

management strategy introduced here can serve as a global optimal solution

when the FCS lifetime is taken into account. Fig. 4.14 illustrates the

comparison between the DP approach and the PMP-based power management

strategy introduced in this subsection. The NEDC 2000 is used here. The

optimal trajectories nearly overlap each other and the fuel consumption

discrepancy is within 0.6%. It can be concluded that the PMP-based power

management strategy introduced here which considers the FCS lifetime can

69

guarantee the global optimality. In the meantime, the PMP-based power

management strategy can save much time compared to the DP approach.

(a)

(b) (c)

Fig. 4.14 Comparison between DP approach and PMP-based power management strategy when

the FCS lifetime is considered: (a) FCS net power trajectories, (b) battery power trajectories, (c)

SOC trajectories

4.3 PMP-based power management strategy considering

battery thermal management

A proton-exchange-membrane (PEM) fuel cell is usually used in vehicular

applications, and the temperature of the PEM fuel cell is controlled properly

within a certain small range by its thermal management system when it is

operating. However, the battery temperature increases during its operation.

0 200 400 600 800 1000 1200-10

0

10

20

30

40

50

Time (s)

FC

S n

et p

ow

er (

kW)

DPPMP

0 200 400 600 800 1000 1200-40

-30

-20

-10

0

10

20

30

Time (s)

Bat

tery

po

wer

(kW

)

DPPMP

0 200 400 600 800 1000 12000.5

0.55

0.6

0.65

0.7

0.75

Time (s)

Bat

tery

SO

C

DPPMP

70

This affects the total fuel consumption, as the battery temperature is related to

the battery efficiency and is further related to the efficiency of the entire

vehicle system. In chapter 3, the battery temperature is assumed to be the same

all the time. However, the battery temperature changes according to equation

(2.15) during its operation, and a good battery thermal management system is

necessary for better fuel economy. In this subsection, the influence of battery

thermal management on the fuel consumption is considered by designating the

battery temperature as a second-state variable of the optimal control problem.

The second battery in Table 2.5 is used in this subsection. Fig. 4.15 illustrates

the OCV, charging internal resistance, and discharging internal resistance of

the battery used here. These data are sourced from an automotive simulation

program called Autonomie.

(a)

(b) (c)

Fig. 4.15 Battery characteristics: (a) OCV, (b) charging resistance, (c) discharging resistance

0

0.5

1

0

20

40

60220

240

260

280

300

Battery SOCBattery temperature (°C)

OC

V (

V)

0

0.5

1

020

4060

0

0.5

1

1.5

2


Ch

arg

ing

re

sist

an

ce (

Oh

m)

0

0.5

1

020

40

60

0

1

2

3

4

5

6


Dis

cha

rgin

g r

esi

sta

nce

(O

hm

)

71

4.3.1 PMP-based power management strategy without considering battery thermal management

In subsection 3.1, we introduced the basic formulation of the PMP-based

optimal control problem for FCHVs, where we assumed that the battery

temperature is always the same. This assumption does not apply here. The

battery temperature changes according to equation (2.15). However, battery

thermal management is not taken into account in this subsection in order to

evaluate its effects on the total fuel consumption in the next subsection.

In this optimal control problem, the battery temperature is specified as a

second-state variable. The state equations, which describe the dynamics of the

state variables, are as follows:

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

, ,

, ,

fcs

fcs

SOC t F SOC t T t P t

T t f SOC t T t P t

=

=

ɺ

ɺ (4.13)

Here, the FCS net power fcsP is the control variable and the battery SOC and

battery temperature are the two state variables. The first state equation is

sourced from equation (2.14), and the second one is sourced from equation

(2.15). The performance measure to be minimized is the total fuel

consumption. Considering the first state equation in (4.13), which is a

constraint on the optimal control problem, the performance measure is as

follows:

( ) ( ) ( )( )2

0

( ) ( ) ( ) ( ), , ( ) ( )ft

fcs h fcs fcstJ P t m P t p t F SOC t T t P t SOC t dt

• • = + ⋅ −

∫ (4.14)

72

The necessary conditions that minimize the performance measure are then as

follows,

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

* * * *

* * * *

* * * * *

, , ,

, , ,

, , , , , ,

fcs

fcs

fcs fcs

HSOC t SOC t T t P t p t

p

Hp t SOC t T t P t p t

SOC

H SOC t T t P t p t H SOC t T t P t p t

•

•

∂=∂∂= −

∂≤

(4.15)

when Hamiltonian is defined as follows:

( ) ( ) ( )( ) ( )( ) ( ) ( )( )2, , ( ), ( ), , ( )hfcs fcs fcsH SOC t T t P t p t m P t p t F SOC t T t P t•

= + ⋅ (4.16)

As stated before, the battery temperature is not considered in the fuel

consumption optimization in this formulation. Thus the second state equation

in (4.13) is not included in the performance measure and in the Hamiltonian,

and equation (4.15) does not give a necessary condition with respect to the

battery temperature.

4.3.2 PMP-based power management strategy considering battery thermal management

In subsection 4.3.1, the battery temperature is not related to the

optimization of the fuel consumption. The battery temperature, however,

affects the total fuel consumption, as it influences the battery efficiency and

further influences the overall efficiency of the vehicle system. Thus, a battery

thermal management system is needed. In this subsection, the PMP-based

optimal control problem introduced in 4.3.1 is reformulated in order to

evaluate the influence of battery thermal management on the total fuel

consumption.

73

The two state equations are identical to those in (4.13). Because the two

state variables are all taken into account in the fuel consumption optimization,

two costates are needed. By introducing the two costates 1p and 2p , the

performance measure to be minimized can be expressed as follows:

( ) { ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )( ) ( )( ) }

20

1

2

( ) ( ) ( ) , ,

( ) , ,

ft

fcs h fcs fcst

fcs

J P t m P t p t F SOC t T t P t SOC t

p t f SOC t T t P t T t dt

•

= + ⋅ −

+ ⋅ −

∫ ɺ

ɺ

(4.17)

The necessary conditions that minimize the performance measure are then as

follows,

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

* * * * * *1 2

1

* * * * * *1 2

* * * * * *1 1 2

* * * * * *2 1 2

* * * * * * * *1 2 1

, , , ,

, , , ,2

, , , ,

, , , ,

, , , , , , , ,

fcs

fcs

fcs

fcs

fcs fcs

HSOC t SOC t T t P t p t p t

p

HT t SOC t T t P t p t p t

p

Hp t SOC t T t P t p t p t

SOCH

p t SOC t T t P t p t p tT

H SOC t T t P t p t p t H SOC t T t P t p t

•

•

•

•

∂=∂

∂=∂

∂= −∂∂= −∂

≤ ( )( )*2p t

(4.18)

when Hamiltonian H is defined as

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )( )

21 2 1

2

, , ( ), , , , ( )

, , ( )

fcs h fcs fcs

fcs

H SOC t T t P t p t p t m P t p t F SOC t T t P t

p t f SOC t T t P t

= + ⋅

+ ⋅

ɺ

(4.19)

Fig. 4.16, Fig. 4.17, and Fig. 4.18 illustrate the simulation results on the

FTP75 urban driving cycle, NEDC 2000, and Japan 1015 driving cycle,

respectively. Here, the initial battery SOC is set to 0.6, and the initial battery

temperature is 25 °C. The result points on each figure are derived from

different values of 1p and 2p . It can be observed that these result points

form a surface in each figure and that each surface can be approximated by

74

two half-planes with similar gradients and which are separated at the battery

SOC of 0.6.

Fig. 4.16 Optimal surface on the FTP75 urban driving cycle when the initial battery SOC is 0.6

and the initial battery temperature is 25 °C

Fig. 4.17 Optimal surface on the NEDC 2000 when the initial battery SOC is 0.6 and the initial

battery temperature is 25 °C

0.4

0.6

0.8

2830

3234

100

120

140

160

180

Final battery SOCFinal battery temperature (°C)

Fu

el c

on

sum

ptio

n (

g)

0.4

0.5

0.6

0.7

0.8

26

28

30

32100

120

140

160

180


Fu

el c

on

sum

ptio

n (

g)

75

Fig. 4.18 Optimal surface on the Japan 1015 driving cycle when the initial battery SOC is 0.6


The three surfaces represent the relationship among the final battery SOC,

the final battery temperature, and the total fuel consumption for the three

driving cycles. Here, the three surfaces are defined as optimal surfaces, as

they are obtained from the extended PMP-based power management strategy.

Any other result points derived from other power management strategies will

always locate above these surfaces on the three driving cycles. It can be seen

that the final battery SOC and final battery temperature simultaneously

influence the total fuel consumption. A higher final battery SOC leads to more

fuel consumption, while a higher final battery temperature results in less fuel

consumption. The shape and location of an optimal surface depend on the

driving cycle and the initial battery conditions as well. Fig. 4.19 illustrates an

optimal surface, which is obtained when the initial battery SOC is 0.6, the

initial battery temperature is 5 °C, and the FTP75 urban driving cycle is

0.4

0.6

0.8

2627

282920

30

40

50

60

70

80


Fu

el c

on

sum

ptio

n (

g)

76

selected. It can be seen that the gradients of the optimal surface and its

location on the three axes are different from those in Fig. 4.16.

Fig. 4.19 Optimal surface on the FTP75 urban driving cycle when the initial battery SOC is 0.6


The optimal trajectories are affected by the battery temperature when the

battery thermal management is taken into account. Fig. 4.20 shows the

optimal trajectories of the battery SOC, the battery temperature, the battery

power, and the FCS net power for different initial battery temperature

conditions when the initial battery SOC and the final battery SOC are both 0.6.

The FTP75 urban driving cycle is used here. This figure shows that the

trajectories of each item are different for each condition. Compared to the

case where the initial battery temperature is 25 °C, the positive battery power

is lower at the beginning and becomes greater along as the battery temperature

increases when the initial battery temperature is 0 °C or 10 °C. In contrast, it

is greater at the beginning and becomes lower gradually due to the increasing

0.4

0.5

0.6

0.7

0.8

141516

1718

19120

130

140

150

160


Fuel c

onsu

mptio

n (g)

77

battery temperature when the initial battery temperature is 25 °C compared to

the other cases.

(a) (b)

(c) (d)

Fig. 4.20 Optimal trajectories for different initial battery temperature conditions: (a) battery

SOC trajectories, (b) battery temperature trajectories, (c) battery power trajectories, (d) FCS net

power trajectories

Fig. 4.21 illustrates the simulation results of the battery SOC, the battery

temperature, the battery power, and the FCS net power over the FTP75 urban

driving cycle both with battery thermal management introduced here and

without thermal management. It is evident that the trajectories of each item

are different for each case. The battery discharges more and charges more in

the case without battery thermal management, causing the battery temperature

increase more in this case. Fig. 4.22 and Fig. 4.23 illustrate the simulation

results on the NEDC 2000 and the Japan 1015 driving cycle, respectively.

0 200 400 600 800 1000 1200 14000.5

0.55

0.6

0.65

0.7

Time (s)

Bat

tery

SO

C

T-initial=0°CT-initial=10°CT-initial=25°C

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

35

Time (s)

Batte

ry te

mper

ature

(°C

)


0 200 400 600 800 1000 1200 1400-25

-20

-15

-10

-5

0

5

10

15

Time (s)

Bat

tery

pow

er (k

W)


0 200 400 600 800 1000 1200 14000

10

20

30

40

50

Time (s)

FC

S n

et p

ow

er (kW

)


78

(a) (b)

(c) (d)

Fig. 4.21 Effects of battery thermal management on the optimal trajectories on the FTP75 urban

driving cycle: (a) battery SOC, (b) battery temperature, (c) battery power, (d) FCS net power

(a) (b)

(c) (d)

Fig. 4.22 Effects of battery thermal management on the optimal trajectories on the NEDC 2000:

(a) battery SOC, (b) battery temperature, (c) battery power, (d) FCS net power

0 200 400 600 800 1000 1200 14000.5

0.52

0.54

0.56

0.58

0.6

0.62

Time (s)

Bat

tery

SO

C

With thermal managementWithout thermal management

0 200 400 600 800 1000 1200 140024

26

28

30

32

34

Time (s)

Bat

tery

tem

per

atu

re (

°C)


0 200 400 600 800 1000 1200 1400-30

-20

-10

0

10

20

Time (s)

Bat

tery

po

wer

(kW

)


0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

35

40

Time (s)

FC

S n

et p

ow

er (

kW)


0 200 400 600 800 1000 12000.5

0.55

0.6

0.65

0.7

0.75

Time (s)

Bat

tery

SO

C

with thermal managementwithout thermal management

0 200 400 600 800 1000 120025

26

27

28

29

30

31

32

Time (s)

Bat

tery

tem

per

atu

re (

°C)


0 200 400 600 800 1000 1200-30

-20

-10

0

10

20

30

Time (s)

Bat

tery

po

wer

(kW

)


0 200 400 600 800 1000 12000

10

20

30

40

50

Time (s)

FC

S p

ow

er (

kW)


79

(a) (b)

(c) (d)

Fig. 4.23 Effects of battery thermal management on the optimal trajectories on the Japan1015

driving cycle: (a) battery SOC, (b) battery temperature, (c) battery power, (d) FCS net power

Fig. 4.24 illustrates the relationship between the optimal surface acquired in

this subsection and the simulation results of subsection 4.3.1 over the FTP75

urban driving cycle. The asterisk indicates the simulation results of subsection

4.3.1. As stated earlier, the asterisk is located above the optimal surface. This

shows the improvement of the fuel economy when battery thermal

management is carried out. Fig. 4.25 and Fig. 4.26 denote this for the NEDC

2000 and the Japan 1015 driving cycle, respectively.

0 100 200 300 400 500 600 700

0.58

0.59

0.6

0.61

0.62

0.63

0.64

0.65

Time (s)

Bat

tery

SO

C

with thermal mangementwithout thermal management

0 100 200 300 400 500 600 70025

25.5

26

26.5

27

27.5

Time (s)

Bat

tery

tem

per

atu

re (

°C)


0 100 200 300 400 500 600 700-10

-5

0

5

10

Time (s)

Bat

tery

po

wer

(kW

)


0 100 200 300 400 500 600 7000

5

10

15

20

25

30

Time (s)

FC

S p

ow

er (

kW)


80

Fig. 4.24 The effect of the battery thermal management on the fuel economy over the FTP75

urban driving cycle

Fig. 4.25 The effect of the battery thermal management on the fuel economy over the NEDC

2000

Fig. 4.26 The effect of the battery thermal management on the fuel economy over the Japan

1015 driving cycle

0.4

0.6

0.8

2830

3234

100

120

140

160

180


Fu

el c

on

sum

ptio

n (

g)

0.4

0.5

0.6

0.7

0.8

26

28

30

32100

120

140

160

180


Fu

el c

on

sum

ptio

n (

g)

0.4

0.6

0.8

2627

282920

30

40

50

60

70

80


Fu

el c

on

sum

ptio

n (

g)

81

Table 4.4 lists detailed data on the effects of the battery thermal

management on the total fuel consumption. The table indicates that the fuel

economy of the FCHV is improved by 2.33%, 4.77%, and 3.02% with battery

thermal management on the FTP75 urban driving cycle, the NEDC 2000, and

the Japan 1015 driving cycle, respectively. The difference in the final battery

temperature for the two cases is removed by considering the relationship

between the final battery temperature and the total fuel consumption for the

same final battery SOC, which is demonstrated on the optimal surface.

Table 4.4 The effects of the battery thermal management on the total fuel consumption

Driving cycle

Fuel consumption (kg/100 km)

Discrepancy (%) Without battery

thermal management

With battery thermal

management

FTP75 urban 1.142 1.116 2.33

NEDC 2000 1.273 1.215 4.77

Japan 1015 1.127 1.094 3.02

4.3.3 Global optimality of the two-state variable PMP-based power management strategy

Based on the previous research [8], the PMP-based power management

strategy introduced in subsection 4.3.2 can guarantee the global optimality if

the two state variables are not coupled to each other and the two costates can

be replaced with two constants. The two state variables, however, are related

to each other by the third and the fourth necessary conditions in (4.18). Thus,

82

a new discussion regarding this case is needed. In the research [52], it was

proved mathematically that the necessary conditions produced by PMP are

sufficient for the global optimal solution under two assumptions: 1) the state

equation is concave and 2) the optimal costate is always negative. This

conclusion is obtained for the one-state variable case for the engine/battery

powered hybrid vehicles. However, we can expand this conclusion to our

problem here. Four assumptions that can guarantee the global optimal solution

in the two-state variable PMP-based optimal control problem are as follows: 1)

the state equation F is concave 2) the costate 1p is always negative 3) the

state equation f is convex 4) the costate 2p is always positive. The first

assumption indicates that the state equation F is concave for the battery

SOC, the battery temperature, and the FCS net power at the same time.

Similarly, the third assumption indicates that the state equation f is convex

for the three parameters simultaneously. We assessed the shape of the two

state equations for each two parameters out of the three parameters based on

the battery characteristics illustrated in Fig. 4.15. Fig. 4.27 illustrates the

images of the state equations F and f . We can see that the concavity and

convexity of the state equations basically satisfy.

83

(a) (b)

(c) (d)

(e) (f)

Fig. 4.27 Concavity and convexity of the state equations based on the characteristics of the

battery used in this dissertation: (a) state equation F versus battery SOC and battery

temperature for different battery power, (b) state equation F versus battery temperature and

battery power for different battery SOC, (c) state equation F versus battery SOC and battery

power for different battery temperature, (d) state equation f versus battery temperature and

battery SOC for different battery power, (e) state equation f versus battery temperature and

battery power for different battery SOC, (f) state equation f versus battery SOC and battery

power for different battery temperature

010

2030

40

0.4

0.5

0.60.7

0.8-5

-4

-3

-2

-1

x 10-3

Battery temperature (°C)Battery SOC

Tim

e d

eriv

ativ

e o

f SO

C (

1/s

)

Pbat=20kW

Pbat=10kW

-40-20

020

40

10

20

30

40-15

-10

-5

0

5

x 10-3

Battery power (kW)Battery temperature (°C)

Tim

e d

iriva

tive

of

SO

C (

1/s

)

SOC=0.8

SOC=0.5

-40-20

020

40

0.4

0.5

0.6

0.7

0.8

-15

-10

-5

0

5

x 10-3

Battery power (kW)Battery SOC

Tim

e d

eriv

ativ

e o

f S

OC

(1

/s)

T=36 °C

T=20 °C

10 15 20 25 30 35 40 45 0.4

0.6

0.8

0

0.5

1

1.5

2

Battery SOC

Battery temperature (°C)

Tim

e d

eriv

ativ

e o

f te

mp

era

ture

(°C

/s)

Pbat=20kW

Pbat=25kW

-40-20

020

40

1020

30400

0.5

1

1.5

2

2.5

3

3.5

Battery power (kW)Battery temperature (°C)

Tim

e d

eriv

ativ

e o

f te

mp

era

ture

(°C

/s)

SOC=0.5

SOC=0.8

-40-20

020

40

0.40.5

0.60.7

0.80

0.5

1

1.5

2

Battery power (kW)Battery SOC

Tim

e d

eriv

ativ

e o

f te

mp

era

ture

(°C

/s)

T=36 °C

T=20 °C

84

Fig. 4.28 illustrates the simulation results derived from DP approach for the

two-state variable case. A short-time driving cycle is used here because of the

long calculation time, which is illustrated in Fig. 4.29.

(a)

(b) (c)

Fig. 4.28 Simulation results derived from DP approach for the two-state variable case

(driving cycle 1): (a) simulation results of the two state variables, (b) simulation result of the

battery SOC, and (c) simulation result of the battery temperature

Fig. 4.29 Driving cycle 1 used in comparison between PMP-based strategy and DP approach

0

50

100

150

0.59

0.592

0.594

0.596

0.598

0.6

27.8

27.9

28

Time (s)Battery SOC

Batte

ry tem

pera

ture

(C

els

ius

degre

e)

0 20 40 60 80 100 120 140 160 1800.59

0.591

0.592

0.593

0.594

0.595

0.596

0.597

0.598

0.599

0.6

Time (s)

Batte

ry S

OC

0 20 40 60 80 100 120 140 160 18027.75

27.8

27.85

27.9

27.95

28

28.05

Time (s)

Batte

ry te

mpera

ture

(C

els

ius

degre

e)

0 50 100 150 2000

5

10

15

20

25

30

35

40

Time (s)

Sp

eed

(km

/h)

85

According to above four assumptions that can guarantee the global optimal

solution in the two-state variable PMP-based optimal control problem, the

battery characteristics illustrated in Fig. 4.27 will make few differences

between the optimal solution derived from the PMP-based strategy introduced

in subsection 4.3.2 and the global optimal solution derived from DP approach.

Fig. 4.30 shows the comparison between DP approach and the PMP-based

power management strategy. We can see that the optimal trajectories overlap

each other most of the time. Simulation result shows that the fuel

consumption discrepancy is within 1.5%. The difference will be disappeared

if a battery, which perfectly satisfies the four assumptions above, is used.

(a) (b)

(c) (d)

Fig. 4.30 Simulation results comparison between PMP-based strategy and DP approach

(driving cycle 1): (a) battery SOC, (b) battery temperature, (c) battery power, and (d) FCS net

power

0 50 100 1500.59

0.595

0.6

0.605

Time (s)

Bat

tery

SO

C

DPPMP

0 50 100 15027.75

27.8

27.85

27.9

27.95

28

28.05

28.1

28.15

Time (s)

Bat

tery

tem

per

atu

re (

Cel

siu

s d

egre

e)

DPPMP

0 20 40 60 80 100 120 140 160 180-6

-4

-2

0

2

4

Time (s)

Bat

tery

po

wer

(kW

)

DPPMP

0 20 40 60 80 100 120 140 160 1800

5

10

15

20

Time (s)

FC

S n

et p

ow

er (

kW)

DPPMP

86

Here, we can conclude that the two-state variable PMP-based power

management strategy introduced in subsection 4.3.2 can guarantee the global

optimality if the battery satisfies the four assumptions above. In the meantime,

the PMP-based strategy can save much time compared to DP approach. The

time saving effect is outstanding in the two-state variable case. For the driving

cycle illustrated in Fig. 4.29, the elapsed time is 13 hours 55 minutes 1 second

for DP approach and is just 36 seconds for the PMP-based strategy. The

battery SOC range and battery temperature range are relatively narrow when

the driving cycle illustrated in Fig. 4.29 is used. Thus, the elapsed time for DP

approach is also relatively short in this case.

Fig. 4.31 shows the simulation results derived from DP approach when the

driving cycle illustrated in Fig. 4.32 is used. Fig. 4.33 shows the comparison

between DP approach and the PMP-based power management strategy for this

driving cycle. The battery SOC rang and battery temperature range are

relatively wide for this driving cycle, thus the time saving effect is more

outstanding in this case compared to the driving cycle illustrated in Fig. 4.29.

The elapsed time is 166 hours 48 minutes 21 seconds for DP approach and is

just 37 seconds for the PMP-based strategy when the driving cycle illustrated

in Fig. 4.32 is used. From the comparisons of the two driving cycles, we can

see that the time saving effect of the PMP-based power management strategy

is more outstanding when the battery SOC range and battery temperature

range, and the driving cycle time increase.

87

(a)

(b) (c)

Fig. 4.31 Simulation results derived from DP approach for the two-state variable case

(driving cycle 2): (a) simulation results of the two state variables, (b) simulation result of the

battery SOC, and (c) simulation result of the battery temperature

Fig. 4.32 Driving cycle 2 used in comparison between PMP-based strategy and DP approach

050

100150

200

0.5750.58

0.5850.59

0.595

27.8

28

28.2

28.4

Time (s)Battery SOC

Ba

ttery

tem

pe

ratu

re (

Ce

lsiu

s d

eg

ree

)

0 50 100 150 2000.575

0.58

0.585

0.59

0.595

0.6

Time (s)

Ba

ttery

SO

C

0 50 100 150 200

27.8

27.9

28

28.1

28.2

28.3

28.4

28.5

Time (s)

Ba

ttery

te

mp

era

ture

(C

els

ius

de

gre

e)

0 50 100 150 200 2500

10

20

30

40

50

60

70

Time (s)

Sp

eed

(km

/h)

88

(a) (b)

(c) (d)

Fig. 4.33 Simulation results comparison between PMP-based strategy and DP approach

(driving cycle 2): (a) battery SOC, (b) battery temperature, (c) battery power, and (d) FCS net

power

4.3.4 Control parameters of the PMP-based power management strategy

Fig. 4.34 illustrates the fuel consumption rate 2hm

•

, time derivative of the

battery SOC SOC•

, time derivative of the battery temperature T•

, and the

Hamiltonian H for the whole range of FCS net power at the calculation

time step when the power required for the motor reqP is 35 kW, the battery

SOC is 0.6, the battery temperature is 30 °C, and the costate 1p and 2p are

set to -90 and 1. The convexity of the Hamiltonian demonstrates that the

0 50 100 150 200

0.58

0.59

0.6

0.61

0.62

Time (s)

Bat

tery

SO

C

DPPMP

0 50 100 150 20027.6

27.8

28

28.2

28.4

28.6

Time (s)

Bat

tery

tem

per

atu

re (

Cel

siu

s d

egre

e)

DPPMP

0 50 100 150 200-10

-8

-6

-4

-2

0

2

4

Time (s)

Bat

tery

po

wer

(kW

)

DPPMP

0 50 100 150 200 2500

5

10

15

20

25

Time (s)

FC

S n

et p

ow

er (

kW)

DPPMP

89

optimal fcsP , which minimizes the Hamiltonian, can be determined for this

calculation time step. The shape of the Hamiltonian depends on the two

costates at each calculation time step, and consequently the optimal fcsP is

also dependent on the two costaes. Therefore, the two costates are called

control parameters.


•, time derivative of the battery SOC SOC

•, time

derivative of the battery temperature T•

, and Hamiltonian H for the whole range of the FCS

net power

0 10 20 30 40 50 600

0.5

1

1.5

2

Fue

l co

nsum

ptio

n r

ate

(g/s

)

0 10 20 30 40 50 60-10

-5

0

5x 10

-3

Tim

e d

eriv

ativ

e of

SO

C (

1/s

)

0 10 20 30 40 50 60-0.2

0

0.2

0.4

0.6

Tim

e d

eriv

ativ

e of

tem

pera

ture

(°C

/s)

0 10 20 30 40 50 600.5

1

1.5

Ham

ilto

nian

(g

/s)

FCS net power (kW)

90

Costate 1p reflects the relationship between the battery SOC and the fuel

consumption, whose unit is g , while costate 2p reflects the relationship

between the battery temperature and the fuel consumption, whose unit is

/g C° . The two costates are determined by the third and the fourth necessary

conditions in (4.18) when their initial values are given. This indicates that the

two costates are related to each other as follows:

* ** * *

1 1 2

* ** * *

2 1 2

F fp p p

SOC SOC

F fp p p

T T

•

•

∂ ∂= − ⋅ − ⋅∂ ∂∂ ∂= − ⋅ − ⋅∂ ∂

(4.20)

Fig. 4.35 illustrates the simulation results of 1p and 2p for different initial

values of 1p and 2p over the FTP75 urban driving cycle. Here, the initial

battery SOC and the final SOC are each 0.6. The initial values of 1p and 2p

should be properly selected based on the physical meanings of the costates

and based on the final battery SOC constraint.

91

(a) (b)

(c) (d)

(e) (f)

Fig. 4.35 Simulation results of two costates for different initial values of them over the FTP75

urban driving cycle

From (4.19), it is clear that the two costates simultaneously influence the

Hamiltonian. Thus, the simulation results depend on both costates. In the

previous subsection 3.2, the influences of the costate value on the final value

of the state variable and on the total fuel consumption were discussed for the

basic PMP-based power management strategy. For the extended PMP-based

strategy presented in subsection 4.3.2, the influences of the costates can be

explained similarly. Greater 1p and 2p values cause a lower final battery

SOC value and a lower final battery temperature value, respectively. Fig. 4.36

0 200 400 600 800 1000 1200 1400-110

-105

-100

Time (s)

p1

(g)

0 200 400 600 800 1000 1200 14001

1.01

1.02

p2

(g/°C

)

p1p2

0 200 400 600 800 1000 1200 1400-105

-100

-95

-90

Time (s)

p1

(g)

0 200 400 600 800 1000 1200 14002

2.01

2.02

2.03

p2

(g/°C

)

p1p2

0 200 400 600 800 1000 1200 1400-100

-90

-80

Time (s)

p1

(g)

0 200 400 600 800 1000 1200 14003

3.02

3.04

p2

(g/°C

)

p1p2

0 200 400 600 800 1000 1200 1400-100

-90

-80

-70

-60

Time (s)p

1 (

g)

0 200 400 600 800 1000 1200 14004

4.01

4.02

4.03

4.04

p2

(g/

°C)

p2p1

0 200 400 600 800 1000 1200 1400-90

-80

-70

-60

Time (s)

p1

(g)

0 200 400 600 800 1000 1200 14005

5.02

5.04

5.06

p2

(g/°C

)

p2p1

0 200 400 600 800 1000 1200 1400-100

-90

-80

-70

-60

-50

Time (s)

p1

(g)

0 200 400 600 800 1000 1200 14006

6.01

6.02

6.03

6.04

6.05

p2

(g/°C

)

p2p1

92

illustrates the relationship between the initial value of 1p and the final

battery SOC and the relationship between the initial value of 2p and the

final battery temperature over three typical driving cycles. The initial values

of 2p are identical for each line in (a), (b), and (c), and the initial values of

1p are identical for each line in (d), (e), and (f).

(a) (b)

(c) (d)

(e) (f)

Fig. 4.36 Relationship between initial value of 1p and final battery SOC and relationship

between initial value of 2p and final battery temperature over three typical driving cycles: (a)

final battery SOC versus initial value of 1p on the FTP75 urban driving cycle, (b) final

-110 -100 -90 -80 -70 -600.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Initial value of p1 (g)

Fin

al b

atte

ry S

OC

p2 initial=1 g/°Cp2 initial=2 g/°Cp3 initial=3 g/°C

-125 -120 -115 -110 -105 -100 -95 -90 -850.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8


Fin

al b

atte

ry S

OC


-140 -130 -120 -110 -100 -90 -80 -70 -600.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8


Fin

al b

atte

ry S

OC


0 2 4 6 8 10 1229.5

30

30.5

31

31.5

32

32.5

33

Initial value of p2 (g/°C)

Fin

al b

atte

ry t

emp

erat

ure

(°C

)

p1 initial=-70 gp1 initial=-100 g

0 5 10 15 2027.5

28

28.5

29

29.5


Fin

al b

atte

ry t

emp

erat

ure

(°C

)

p1 initial=-70 gp1 initial=-100 g

0 2 4 6 8 1026

26.5

27

27.5

28

28.5

29


Fin

al b

atte

ry t

emp

erat

ure

(°C

)

p1 initial=-70 gp1 initial=-100 gp1 initial=-130 g

93

battery SOC versus initial value of 1p on the NEDC 2000, (c) final battery SOC versus initial

value of 1p on the Japan 1015 driving cycle, (d) final battery temperature versus initial vale

of 2p on the FTP75 urban driving cycle, (e) final battery temperature versus initial vale of

2p on the NEDC 2000, (f) final battery temperature versus initial vale of 2p on the Japan

1015 driving cycle

4.4 Discussions on the combined case

Previously, we introduced the PMP-based power management strategy

considering three important factors which are the battery SOC constraint, the

FCS lifetime, and the effect of the battery thermal management on the fuel

economy. These three factors are separately considered in the previous three

subsections, and now the case in which the three factors are all taken into

account is discussed.

In this case, there are two state variables and three cost functions. Two state

equations are the same with (4.13). The Hamiltonian in this case can be

defined as follows:

( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )

21 2

1 2

, ( ), ( ), , ( ) ( )

, , ( ) , , ( )

fcs h fcs fcs fcs

fcs fcs

H SOC t T t P t p t p t A m P t B S P t C L P t

p t F SOC t T t P t p t f SOC t T t P t

•= ⋅ + ⋅ + ⋅

+ ⋅ + ⋅(4.21)

Here, coefficients A, B, and C are weighting factors which have the following

relationship:

1A B C+ + = (4.22)

94

The value of each coefficient is determined based on the main goal of the

control. For example, C is greater than A and B if the FCS lifetime is the main

factor to be considered.

The necessary conditions that obtain the optimal solution here are as

follows:

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

* * * * * *1 2

1

* * * * * *1 2

* * * * * *1 1 2

* * * * * *2 1 2

* * * * * * * *1 2 1

, , , ,

, , , ,2

, , , ,

, , , ,

, , , , , , , ,

fcs

fcs

fcs

fcs

fcs fcs

HSOC t SOC t T t P t p t p t

p

HT t SOC t T t P t p t p t

p

Hp t SOC t T t P t p t p t

SOCH

p t SOC t T t P t p t p tT

H SOC t T t P t p t p t H SOC t T t P t p t

•

•

•

•

∂=∂

∂=∂

∂= −∂∂= −∂

≤ ( )( )*2p t

(4.23)

This looks identical to (4.18), but the optimal trajectories derived here will be

different from previous three cases because of the added cost functions.

95

Chapter 5 Concluding remarks

5.1 Conclusion

A PMP-based power management strategy for FCHVs is presented and

mathematically extended by considering three important factors which are

limitations on the battery SOC usage, FCS lifetime, and effects of battery

thermal management on the fuel economy. These extensions are useful for

realization of the PMP-based strategy, as these extensions are closer to the

reality. Global optimality is discussed for the three extended cases and

simulation time consumed in the PMP-based strategy and DP approach is

compared for two-state variable cases. The following points are drawn from

this dissertation.

(1) In order to overcome drawbacks of the existing method, which

considers the battery SOC boundary while minimizing the fuel

consumption but makes the battery SOC trajectory fluctuating, we

introduce a new cost function to the basic PMP-based optimal control

problem to consider the battery SOC constraint factor. Simulation

results illustrate that there is no fluctuation in the optimal battery SOC

trajectory after the reformulation and the tendency of the battery SOC

trajectory for the reformulation is similar to that for the case where the

battery SOC constraint is not considered. Simulation results also show

96

that the fuel economy is worse when considering the limitations of the

battery SOC usage in the PMP-based power management strategy.

(2) A second cost function is defined and introduced to the basic PMP-

based optimal control problem in order to take into account the FCS

lifetime while optimizing the fuel economy of the FCHV. Simulation

results illustrate that the optimal trajectory of the FCS net power

becomes smooth through the reformulation. The power changing rate

of the FCS is also smaller for the case of reformulation compared to the

basic PMP-based power management strategy. However, there is a

tradeoff between the FCS lifetime and the fuel economy. The optimal

line moves up when the FCS lifetime is considered and the gap

between the original optimal and the new optimal lines indicates the

amount of fuel usage attributed to the new cost function. Prolonging the

FCS lifetime will be significant because of the high-priced FCS, even

though there is a tradeoff between the FCS lifetime and the fuel

economy.

(3) The effect of battery thermal management on the fuel economy is

evaluated by designating the battery temperature as a second state

variable other than the battery SOC and adding it mathematically to the

PMP-based optimal control problem. An optimal surface is defined for

this reformulation which represents the relationship among the final

battery SOC, the final battery temperature, and the total fuel

97

consumption. For a driving cycle, any other simulation result points

derived from other power management strategies will always locate

above the optimal surface, as the optimal surface is obtained from the

extended PMP-based power management strategy. The optimal surface

indicates that the final battery SOC and final battery temperature

simultaneously influence the total fuel consumption. A higher final

battery SOC causes more fuel consumption, and a higher final battery

temperature results in less fuel consumption. The battery discharges

more and charges more in the case without battery thermal

management. From the relationship between the optimal surface and

the simulation results derived when the battery thermal management is

not taken into account, it could be concluded that the fuel economy of

the FCHV is improved by 2.33%, 4.77%, and 3.02% with the battery

thermal management on the FTP75 urban driving cycle, NEDC 2000,

and Japan 1015 driving cycle, respectively.

(4) The PMP-based power management strategy which considers the

battery SOC constraint guarantees the global optimality under the

assumption that the battery OCV and internal resistance are not

dependent on the battery SOC. The PMP-based power management

strategy which takes into account the FCS lifetime also guarantees the

global optimality under the same assumption. The PMP-based power

management strategy which considers the effect of battery thermal

98

management on the fuel economy guarantees the global optimality

under the assumption that the two state equations satisfy convexity and

concavity respectively, and the two costates are always negative and

always positive respectively. Simulation results derived from the PMP-

based strategy and DP approach show consistency for the three cases.

The elapsed time of the PMP-based power management strategy and

DP approach is compared for the two-state variable case in order to

prove the effectiveness of the PMP-based strategy. The comparison

result shows that the time which the DP approach consumed is way

longer than the PMP-based strategy’s while the two strategies obtain

the same simulation results. The time saving effect of the PMP-based

power management strategy is more outstanding when the battery SOC

range and the battery temperature range, and the driving cycle time

increase.

5.2 Future work

Future work of this dissertation is to realize the presented PMP-based power

management strategy to real FCHVs. In order to achieve this goal, a pre-

process on the presented strategy is needed to make the strategy applicable to

real vehicles and to establish the real-time control concept. This control will be

a map-based control and it mainly deals with the decision of control

parameters. A pre-experiment also needs to be designed and carried out. This

99

is a complicated process from a technological viewpoint and from an

economic viewpoint, as there are lots of issues regarding the real vehicle

control and the test bench cost is high. However, it will be a big success in the

automotive area if the realization of the presented power management strategy

is completed. It can be started from a reduced-scale of an FCHV powertrain. I

believe that the presented power management strategy will be applied to real

vehicles as a powerful solution in the near future.

100

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107

국문초록

하이브리드 차량의 연비와 성능 향상을 위하여 지난 몇 십년

동안 많은 연구가 활발히 진행되었다. 본 논문에서는 연료전지

하이브리드 차량을 대상으로 폰트리아진의 최적 원리 기반의

동력관리 전략을 제기하였다. 또한 연료전지 하이브리드 차량의 세

가지 중요한 요소를 고려하기 위하여 이 전략을 수학적으로

확장하였다. 이 세 가지 요소들은 각각 배터리 SOC의 사용범위에

대한 구속, 연료전지 시스템의 수명, 배터리의 열관리가 연비에

미치는 영향이다. 제기된 전략은 각 상황에 대하여 컴퓨터

시뮬레이션으로 구현되었다.

배터리 SOC에 대한 구속 문제는 연료소모율과 별도로 새로운

cost function을 폰트리아진의 최적 원리 기반의 최적 제어 문제에

도입하는 것으로 해결하였다. 이 해결책으로 인해 연료소모를

최소화하는 동시에 배터리 SOC에 대한 구속 요구를 만족시킬 수

있다. 연료전지 시스템의 수명을 연료소모 최소화 문제와 함께

고려하기 위하여 새로운 cost function을 정의하여 폰트리아진의

최적 원리 기반의 최적 제어 문제에 추가하였는데 이 새로운 cost

function은 연료전지 시스템의 파워 변화율과 연관되어 있다.

전략의 새로운 변형을 통하여 연료전지 시스템의 수명이 연장될 수

있음을 시뮬레이션 결과에서 확인할 수 있다. 또한 연료전지

시스템의 수명을 고려함으로 인해 연비 면에서는 손실을 본다는

108

것도 확인할 수 있다. 배터리의 열관리가 연비에 미치는 영향은

폰트리아진의 최적 원리 기반의 최적 제어 문제에서 배터리 온도를

배터리 SOC와 별도로 두 번째 상태변수로 지정하는 것을 통하여

고려하였다. 이 상황에서 시뮬레이션을 통하여 배터리 최종 SOC,

배터리 최종 온도, 그리고 전체 연료소모량 사이의 관계를

나타냈는데 이 관계는 하나의 면으로 표현될 수 있으며 이 면은

기울기가 비슷한 두 개의 서로 교차되는 반 평면으로 구성되었음을

확인할 수 있다. 본 논문에서는 이 면을 최적 면이라고 정의하는데

이는 이 면이 폰트리아진의 최적 원리 기반의 동력관리 전략에서

도출되었기 때문이다. 배터리의 열관리로 인한 연비 향상 가능성은

시뮬레이션을 통하여 보여주었는데 연료전지 하이브리드 차량의

연비는 주행싸이클에 따라 최대로 4.77%까지 향상될 수 있다. 본

논문에서는 또한 위에서 제기된 세 가지 요소들을 동시에 고려한

경우에 대하여 토론하였다.

위의 세 가지 확장된 상황에 대하여 본 논문에서는 폰트리아진의

최적 원리 기반의 동력관리 전략의 global optimality에 대해

토론하였다. 또한 이 세 가지 상황에서 폰트리아진의 최적 원리

기반의 동력관리 전략에서 도출된 결과를 다이나믹 프로그래밍에서

도출된 결과와 비교하였다. 폰트리아진의 최적 원리 기반의

동력관리 전략은 배터리의 일정한 가정하에 global optimality를

보장하면서 또한 다이나믹 프로그래밍에 비해 많은 시간을 절약할

수 있는데 그 시간 절약 효과는 특히 상태변수가 두 개인

109

시스템에서 더 뛰어나다.

주요어주요어주요어주요어:::: 연료전지 하이브리드 차량, 폰트리아진의 최적화 원리,

동력관리 전략, 수학적 확장, 시간 절약 효과

학번학번학번학번:::: 2007-31059

110

감사의 글

한국에 온지도 벌써 5 년이란 시간이 흘렀습니다. 처음에는 이

5 년이 언제 다 지날까 싶었는데 막상 졸업하려니 시원섭섭합니다.

졸업을 하며 그 동안 저의 연구와 생활에 도움과 조언을 아끼지

않으셨던 모든 분들께 진심으로 감사를 드립니다. 우선, 5 년 전

낯선 한국 서울대에 와서 연구실을 찾아 방황하고 있는 저를 흔쾌히

받아주셨던 저의 지도교수 차석원 교수님께 깊은 감사의 뜻을

전하고 싶습니다. 그때 재생에너지변환연구실에 들어왔기에 오늘의

제가 있을 수 있지 않았나 싶습니다. 차석원 교수님의 가르침은

저의 인생에 있어 값진 경험이 될 것 같습니다. 그리고 프로젝트를

지도해 주시면서 소중한 가르침을 주셨던 서울산업대학교

박영일교수님과 임원식교수님께도 진심으로 감사를 드립니다. 또한

저의 박사학위논문을 심사해 주시면서 조언을 아끼지 않으셨던

서울대학교 안성훈교수님과 윤병동교수님께도 감사를 드립니다.

논문을 준비하면서 연구실 졸업 선배이신 김남욱 박사님한테서

조언을 많이 구했습니다. 미국에 계심에도 불구하고 매번 이메일로

저의 의문점에 답해주시느라 수고 많으셨습니다. 연구실 저의

하이브리드 팀의 팀원인 대흥이 오빠, 창우 오빠, 호원이, 종렬이,

현섭이, 형균이, 종대에게도 고맙다는 말을 전하고 싶습니다. 매 번

학회도 같이 다니고 하면서 재미있는 일도 많았는데 좋은 추억들로

남아있을 것입니다. 그리고 또 다른 한 팀인 연료전지 팀의

111

팀원들에게도 감사를 드립니다. 무엇보다 저의 부모님한테 감사의

마음을 전하고 싶습니다. 중국 어느 시골 동네에서 저를 낳아

서울대 박사로 키워주시느라 지금까지 참 수고가 많으셨습니다.

앞으로 더 잘 클 것이니 기대하십시오. 그리고 오랜 시간 동안

저한테 정신적 힘을 주신 분들에게도 감사를 드립니다. 모두들

수고하셨습니다.

끝으로 지금까지 저한테 도움을 주신 모든 분들의 행복을 기원하며

이 논문을 저의 부모님께 바칩니다.

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