TOWARDS ELECTRIC
POWER TRAINS
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Contents
Model based engineering and design of electric power trains
4.
Analytic solution to the energy management problem for fuel cell hybrid vehicles
11.
Distribution of the power for traction for fixed gear electric propulsion systems
21.
Rapid control prototyping platform for electric and hybrid vehicle drive lines
32.
A prototype emission free cooling trailer
40.
Design and test of a battery pack simulator
47.
The modelling of the temperature at the poles and core of large prismatic LiFePO4 cells
62.
AUTOMOTIVE
3
Model based engineering and design
of electric power trains
Approach
Bram Veenhuizen
HAN University of Applied Science, Arnhem, The Netherlands
Introduction
Since some decades now, the automotive industry focuses its development efforts on the reduction of
exhaust gas emissions and fuel consumption. Over time remarkable results have been obtained, especially regarding the emissions of soot, NOx and hydrocarbons. Improvements in the combustion process
for diesel and gasoline engines contributed to this success. Still, the efficiency of the total drive train of a
regular vehicle leaves ample room for improvement. In city driving only about 10% of the energy available in the fuel is used for propulsion and power generation; the rest is wasted as heat. This is mainly
caused by low part load engine efficiency, inefficient driving (late up shift), waste of brake energy and
inefficient transmissions.
Since two decades, more and more OEM’s develop hybrids to address these shortcomings. By making
use of the high efficiency, high torque capacity and power density of electric motors, the efficiency of
the drive trains was improved. Nowadays, also the electric vehicle, powered only by means of a battery,
has become popular again.
As a general trend we see a continuous increase in the electrification of the automotive power train.
This is accompanied with a steady increase in the electrification of the auxiliary systems, like the water
pump and parts of the HVAC system. This holds for both light duty and heavy duty vehicles.
Recognizing this trend led HAN Automotive Research (HANAR) to initiate the RAAK-PRO Electric Power
Train (EPT) project. The goal of this project was to answer the central research question: How can we
model the components and subsystems of electric, hybrid and fuel cell vehicles such that the realization process of these vehicle can be accelerated and improved and how can these models assist in the
development of the system control software? The modelling efforts should also pay off for our lecture
material: cases treated in the classroom find their origin in the results of the EPT project.
This booklet gives an overview of the project activities that were undertaken to accomplish these goals.
This chapter gives a management overview of all project activities, whereas the succeeding chapters
give more in-depth detail on various subprojects.
Figure 1. EPT subprojects projected on the V-cycle method for product development
Within the framework of the EPT project, several subprojects were undertaken, ranging from design
studies of vehicle power trains and the realization of several test vehicles to a PhD research on the
energy management of a fuel cell hybrid drive train. The EPT project was also used to assist other projects in order to improve the quality of the overall outcome. Examples are the Pluto project, for which
models were constructed for the entire test rig and the H2ANCAR project, which produced the fuel cell
hybrid Fiat HyDoblo vehicle. For this vehicle the energy management system was designed and tested
successfully. These projects themselves provided the test facilities, needed for validation of the models
created within the EPT project.
The subprojects of the EPT project are positioned inside the dotted box indicated by “Scope of EPT project” in Figure 1, which also shows the V-cycle method for product development in a simplified representation. This figure shows the relationship with other projects, which will all be described shortly in
the following sections.
Overview of subprojects and related projects
Energy management for fuel cell hybrid vehicles
The motive for starting a PhD research on the energy management of fuel cell hybrid drive trains was
two-fold:
i.
ongoing worldwide developments in the field of the vehicle power train in the direction of
further electrification (hybrids and electric vehicles);
ii.
a number of companies in the region active in the field of fuel cells and hydrogen. Also the
municipality of Arnhem advocates itself as “hydrogen city”.
The subproject was undertaken in collaboration with the Faculty of Electrical Engineering of the University of Eindhoven, providing the promotor. Nedstack, a fuel cell manufacturing company in Arnhem,
4
5
supplied the fuel cells for testing and model validation, as well as in depth technical information. Silent
Motor Company was a partner during the design and realization phase of the fuel cell stack systems.
The subproject has resulted in a PhD thesis [1], a number of journal publications and several congress
contributions (see references in [1] and in chapter 2). The project activities have greatly enhanced the
knowledge level at the HAN University with respect to fuel cell technology and fuel cell hybrid drive
trains. The acquired knowledge was used for making new lecture material for both bachelor and master
students. The development of the ARVAL Inspire II [2] by HAN bachelor students, greatly benefited from
this research. The HAN organized two workshops on the basis of the acquired knowledge.
Chapters 2 and 3 present an overview of the scientific results that were obtained.
Results of this research were used for the realization the
HAN HyDoblo Fuel cell vehicle. The vehicle was then used
for validating the energy management strategy. The strategy
was implemented in the vehicle controller by using the software development tools HANcoder and HANtune resulting
from another subproject of the EPT project, described in the
next section and in chapter 4.
These activities have formed the basis for the follow-up project Fast & Curious and, more recently,
SMARTcode. An ever increasing number of SME’s use the tools and collaborate in a community. This
community allows for a quick exchange of ideas and market driven product developments.
Duvel, CVT drive train modelling
In a later phase of the EPT project, it was decided to apply the tools and ideas developed during the first
phase in a project together with Bosch Transmission Technology. The main research question was: what
role can the Continuously Variable Transmission (CVT) play for the development of electric vehicles? The
answer to this question was to be obtained by a modelling and simulation study, thereby making the
relevance for the EPT project obvious. Several student groups (bachelor, master for thesis assignments
and two minor groups) have been active for this research assignment. By using the AMEsim simulation
package, students were able to simulate the various drive train variants with relative ease. Detailed efficiency models of CVT variator, electric motors, batteries and hydraulic systems were available, leaving
the parameterization work and model structure design for the students.
The project work together with Bosch TT is part of a more comprehensive multi-year collaboration
between a big TIER 2 supplier and the Automotive Centre of Expertise (ACE). This collaboration focuses
on project work and on improved contact between the company and the students, being their potential
future staff members.
During the research work on the energy management for
fuel cell hybrid drive lines, a description of the power required for propulsion of the vehicle was needed. Simulations
Figure 2. Fiat HyDoblo test vehicle
on hybrid propulsion systems generally use driving cycles to
define the speed sequence of the vehicle. Disadvantages of this deterministic approach are the limited
value of just one driving cycle to represent real-life conditions and the risk of ’cycle beating’ in optimizations. Chapter 3 presents the results of a study into the modelling of the vehicle load in terms of normal
distributions.
Battery hybrid electric vehicle rapid control prototyping platform
During the first phase of the EPT project, an already existing rapid control prototyping (RCP) platform
initiative was further developed to become a safe, universal RCP platform for BHEV applications. This
development was very well suited to be incorporated within EPT, since it is clearly based on model
based design and resulted in tools that were of great value for implementing control strategies (like the
above mentioned energy management strategy) in both vehicles and test rigs.
Partners during this phase of development were companies like Drive Train Innovations (now Punch Powertrain) for applications on the Pluto test rig and Silent Motor Company for fuel cell system applications.
The subproject resulted in the development of the ‘HANcoder RC30 Target’, a so called ‘embedded target’ for generating code from MATLAB/Simulink for a Rexroth RC30 series automotive specified controller (ECU). Also the development of an application engineering tool, named HANtune, to serve as a real
time dashboard for the algorithm running in the RC30 ECU, was a result of this subproject. A pragmatically oriented handbook for applying the functional safety related procedures and methods as ISO26262
in electric power train control systems also originates from this subproject. The subproject also has
important spin-off for education, since the tools are also used extensively in the minor ‘Autotronica’.
Students are much less involved with direct coding and debugging of controller software, thereby saving
time for model based design in real life applications.
More details can be found in chapter 4.
6
Figure 3. Typical example of an AMEsim model of a CVT drive line for an electric vehicle
The project resulted in a number of (thesis) reports that clarified the possible (dis-)advantages of a
CVT for electric vehicle application. The reports also gave recommendations for improving the CVT
performance on certain aspects, thereby making the CVT a more viable option. A typical example of a
model used by some students is shown in Figure 3 [3]. In this figure, the battery, variator, electric motor,
hydraulics, auxiliaries and various controllers can be identified.
The project activities have been reviewed intensively with Bosch engineers on a regular basis. The activities have led to the start of a follow-up project: noise measurement on a CVT on the Pluto test rig.
A prototype emission free cooling trailer
HAN Automotive Research was asked to contribute to the New Cool project, aiming at the development
of an autonomous cooling trailer. The project was initiated by Twan Heetkamp Trailers in collaboration
with TMC, TRTA, TPTS, VALX axles and TRS cooling systems. The New Cool trailer (Figure 4) powers the
7
cooling compressor from brake energy, solar power and from the grid. Both students and HANAR staff
contributed to this development in the fields of energy management and rapid control prototyping.
Also during the testing phase HANAR was involved. Experience and knowledge obtained during the EPT
project proved to be of great value for this development. The project resulted in a fully functional trailer,
showing good performance characteristics. Aspects for improvement, like component sizing and optimized energy management, were also identified. Results have been presented at the FISITA 2014 congress
[4]. A summary of this paper is included in chapter 5.
Figure 4. The New Cool trailer
Battery modelling and simulation
Batteries play a central role in the drive lines of electric and hybrid vehicles. They are bulky, heavy,
expensive, constitute a potential hazard, are very sensitive to temperature and have a limited durability,
which in turn depends on aspects like depth of discharge, C-rate in charging and discharging and, again,
temperature. All this calls for an in-depth study into the functional behaviour of automotive batteries.
In collaboration with KEMA (now DNV-GL) and the HAN engineering department a research project was
initiated, aiming at obtaining functional models of Lithium batteries. The research was focused on the
electric and thermal behaviour. This work resulted in validated models in both MATLAB/Simulink and
AMEsim simulation environments. A battery simulator was developed (see chapter 6), which can be
used for testing of battery management systems. The thermal modelling, described in detail in chapter
7, focused on the thermal behaviour on cell level, in interaction with its surroundings. As a result of this
work we are now in a better position to design and test battery systems on board of vehicles and to
make an assessment of the commercial viability of various solutions. As an example may serve the study
on the efficiency of regenerative braking within the research project ‘eMobility-Lab’ of the Hogeschool
Rotterdam [5].
Pluto test facility
The Pluto test rig (see Figure 5) has been designed and realised
within the RAAK Internationaal project “Pluto”. This rig is very
well suited to test all kinds of light duty electric drive trains.
Within the EPT project we have focused on generating models
for the test facility, that allow for the development of control
software for the rig. During the control system development
in Pluto, plant models have been developed for power train
components and vehicle behaviour. These models have been
used in various stages of the development, when no real power
Figure 5. Pluto test facility
train hardware was available yet: at simulation level for early
development of algorithms and in HIL simulation to test and optimize real-time behaviour of the control
algorithms. Some more details can be found in chapter 4. Collaborating with companies like Drive Train
Innovation (now Punch Powertrain) and Bosch Transmission Technology, we were able to finish this
development and attract research assignments from these industries and others. 8
Colt test vehicle
An electric vehicle (Figure 6) has been developed. The
municipality of Helmond financially supported this activity. Partners during this development were: HAN-Automotive, Fontys, ACE, ROC ter AA, TU-Eindhoven and the
companies All Green Vehicles, Nedcar, Panasonic, Sterr
Autoschade, Aweflex, NXP, Van den Heuvel Motorsport
and TNO Automotive. The vehicle was used by TNO
for the test of measuring equipment. NXP used the
vehicle to test their new Battery Management System.
The vehicle has been optimized for use in bachelor and
master courses. Several students have participated in
the development activities. The vehicle was also used
Figure 6. EuroColt Electric vehicle
at several occasions for promotional purposes.
Conclusions and outlook
The RAAK-PRO Electric Power Train project has been very successful in providing the means to extend
and disseminate our knowledge in the field of battery electric, hybrid and fuel cell drive trains. Results
of the project have been transferred to numerous projects with many partners. Several examples of
these projects are shown in this booklet.
By using high level simulation tools like MATLAB/Simulink and AMEsim, models were derived for fuel
cells, batteries, electric and hybrid vehicles and the Pluto hybrid drive train test facility. These models
were very important during both the design and the testing phases of the various products we worked on.
The models have found their way also in the curriculum offered to bachelor and master students. In
turn, many students contributed to the realization of the models during internships and thesis work.
Results from the RAAK-PRO Electric Power Train project will be used in ongoing and future projects with
a focus on power train development. An extension of the application area to industrial applications
has been undertaken in running projects in the field of model based design (Fast & Curious, SMARTcode). Another extension towards agricultural applications is currently under investigation. Activities for
automotive companies (DAF Trucks, Bosch TT, Bosal, etc.) are carried out, which greatly benefit from the
knowledge acquired and hardware developed during the EPT project.
The RAAK-PRO Electric Power Train project has made the HAN Automotive Research group a more valuable partner for all of its stakeholders: teachers, companies, staff and students.
Acknowledgements
The work described in this booklet would not have been possible without the support of numerous
students, who contributed during minor projects, internships and thesis assignments. The financial support from the RAAK-PRO program is gratefully acknowledged. Numerous companies and institutes have
contributed to the EPT project: Nedstack, Bosch Transmission Technology, Drive Train Innovations (now
Punch Powertrain), Silent Motor Company, APTS, e-Traction, Spijkstaal Electro BV, TU Eindhoven, Hogeschool Rotterdam, Fontys Hogescholen, KEMA (now DNV-GL), LMS International (now Siemens) and
Dutch-INCERT. Their contribution, either as member of the project consortium or by providing technical
support is gratefully acknowledged.
9
References
[1]Edwin Tazelaar, Energy Management and Sizing of Fuel Cell Hybrid Propulsion Systems, PhD Thesis., 2013.
[2]HAN University of Applied Science. (2015, Feb.) [Online].
https://www.facebook.com/hanhydromotivenl
[3] Gino van Luijn, “Feasibility of the application of a CVT in an electric vehicle,” 2014.
[4]Bram Veenhuizen, Menno Merts, Karthik Venkatapathy, and Erik Vermeer, “A prototype Emission
Free cooling trailer,” in FISITA 2014, Maastricht, 2014.
[5]Stefan van Sterkenburg, “Analysis of regenerative braking efficiency -A case study of two electric vehicles operating in the Rotterdam area,” in IEEE Vehicle Power and Propulsion Conference
(VPPC) 2011.
[6]Dennie Craane and Willem Roovers, “Feasibility study for CVT in EV; DUVEL project, Final report,”
Bosch, Tilburg, 2013.
Analytic solution to the energy management
problem for fuel cell hybrid vehicles
Edwin Tazelaar
HAN University of Applied Sciences, Arnhem, The Netherlands
Abstract
As fuel cell hybrid propulsion systems comprise energy storage, an energy management strategy is needed for proper control. Objective of such strategy is to deliver the demanded power for traction
with a minimum hydrogen consumption. This paper discusses an analytic solution to this optimization
problem, based on the minimization of losses in the propulsion system. This analytic solution enables a
real-time implementation. It has been validated through simulations and measurements on a 10 kW test
facility and on road in a fuel cell hybrid vehicle. Simulations show the presented solution performs within
2% of a benchmark derived off-line by dynamic programming and brute force calculations. Experiments
confirm this fuel consumption and demonstrate its practical value.*)
1. Introduction
Fuel cell hybrid power trains comprise an energy storage to supply peaks in the power demand and to
facilitate regenerative braking. In terms of control systems, the presence of storage provides additional
freedom to minimize the vehicles’ fuel consumption. The approach to this control challenge is generally
referred to as the Energy Management Strategy (EMS).
Several energy management strategies are proposed in literature, based on fuzzy logic [1], efficiency
maps [2], classic control [3] or rule-based [4]. Also efforts are made to model the fuel consumption as a
cost function and to find the minimum in this cost function using different techniques, such as dynamic
programming [5]. Analytic solutions are found for comparable applications as alternator optimization
[6] or internal combustion engines [7, 8]. These analytic solutions represent the power from the battery
with equivalent costs, resulting in an Equivalent Consumption Minimization Strategy (ECMS). As models,
an efficiency as zeroth order battery model, and a quadratic representation of the fuel consumption, are
used [6, 7, 8]. In [9] this approach is extended to a first order battery model, with a cost function based
on physics to express the fuel consumption. An extensive overview of publications on EMSs for fuel cell
hybrid vehicles is provided for in [10].
This study minimizes the fuel consumption using an alternative approach. Where most studies focus on
the level of power, this study shifts one level of detail further, to voltages and currents. In addition, not
the fuel consumption is minimized, but the losses in the power train. A minimum in losses is the equivalent to a minimum in fuel consumption. The result provides an analytic solution expressed in measurable variables and physical parameters.
The solution found corresponds to the solution presented in [14, 15]. This paper presents the derivation
of the analytic solution to the energy management problem in terms of voltages and currents, and its
validation through simulation and measurements.
10
*)This paper contains parts and partly summarizes content of [14] and [15].
11
2. Problem definition
Objective
of the
energy
management
strategy
to minimize
thevehicle.
losses in
theminimum
fuel cell hybrid
propulsion
system,
without
compromising
theisdrivability
of the
This
in losses
propulsion
system,
without
compromising
drivability
of the vehicle.
in a
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considered
topology
offuel
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in
matches the
in fuelsource,
consumption.
subsystem
as minimum
primary power
a battery for energy storage, a DC/DC converter to match voltage
The and
considered
topology
of thealternating
propulsion current
system to
is presented
figure
It comprises
a fuel cell
levels
an inverter
to provide
the motor.inThe
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stack subsystem
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considered
topology
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the
propulsion
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is
presented
in
figure
1.
It
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fuel
cell
stack
subsystem
as
primary
power
source,
a
battery
for
energy
storage,
a
DC/DC
converter
to
match
des auxiliaries as an air compressor, hydrogen control valve, recirculation pump, humidifier and cooling
stack
subsystem
as
primary
power
source,
a battery storage
for
energy
storage,
a DC/DC
match
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levels and
to provide
current
to the
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and
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cooling auxiliaries
system. Hydrogen
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Figure 1. Topology fuel cell hybrid propulsion system.
Figure 1
Figure 1
Topology fuel cell hybrid propulsion system.
Topology fuel cell hybrid propulsion system.
MODELS
3.3 Models
3
MODELS
The fuel cell stack converts hydrogen into electric power. A polarization curve defines the relation betThe fuel cell stack converts hydrogen into electric power. A polarization curve defines the relation
ween stack voltage and stack current. Figure 2 presents a typical polarization curve. Under normal opeThe
fuel cell
stack
converts
hydrogen
into Figure
electric2power.
A polarization
curve defines
theUnder
relation
between
stack
voltage
and stack
current.
presents
a typical polarization
curve.
normal
rating conditions, both the activation area (A) and the concentration area (C) are avoided. In area (B),
between
stack
voltage and
current. Figure
2 presents
a typical polarization
Under normal
operating
conditions,
bothstack
the activation
area (A)
and the concentration
area (C)curve.
are avoided.
In area
the ohmic losses dominate the total internal losses. For this operating area, the fuel cell stack acts as an
operating
conditions,
both
the activation
(A) and
the concentration
areaarea,
(C) are
areaacts
(B), the ohmic
losses
dominate
the totalarea
internal
losses.
For this operating
theavoided.
fuel cell In
stack
internal voltage source VFC0 with an internal resistance RFC, resulting in a voltage VFC at the terminals
(B),
ohmic losses
dominate
internal
losses.
For this
the fuel V
cell
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internal
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, resultingarea,
in a voltage
the acts
as the
an internal
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source Vthe
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FC at
of the stack:
asterminals
an internal
voltage
source VFC0 with an internal resistance RFC, resulting in a voltage VFC at the
of the
stack:
terminals of the stack:
𝑉𝑉!" = 𝑉𝑉!"! − 𝑅𝑅!" 𝐼𝐼!"
(1)
𝑉𝑉!" = 𝑉𝑉!"! − 𝑅𝑅!" 𝐼𝐼!"
(1)
!
Therefore, the marginal losses in the fuel cell stack in the ohmic operating area resemble 𝐼𝐼!"
𝑅𝑅!" .
!
Therefore,the
the marginal
marginal losses
lossesininthe
the fuel
fuel cell
cell stack
stackin
inthe
the ohmic
ohmic operating
operating area resemble 𝐼𝐼!" 𝑅𝑅!" .
Therefore,
12
polarization
N ·Ecurve
FC
VFC0
0
NFC·E0
NFC·E0
VFC0
Voltage
Voltage
[V]Voltage
[V]Voltage
Voltage
[V][V] [V]
2
PROBLEM
DEFINITION
Objective
of the energy
management strategy is to minimize the losses in the fuel cell hybrid propulsion
2
PROBLEM
DEFINITION the drivability of the vehicle. This minimum in losses matches the minisystem,
without compromising
mum
in fuelof
consumption.
Objective
the energy management strategy is to minimize the losses in the fuel cell hybrid
Voltage [V]
NFC∙E0
NFC
V·E
FC00
polarization curve
polarization curve
polarization curve
VFC = VFC0 N-FCVR·E
0 IFC
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FC
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= VFC0 - RFC
IFC
polarization
curve
polarization
curve
B
A
B
VFC = VFC0 - RFCIFC
VFC0
VFC = VFC0 - RFCIFC
VFC0
A
B [A]
Current
Current
[A]
A
B
VFC = VFC0 - RFCIFCCurrent [A]
Current [A]
VFCpolarization
= VFC0 - RFCIFC
Figure
2. Typical
stack polarization
curve.
Figure
2 fuel cell
Typical
fuel cell stack
curve.
A
B
A
B
Figure 2
Typical fuel cell stack polarization curve.
Current
Figure 2
Typical fuel cell stack polarization curve. [A]
Current [A]
A
C
C
C
C
C
C
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to to
thethe
fuel
cell
stack,
thethe
battery
is represented
byby
anan
ideal
voltage
source
VS0Vwith
anan
internal
Analogue
fuel
cell
stack,
battery
is represented
ideal
voltage
source
S0 with
resistance
RS, as indicated
inindicated
figure
1.inThe
resulting
at the
terminals
the battery
equals:
internal
RS, asfuel
figure
1. Thevoltage
resulting
voltage
at the of
terminals
of the
battery
Figure 2resistance
Typical
cell stack
polarization
curve.
Analogue to the fuel cell stack, the battery is represented by an ideal voltage source VS0 with an
Figure
cellthe
stack
polarization
curve. by an ideal voltage source VS0 with an
equals:2 to the Typical
Analogue
fuel cellfuel
stack,
battery
is represented
internal resistance RS, as indicated in figure 1. The resulting voltage at the terminals of the
battery
internal resistance RS, as indicated in figure 1. The resulting voltage at the terminals of the battery
equals:
𝑉𝑉!! − 𝑅𝑅! 𝐼𝐼! by an ideal voltage source VS0 with(2)
equals: to the fuel cell stack, the battery 𝑉𝑉is! =
Analogue
represented
an
Analogue to the fuel cell stack, the battery is represented by an ideal voltage source VS0 with an
internal resistance RS, as indicated in figure𝑉𝑉1.=The
resulting
voltage
at
the
terminals
of
the
battery
𝑉𝑉!!!− 𝑅𝑅! 𝐼𝐼!
(2)
! The
internal
resistance
RS, as
indicated
figure𝑉𝑉equal
1.
voltage at the terminals of the battery
As
a result,
the power
losses
in theinbattery
𝐼𝐼 resulting
𝑅𝑅 .
(2)
equals:
! = 𝑉𝑉!!! −!𝑅𝑅! 𝐼𝐼!
equals:
!
AsAs
a result,
the
power
losses
in
the
battery
equal
a result, the power losses in the battery equal 𝐼𝐼!! 𝑅𝑅! .
The
converter
matches
the battery
fuel cell𝑉𝑉equal
stack𝐼𝐼 voltage
VFC with the bus voltage defined by(2)
the
As aDC/DC
result, the
power losses
in the
𝑅𝑅 .
! = 𝑉𝑉!!! −!𝑅𝑅! 𝐼𝐼!
𝑉𝑉
=
𝑉𝑉
−
𝑅𝑅
𝐼𝐼
(2)
V
.
This
conversion
introduces
a
small
loss,
expressed
in
a
converter
power
efficiency
battery
voltage
!!
!VFC
! with the bus voltage defined by the batS matches the fuel cell stack
The
DC/DC
converter
voltage
The
DC/DC
converter
matches the fuel cell!stack
voltage
VFC with the bus voltage defined by the
!
η
(<1):
The
DC/DC
converter
matches
the
fuel
cell
stack
voltage
VFC with
bus voltage
defined
by theη
cnv
As
a
result,
the
power
losses
in
the
battery
equal
𝐼𝐼
𝑅𝑅
.
!! ! expressed
tery
voltage
VS. This
introduces
a small
loss,
in the
a converter
power
efficiency
VS. conversion
This conversion
introduces
a small
loss, expressed
in a converter
power
efficiency
battery
voltage
cnv
As a result,
the Vpower
losses
in
the
battery
equal
𝐼𝐼
𝑅𝑅
.
!
!
.
This
conversion
introduces
a
small
loss,
expressed
in
a
converter
power
efficiency
battery
voltage
S
(<1):
ηcnv (<1):
𝐼𝐼!"# 𝑉𝑉
ηcnv (<1):
The
DC/DC converter matches the fuel cell
stack
VFC with the bus voltage defined by(3)
the
! = 𝜂𝜂voltage
!"# 𝐼𝐼!" 𝑉𝑉!"
The
DC/DC
converter
matches
the
fuel
cell
stack
voltage
V
FC with the bus voltage defined by the
expressed in a converter power efficiency
battery voltage VS. This conversion introduces
(3)
𝐼𝐼!"# 𝑉𝑉!a=small
𝜂𝜂!"# 𝐼𝐼loss,
!" 𝑉𝑉!"expressed in a converter power efficiency
battery voltage VS. This conversion introduces
(3)
𝐼𝐼!"# 𝑉𝑉!a=small
𝜂𝜂!"# 𝐼𝐼loss,
ηcnv (<1):
!" 𝑉𝑉!"
ηcnv (<1):
The
converter converts the fuel cell stack current IFC to a net current IFCn according to a ratio r:
𝐼𝐼!"#
𝑉𝑉! =
𝑉𝑉net
The
converter
converts
fuel
cell
stack
current
anet
currentIIFCn according
to a ratio r(3)
:
!" current
The
converter
converts
thethe
fuel
cell
stack
current
I I𝜂𝜂𝜂𝜂FC!"#
toto𝐼𝐼𝐼𝐼a!"
FCn according to a ratio r:
(3)
𝐼𝐼 𝑉𝑉 =FC
𝑉𝑉
𝐼𝐼!"#
=
𝑟𝑟𝐼𝐼to!"!"
The converter converts the fuel cell stack !"#
current
IFC!"#
a net
:
!
!" current IFCn according to a ratio r(4)
(4)
𝐼𝐼!"# = 𝑟𝑟𝐼𝐼!"
bythe
thefuel
voltages
and current
converter
efficiency
according:
with
ratio r is defined
𝐼𝐼!"# =
The converter
converts
cell stack
IFC𝑟𝑟𝐼𝐼
to!"a net current
IFCn according to a ratio r(4)
:
The converter converts the fuel cell stack current IFC to a net current IFCn according to a ratio r:
with ratio r is defined by the voltages and converter efficiency according:
𝑉𝑉!"
𝑉𝑉!"! − 𝐼𝐼!"according:
𝑅𝑅!"
efficiency
with ratio r is defined by the voltages and converter
(4)
= 𝜂𝜂!"# 𝐼𝐼!"#
= 𝜂𝜂= 𝑟𝑟𝐼𝐼!"
(5)
with ratio r is defined by the voltages𝑟𝑟 and
converter
efficiency
according:
(4)
𝑉𝑉!𝐼𝐼!"# =!"#
𝑉𝑉
−
𝐼𝐼
𝑅𝑅
𝑟𝑟𝐼𝐼
!!
!
!
!"
𝑉𝑉!"
𝑉𝑉!"! − 𝐼𝐼!" 𝑅𝑅!"
𝑟𝑟
=
𝜂𝜂
=
𝜂𝜂
(5)
𝑉𝑉!"
𝑉𝑉!"! − 𝐼𝐼!"according:
𝑅𝑅
!"#converter
and
efficiency
with ratio r is defined by the voltages
𝑉𝑉! = 𝜂𝜂!"#
𝑉𝑉!! − 𝐼𝐼! 𝑅𝑅!!"
𝑟𝑟
=
𝜂𝜂
(5)
!"#
!"#
Over
time,r the
average
currentand
hasconverter
to! approach
toaccording:
avoid depletion and overcharging.
is defined
bybattery
the voltages
efficiency
with ratio
𝑉𝑉
𝑉𝑉zero
!! − 𝐼𝐼! 𝑅𝑅!
Assuming sufficiently small excursions for the
fuel cell𝑉𝑉stack
and battery current, the ratio r is
− 𝐼𝐼current
Over time, the average battery current has𝑉𝑉to
zero
to 𝑅𝑅avoid
depletion and overcharging.
!" approach
!"!
!"
!"
𝑟𝑟
=
𝜂𝜂
=
𝜂𝜂
(5)
𝑉𝑉
𝑉𝑉
−
𝐼𝐼
𝑅𝑅
!"#
!"#
approximated
Over time, the with:
average battery current has 𝑉𝑉
to
zero
to
avoid
depletion and overcharging.
!" approach
!"!
!"
!"
𝑉𝑉!! − current
𝐼𝐼! 𝑅𝑅!
Assuming sufficiently small excursions
cell stack
and battery current, the ratio
! fuel
𝑟𝑟 = 𝜂𝜂for
= 𝜂𝜂!"#
(5) r is
!"# the
𝑉𝑉! fuel cell stack
𝑉𝑉!! − current
𝐼𝐼! 𝑅𝑅!
Assuming sufficiently small excursions for the
and battery current, the ratio r is
approximated with:
𝑉𝑉
−
𝐼𝐼
𝑅𝑅
approximated
Over time, the with:
average battery current has to approach
zero
!"!
!"
!"to avoid depletion and overcharging.
(6)
𝑟𝑟 ≈ to
𝜂𝜂!"#
Over
time, sufficiently
the average
battery
current has
approach
zero current
to avoidand
depletion
and
overcharging.
𝑉𝑉
Assuming
small
excursions
for
the
fuel
cell
stack
battery
current,
the ratio r is
!!
Over time, the average battery current has to approach
avoid
depletion
and
overcharging.
𝑉𝑉!"! −zero
𝐼𝐼!" 𝑅𝑅to
!"current and battery current, the ratio r is
Assuming sufficiently
small excursions for
fuel cell
(6)
≈the
𝜂𝜂fuel
− stack
𝐼𝐼!" 𝑅𝑅current
!"# 𝑉𝑉
approximated
with: small
!"! stack
!"
Assuming
sufficiently
excursions for𝑟𝑟𝑟𝑟the
cell
and battery current, the ratio
𝑉𝑉
(6) r is
!!
≈
𝜂𝜂
!"#
approximated
with:
r is a constant in the optimization.
For
the
experiments
discussed
later,
the
maximum
As a result, ratio
𝑉𝑉
!!
approximated with:
instantaneous error made due to this approximation
is
13%, at the upper boundary of the
𝑉𝑉!"!For
− 𝐼𝐼maximum
𝑅𝑅!"
the
experiments discussed later, the maximum
As a result, ratio r is a constant in the optimization.
!"
(6)
𝑟𝑟 ≈ 𝜂𝜂!"# 𝑉𝑉!"!
− 𝐼𝐼!"
𝑅𝑅!"
propulsion
window
(maximum
battery
current).
a constant
in the optimization.
For
the
experiments
discussed later, the maximum
As
a result,systems
ratio r isoperating
instantaneous error made due to this approximation
is𝑉𝑉!!maximum 13%, at the upper boundary(6)
of the
𝑟𝑟 ≈ 𝜂𝜂!"#
instantaneous error made due to this approximation is𝑉𝑉!!maximum 13%, at the upper boundary of the
propulsion systems operating window (maximum battery current).
Id, the
needed
for
The
current
by the
stack and
battery
has to provide
the current
propulsion
systems
window
(maximum
battery
current).
isoperating
a constant
in fuel
the cell
optimization.
For
the
experiments
discussed
later,
maximum
As
atotal
result,
ratio rdelivered
r
is
a
constant
in
the
optimization.
For
the
experiments
discussed
later,
the
maximum
As
a
result,
ratio
propulsion
anderror
the current
Iaux, to
needed
for the auxiliaries.
Especially,
the
air compressor
significantly
instantaneous
made due
is maximum
at the
boundary
of the
Id, needed
for
The total current
delivered
by
the this
fuel approximation
cell stack and battery
has to13%,
provide
theupper
current
instantaneous
error
made
due
to
this
approximation
is
maximum
13%,
at
the
upper
boundary
of the
I
,
needed
for
The
total
current
delivered
by
the
fuel
cell
stack
and
battery
has
to
provide
the
current
d
propulsion
systems
operating
window
(maximum
battery
current).
propulsion and the current Iaux, needed for the auxiliaries. Especially, the air compressor significantly
propulsion
systems
operating
window
(maximum
battery
current).
propulsion and the current Iaux, needed for the auxiliaries. Especially, the air compressor significantly13
The total current delivered by the fuel cell stack and battery has to provide the current Id, needed for
The total current delivered by the fuel cell stack and battery has to provide the current Id, needed for
propulsion and the current Iaux, needed for the auxiliaries. Especially, the air compressor
significantly
!!
As a result, ratio r is a constant in the optimization. For the experiments discussed later, the maximum
instantaneous error made due to this approximation is maximum 13%, at the upper boundary of the
propulsion systems operating window (maximum battery current).
The total current delivered by the fuel cell stack and battery has to provide the current Id, needed for
propulsion and the current Iaux, needed for the auxiliaries. Especially, the air compressor significantly
contributes to the current demand of the auxiliaries. As the oxygen consumption of the fuel cell system
contributes to the current demand of the auxiliaries. As the oxygen consumption of the fuel cell system
is proportional
to current
the stack
current,
compressor
current proportional
air mass
contributes
to the
demand
ofand
the assuming
auxiliaries.an
Asairthe
oxygen consumption
of the fuel to
cellitssystem
is proportional to the stack current, and assuming an air compressor current proportional to its air
flow,
the current
to operate
theand
auxiliaries
is approximated
by: current proportional to its air
is
proportional
to Ithe
current,
assuming
an air compressor
aux,stack
mass flow, the current Iaux, to operate the auxiliaries is approximated by:
mass flow, the current Iaux, to operate the auxiliaries is approximated by:
contributes to the current demand of the auxiliaries. As the oxygen consumption of the fuel cell system
𝐼𝐼!"# = 𝛾𝛾𝐼𝐼!" + 𝐼𝐼!"#!
(7)
is proportional to the stack current, and assuming
an+air
compressor current proportional to its
𝐼𝐼!"# = 𝛾𝛾𝐼𝐼!"
𝐼𝐼!"#!
(7)air
mass flow, the current Iaux, to operate the auxiliaries is approximated by:
Here, Iaux0 defines the offset and γ the proportional part of the stack current needed for the auxiliaries.
Here, Iaux0 defines the offset and γ the proportional part of the stack current needed for the auxiliaries.
This relation is verified on the test facility as an acceptable approximation [14].
This
is verified
on theand
testγfacility
[14].needed for the(7)
𝐼𝐼as
= acceptable
𝛾𝛾𝐼𝐼!" +
𝐼𝐼!"#!
Here,relation
I
defines
the offset
the proportional
part
ofapproximation
the stack current
auxiliaries.
!"#an
aux0
This relation is verified on the test facility as an acceptable approximation [14].
Here, Iaux0 defines the offset and γ the proportional part of the stack current needed for the auxiliaries.
This relation is verified on the test facility as an acceptable approximation [14].
4
ANALYTIC SOLUTION
4
ANALYTIC SOLUTION
4. Analytic solution
The losses in the propulsion system are dominated by the internal losses in the fuel cell stack, the
The
lossesin
in the
the propulsion
propulsion system
system are dominated
dominated by the
the internal losses
in the
cell
stack, the
The losses
lossesequations
in
the fuel
fuelof
cell
the loslosses
in the battery
and the losses inare
the converter.by
Basedinternal
on the model
thestack,
previous
4
ANALYTIC
SOLUTION
losses
in
the
battery
and
the
losses
in
the
converter.
Based
on
the
model
equations
of
the
previous
ses in thethese
battery
and are
the expressed
losses in the
Based
onInthe
equations
oflosses
the previous
chapter,
losses
in converter.
a cost function
(8).
thismodel
cost function,
the
in the chapter,
chapter, these losses are expressed in a cost function (8). In this cost function, the losses in the
these
losses
are
expressed
in
a
cost
function
(8).
In
this
cost
function,
the
losses
in
the
DC/DC
DC/DC converter are implicitly included in the battery current, and the current to the auxiliaries converter
is
The losses
in the propulsion
system
areindominated
the internal
losses
in the
cell stack, is
the
DC/DC
converter
are implicitly
included
the batterybycurrent,
and the
current
to fuel
the auxiliaries
are implicitly
included
in thecurrent
battery
current,The
andcontrol
the current
to the
auxiliaries
is considered
part
considered
part
of the total
demand.
variable
in this
cost function,
available
to of the
losses in thepart
battery
thecurrent
lossesdemand.
in the converter.
Based
on the
equations
the previous
considered
of theand
total
The control
variable
inmodel
this cost
function,ofavailable
to
total current
The control
variable
in this
cost function,
available
minimize
the
function,
IFC , defined
by the to
control
signal
to cost
the DC/DC
minimize
the demand.
cost function,
is the fuel
cell stack
current
chapter, these
losses
are expressed
a cost
this cost
thesignal
losses
theDC/DC
IFC ,Indefined
by function,
the control
tointhe
minimize
the cost
function,
is the fuelin
cell
stackfunction
current(8).
is the fuel cell stack current IFC , defined by the control signal to the DC/DC converter.
converter.
DC/DC converter are implicitly included in the battery current, and the current to the auxiliaries is
converter.
considered part of the total current demand. The control variable in this cost function, available to
!!
minimize the cost function, is the fuel cell stack!! current IFC , defined by the control signal to the DC/DC
!
!
(8)
𝐽𝐽(𝐼𝐼!" ) =
𝐼𝐼!"
! 𝑅𝑅!" + 𝐼𝐼!! 𝑅𝑅! 𝑑𝑑𝑑𝑑
converter.
(8)
𝐽𝐽(𝐼𝐼!" ) =
𝐼𝐼!"
𝑅𝑅!" + 𝐼𝐼! 𝑅𝑅! 𝑑𝑑𝑑𝑑
!!!
!!!
!!
This cost function is subject to two constraints: the total current demand should be met (9), and the
! total current
This cost function is subject to two constraints:
the
demand should be met (9), and
(8) the
𝐽𝐽(𝐼𝐼!" ) =
𝐼𝐼!"
𝑅𝑅!"latter
+ 𝐼𝐼!! 𝑅𝑅constraint
! 𝑑𝑑𝑑𝑑
battery
not is
be depleted
nor overcharged.
This
is stated
more
stringent
as athe
This costshould
function
to two
constraints: the
total
current
demand
should
be met
(9), and
battery
should
not besubject
depleted
nor overcharged.
This
latter
constraint
is
stated
more
stringent
as a
!!!
zero
difference
in
the
State
Of
Charge
(SOC)
between
the
end
and
the
start
of
the
driving
cycle
(10).
battery
should
not
be
depleted
nor
overcharged.
This
latter
constraint
is
stated
more
stringent
a zero
zero difference in the State Of Charge (SOC) between the end and the start of the driving cycle as
(10).
This
guarantees
a
continuous
valid
operation
of
the
battery,
and
enables
comparison
between
difference
in the aState
Of Charge
between
the
endcurrent
andand
the
start ofshould
the driving
cycle
This guaThis
cost
function
iscontinuous
subject
to valid
two(SOC)
constraints:
total
demand
be met
(9), (10).
and the
guarantees
operation
ofthe
the
battery,
enables
comparison
between
different energy management strategies.
rantees
a
continuous
valid
operation
of
the
battery,
and
enables
comparison
between
different
energy
battery
should
not
be
depleted
nor
overcharged.
This
latter
constraint
is
stated
more
stringent
as
a
different energy management strategies.
zero
differencestrategies.
in the State Of Charge (SOC) between the end and the start of the driving cycle (10).
management
𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" + 𝐼𝐼! − 𝐼𝐼! + 𝐼𝐼!"#! = 0
(9)
This guarantees a continuous valid𝑟𝑟operation
and
comparison between
− 𝛾𝛾 𝐼𝐼!" +of𝐼𝐼! the
− 𝐼𝐼battery,
= enables
0
(9)
! + 𝐼𝐼!"#!
different energy management strategies.
!!
𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" +!!𝐼𝐼! − 𝐼𝐼! + 𝐼𝐼!"#! = 0
𝐼𝐼! 𝑑𝑑𝑑𝑑 = 0
𝐼𝐼! 𝑑𝑑𝑑𝑑 = 0
!!!
!!!
!!
(9)
(10)
(10)
As constraint (9) represents a linear relation, it is used to reduce the number of equations. The
As constraint (9) represents a linear relation, it is
the number of equations. The
(10)
𝐼𝐼! used
𝑑𝑑𝑑𝑑function
= to
0 reduce
optimization problem of finding the minimum in cost
(8) subject to constraints (9) and (10) is
optimization problem of finding the minimum!!!
in cost function (8) subject to constraints (9) and (10) is
solved analytically by converting cost function J into a Lagrangian L by combining the constraint (10)
solved analytically by converting cost function J into a Lagrangian L by combining the constraint (10)
on the SOC with cost function (8) using a Lagrange multiplier λ [11, 12]:
As
constraint
(9) represents
a linear
relation,
it is used
to reduce
the12]:
number of equations. The
on the
SOC with
cost function
(8) using
a Lagrange
multiplier
λ [11,
optimization problem of finding the minimum in cost function (8) subject to constraints (9) and (10) is
!!
!! function J into a Lagrangian L by combining the constraint (10)
solved analytically by converting cost
!
!
𝐿𝐿
𝐼𝐼
,
𝜆𝜆
=
𝐼𝐼!"
+ 𝐼𝐼! + multiplier
𝐼𝐼!"#! − 𝑟𝑟λ−[11,
𝛾𝛾 𝐼𝐼!"
! a𝑅𝑅Lagrange
! 𝑅𝑅! 𝑑𝑑𝑑𝑑
on the SOC with cost function
(8) using
12]:
𝐿𝐿 𝐼𝐼!"
𝐼𝐼!"
𝑅𝑅!"
!" , 𝜆𝜆 =
!" + 𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" 𝑅𝑅! 𝑑𝑑𝑑𝑑
!!!
(11)
!!
!!!
(11)
!!
!
14
𝐿𝐿 𝐼𝐼!" , 𝜆𝜆 =
!
!!!
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" 𝑑𝑑𝑑𝑑
+ 𝜆𝜆
! + 𝜆𝜆
!
𝐼𝐼!"
𝑅𝑅!" + 𝐼𝐼!𝐼𝐼!++𝐼𝐼!"#!
𝐼𝐼!"#!−− 𝑟𝑟𝑟𝑟−−𝛾𝛾𝛾𝛾 𝐼𝐼!"
𝐼𝐼!" 𝑑𝑑𝑑𝑑
𝑅𝑅! 𝑑𝑑𝑑𝑑
!!!
!!!
!
(11)
!!!
𝐼𝐼! 𝑑𝑑𝑑𝑑 = 0
(10)
As constraint (9) represents a linear relation, it is used to reduce the number of equations. The optimiAs
constraint
(9)ofrepresents
linear relation,
it is
used to(8)
reduce
theto
number
of equations.
Theis solved
zation
problem
finding thea minimum
in cost
function
subject
constraints
(9) and (10)
optimization
problem
of
finding
the
minimum
in
cost
function
(8)
subject
to
constraints
(9)
and
is
analytically by converting cost function J into a Lagrangian L by combining the constraint (10)(10)
on the
solved analytically by converting cost function J into a Lagrangian L by combining the constraint (10)
SOC with cost function (8) using a Lagrange multiplier λ [11, 12]:
on the SOC with cost function (8) using a Lagrange multiplier λ [11, 12]:
𝐿𝐿 𝐼𝐼!" , 𝜆𝜆 =
!!
!!!
!
𝐼𝐼!"
𝑅𝑅!" + 𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ! 𝑅𝑅! 𝑑𝑑𝑑𝑑
+ 𝜆𝜆
!!
!!!
(11)
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" 𝑑𝑑𝑑𝑑
To ensure that a minimum is found in the Lagrangian L(IFC , λ), small random deviations towards variables
shouldthat
notaresult
in deviations
theLagrangian
Lagrangian.L(I
AsFCnecessary
for this minimum,
To ensure
minimum
is found ininthe
, λ), small conditions
random deviations
towards the
derivatives
towards
and the introduced
Lagrange
have to befor
zero:
variables should
notvariable
result inIFC
deviations
in the Lagrangian.
Asmultiplier
necessaryλ conditions
this minimum,
IFC in
and
introducedL(I
Lagrange
multiplier
have to betowards
zero:
the ensure
derivatives
variable
To
that towards
a minimum
is found
thethe
Lagrangian
randomλ deviations
FC, λ), small
variables should not result in deviations in the Lagrangian. As necessary conditions for this minimum,
𝜕𝜕𝜕𝜕(𝐼𝐼
To
that towards
a minimum
is found
thethe
Lagrangian
randomλ deviations
IFC in
and
introduced
Lagrange
multiplier
have to betowards
zero:
the ensure
derivatives
variable
!" , 𝜆𝜆) L(I
FC, λ), small
=0
(12)
variables should not result in deviations in the 𝜕𝜕𝐼𝐼
Lagrangian.
As necessary conditions for this minimum,
!"
𝜕𝜕𝜕𝜕(𝐼𝐼
IFC in
and
introduced
Lagrange
multiplier
have to betowards
zero:
the ensure
derivatives
variable
!" , 𝜆𝜆) L(I
To
that towards
a minimum
is found
thethe
Lagrangian
, λ), small
randomλ deviations
= 0FC
(12)
𝜕𝜕𝜕𝜕(𝐼𝐼
,
𝜆𝜆)
!"
!"
variables should not result in deviations in the 𝜕𝜕𝐼𝐼
Lagrangian.
(13)
= 0 As necessary conditions for this minimum,
𝜕𝜕𝜕𝜕
!" , 𝜆𝜆)
introduced
Lagrange multiplier λ have to be zero:(12)
the derivatives towards variable IFC and the𝜕𝜕𝜕𝜕(𝐼𝐼
𝜕𝜕𝜕𝜕(𝐼𝐼
, 𝜆𝜆) = 0
𝜕𝜕𝐼𝐼!"
!"
(13)
0
The second necessary condition (13) just returns the =
constraint
on the SOC (10). The first necessary
𝜕𝜕𝜕𝜕 , 𝜆𝜆)
𝜕𝜕𝜕𝜕(𝐼𝐼
!" , as function of the Lagrange multiplier:
condition (12) provides the optimal solution 𝜕𝜕𝜕𝜕(𝐼𝐼
IFC*(λ)
=0
(12)
, 𝜆𝜆)constraint
!"
The second necessary condition (13) just returns
the
on the SOC (10). The first necessary
con𝜕𝜕𝐼𝐼
(13)
!"
=
0
The second necessary condition (13) just returns
the
constraint
on
the
SOC
(10).
The
first
necessary
𝜕𝜕𝜕𝜕
dition (12) provides the optimal solution
IFC*(λ),
as function of the Lagrange multiplier:
!!
condition (12) provides the optimal
solution
IFC*(λ), as function of the Lagrange multiplier:
𝜕𝜕𝜕𝜕(𝐼𝐼
𝜕𝜕𝜕𝜕(𝐼𝐼!" , 𝜆𝜆)
!" , 𝜆𝜆)
(13)
=
0
The second necessary condition
constraint
on the SOC (10). The first necessary
= (13)
2𝐼𝐼!"just
𝑅𝑅!" returns
− 2 𝑟𝑟 −the
𝛾𝛾 [𝐼𝐼
! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅! 𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
𝜕𝜕𝐼𝐼
!! solution I *(λ), as function of the Lagrange multiplier:
condition (12) provides !"
the optimal
FC
!!!
(14)
!!
𝜕𝜕𝜕𝜕(𝐼𝐼!" , 𝜆𝜆)
The second necessary condition
on 𝑟𝑟the
(10).
= (13)
2𝐼𝐼!"just
𝑅𝑅!" returns
− 2 𝑟𝑟 −the
𝛾𝛾 [𝐼𝐼constraint
− 𝛾𝛾SOC
𝐼𝐼!" ]𝑅𝑅
! + 𝐼𝐼!"#! −
! 𝑑𝑑𝑑𝑑The first necessary
𝜕𝜕𝐼𝐼!"
!! + 𝜆𝜆
− 𝑟𝑟 − 𝛾𝛾 𝑑𝑑𝑑𝑑
condition (12) provides the optimal
!!! solution IFC*(λ), as function of the Lagrange multiplier:
(14)
!!
𝜕𝜕𝜕𝜕(𝐼𝐼!" , 𝜆𝜆)
!!!
=
2𝐼𝐼!" 𝑅𝑅!" − 2 𝑟𝑟 − 𝛾𝛾 [𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅! 𝑑𝑑𝑑𝑑
𝜕𝜕𝐼𝐼!"
!! + 𝜆𝜆
− 𝑟𝑟 − 𝛾𝛾 𝑑𝑑𝑑𝑑
!!!
(14)
!!
A solution for which𝜕𝜕𝜕𝜕(𝐼𝐼
this!"necessary
condition
(14) is zero equals:
, 𝜆𝜆)
=
2𝐼𝐼!" 𝑅𝑅!!!
!" − 2 𝑟𝑟 − 𝛾𝛾 [𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅! 𝑑𝑑𝑑𝑑
𝜕𝜕𝐼𝐼!"
− 𝑟𝑟 − 𝛾𝛾 𝑑𝑑𝑑𝑑
+ 𝜆𝜆
!!!
(14)
2𝐼𝐼!"necessary
𝑅𝑅!" − 2
𝑟𝑟 −condition
𝛾𝛾 [𝐼𝐼!!! + 𝐼𝐼(14)
𝑟𝑟 − 𝛾𝛾equals:
𝐼𝐼!" ]𝑅𝑅! − 𝜆𝜆(𝑟𝑟 − 𝛾𝛾) = 0
(15)
A solution for which this
is zero
!"#! −
!!!
A solution for which this necessary condition
− 𝑟𝑟(14)
− 𝛾𝛾is zero
𝑑𝑑𝑑𝑑 equals:
+ 𝜆𝜆
*
This
relatesforthe
optimal
as:𝜆𝜆(𝑟𝑟 − 𝛾𝛾) = 0
A
solution
which
this
necessary
is zero
2𝐼𝐼!"stack
𝑅𝑅!" −current
2 𝑟𝑟 −condition
𝛾𝛾IFC[𝐼𝐼!to+Lagrange
𝐼𝐼(14)
𝑟𝑟 multiplier
− 𝛾𝛾equals:
𝐼𝐼!" ]𝑅𝑅λ! −
!"#! −
!!!
(15)
!
This
relatesforthe
optimal
as:
𝑟𝑟𝜆𝜆(𝑟𝑟
− 𝛾𝛾− 𝛾𝛾) = 0
𝑟𝑟condition
−
𝛾𝛾[𝐼𝐼*!𝑅𝑅to!+Lagrange
2𝐼𝐼∗!"stack
𝑅𝑅!" −current
2 𝑟𝑟 −
𝛾𝛾IFC
𝐼𝐼(14)
𝑟𝑟 multiplier
− 𝛾𝛾equals:
𝐼𝐼!" ]𝑅𝑅λ! −
(15)
!
A solution
which
this
is+zero
!"#! −
(16)
𝐼𝐼!" necessary
𝜆𝜆 =
𝐼𝐼
𝐼𝐼
+
𝜆𝜆
!
!"#!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
This relates the optimal
stack current
𝑟𝑟𝜆𝜆(𝑟𝑟
− 𝛾𝛾− 𝛾𝛾) = 0
𝑟𝑟 −IFC
𝛾𝛾 * 𝑅𝑅to Lagrange multiplier λ !!as:
2𝐼𝐼
(15)
∗!" 𝑅𝑅!" − 2 𝑟𝑟 − 𝛾𝛾 [𝐼𝐼! !+ 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅! −
(16)
𝐼𝐼
𝜆𝜆
=
𝐼𝐼! + 𝐼𝐼(16)
+
𝜆𝜆
!"
!"#! provides
! 𝑅𝑅relation
Combining both constraints (9)𝑅𝑅and
(10)
with
an
analytic
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾 ! 𝑅𝑅! solution for λ:
!"
!
!"
This relates the optimal stack current IFC* to Lagrange multiplier λ! as:
𝑟𝑟 − 𝛾𝛾
𝑟𝑟 − 𝛾𝛾 * 𝑅𝑅! Lagrange multiplier λ !as:
This relates the optimal
∗ stack current IFC to
(16)
𝐼𝐼
𝜆𝜆
=
𝐼𝐼! + 𝐼𝐼(16)
+
𝜆𝜆
!!
!"
!"#! provides
! 𝑅𝑅relation
Combining both constraints (9)𝑅𝑅and
(10)
with
an
analytic
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
+
𝑟𝑟 −
𝛾𝛾 ! 𝑅𝑅! solution for λ:
!"
!
!"
∗
(17)
− 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!"
(𝜆𝜆) 𝑑𝑑𝑑𝑑 =! 0𝑟𝑟 − 𝛾𝛾
! +
𝑟𝑟𝐼𝐼−
𝛾𝛾 𝐼𝐼𝑅𝑅!"#!
!
!
∗
(16)
!
𝐼𝐼
𝜆𝜆
=
𝐼𝐼
+
𝐼𝐼
+
𝜆𝜆
!
!"
!
!"#! provides an analytic
Combining both constraints
(9)𝑅𝑅and
(16)
! relation
!!! (10) with
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! solution for λ:
!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
∗
(17)
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!"
(𝜆𝜆) 𝑑𝑑𝑑𝑑 = 0
!!
Subsequently:
Combining both constraints (9) and
!!! (10) with relation (16) provides an analytic solution for λ:
∗
(17)
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!"
(𝜆𝜆) 𝑑𝑑𝑑𝑑 = 0
Subsequently:!!
!
!
!!!!
𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
! 𝑟𝑟 − 𝛾𝛾
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 [
𝐼𝐼! +
𝜆𝜆] 𝑑𝑑𝑑𝑑
∗ 𝐼𝐼!"#! +
(17)
!
(18)
𝐼𝐼
+
𝐼𝐼
−
𝑟𝑟
−
𝛾𝛾
𝐼𝐼
(𝜆𝜆) 𝑑𝑑𝑑𝑑
=
0
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
𝑅𝑅
+
𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
!
!"#!
!"
!"
!
!"
!
Subsequently:
!!!!
!
!!!
𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
! 𝑟𝑟 − 𝛾𝛾
𝐼𝐼! + 𝐼𝐼!"#! − =
𝑟𝑟 −0 𝛾𝛾 [
𝐼𝐼! + 𝐼𝐼!"#! +
𝜆𝜆] 𝑑𝑑𝑑𝑑
! 𝑅𝑅
(18)
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
+
𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
!"
!
!"
!
15
*
This relates the optimal
stack
as:𝜆𝜆(𝑟𝑟 − 𝛾𝛾) = 0
2𝐼𝐼
𝑅𝑅!" −current
2 𝑟𝑟 − 𝛾𝛾IFC[𝐼𝐼!to+Lagrange
𝐼𝐼!"#! − 𝑟𝑟 multiplier
− 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅λ! −
2𝐼𝐼!"
!" 𝑅𝑅!" − 2 𝑟𝑟 − 𝛾𝛾 [𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" ]𝑅𝑅! − 𝜆𝜆(𝑟𝑟 − 𝛾𝛾) = 0
(15)
(15)
𝑟𝑟 − 𝛾𝛾
𝑟𝑟 −I 𝛾𝛾 ** 𝑅𝑅to
This
stack current
multiplier
λλ !!as:
! Lagrange
∗ stack current IFC
This relates
relates the
the optimal
optimal
Lagrange
multiplier
as:
FC to
(16)
𝐼𝐼!"
𝜆𝜆 =
𝐼𝐼
+
𝐼𝐼
+
𝜆𝜆
!
!"#!
!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
!
−
!
𝑟𝑟𝑟𝑟 −
! 𝑟𝑟
∗
− 𝛾𝛾
𝛾𝛾
− 𝛾𝛾
𝛾𝛾 𝑅𝑅
𝑅𝑅!!
! 𝑟𝑟an
𝐼𝐼
𝜆𝜆
=
𝐼𝐼
+
𝐼𝐼
+
𝜆𝜆
∗
! + 𝐼𝐼(16)
!"#!
Combining
(9)
and
(10)
with
relation
(16)
provides
analytic
solutionfor
forλλ:
(16)
Combining both
both constraints
constraints
(10)
with
relation
provides
an
analytic
: (16)
!
!
𝐼𝐼!"
𝜆𝜆
=
𝐼𝐼
+
𝜆𝜆solution
!"
!
!"#!
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
𝑅𝑅
+
𝑟𝑟
−
!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾
𝛾𝛾 ! 𝑅𝑅
𝑅𝑅!
!"
!
!"
Subsequently:
Subsequently:!
!
Subsequently:
Subsequently:
!!
!!!
!!
and
and:
and
and
!!!
!!!
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 [
𝑅𝑅!"
=0
𝐼𝐼𝐼𝐼! +
𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 [
! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 [𝑅𝑅!"
𝑅𝑅!"
!!!
!!!
=
0
=0
!!
!!
!!!
!!
result in:
∗
𝐼𝐼𝐼𝐼! +
𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!"
∗ (𝜆𝜆) 𝑑𝑑𝑑𝑑 = 0
! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" (𝜆𝜆) 𝑑𝑑𝑑𝑑 = 0
!!!
!!!
𝑅𝑅!"
𝑅𝑅
𝑅𝑅!"
!"
!
!
𝑟𝑟 − 𝛾𝛾
𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
𝐼𝐼! + 𝐼𝐼!"#! +
𝜆𝜆] 𝑑𝑑𝑑𝑑
!
+ 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
!
− 𝛾𝛾
!
𝑟𝑟𝑟𝑟 −
! 𝑟𝑟
− 𝛾𝛾
𝛾𝛾 𝑅𝑅
𝑅𝑅!!
! 𝑟𝑟 − 𝛾𝛾
𝐼𝐼
+
𝐼𝐼
+
𝜆𝜆] 𝑑𝑑𝑑𝑑
! + 𝐼𝐼!"#! +
!
𝐼𝐼
!"#!
+
𝑅𝑅
+ 𝑟𝑟 − 𝛾𝛾 !! 𝑅𝑅! 𝜆𝜆] 𝑑𝑑𝑑𝑑
+ 𝑟𝑟𝑟𝑟 −
− 𝛾𝛾
𝛾𝛾 ! 𝑅𝑅
𝑅𝑅!! !
𝑅𝑅!"
!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
!
!
𝑅𝑅!"
! 𝑟𝑟 − 𝛾𝛾
𝐼𝐼
+
𝐼𝐼
−
𝜆𝜆 𝑑𝑑𝑑𝑑 = 0
!"#!
+ 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! !
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
!
!
!
𝑅𝑅
! 𝑟𝑟
𝑟𝑟 −
− 𝛾𝛾
𝛾𝛾 !
𝑅𝑅!"
!"
!
𝐼𝐼
+
𝐼𝐼
−
𝜆𝜆
!
𝐼𝐼! + 𝐼𝐼!"#!
𝜆𝜆 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑 =
=0
0
!"#! − 𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 !
+
+ 𝑟𝑟𝑟𝑟 −
− 𝛾𝛾
𝛾𝛾 ! 𝑅𝑅
𝑅𝑅!! !
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅
𝑅𝑅!!
!!
!!!
!
!
!
their!
!!!
!!
𝑟𝑟 − 𝛾𝛾 ! 𝜆𝜆 𝑑𝑑𝑑𝑑 =
!!
!!
!!!
𝑅𝑅!" (𝐼𝐼! + 𝐼𝐼!"#! ) 𝑑𝑑𝑑𝑑
∗
𝐼𝐼!"
∗
𝐼𝐼!"
=
=
!
𝑟𝑟 − 𝛾𝛾1 𝑅𝑅!
𝑟𝑟 − 𝛾𝛾!!!
𝑅𝑅!𝑅𝑅!"
∗
!
𝐼𝐼
+
𝐼𝐼
+
𝐼𝐼!"
=
𝐼𝐼
+
𝐼𝐼
+
𝐼𝐼! − 𝐼𝐼𝐼𝐼!! + 𝐼𝐼!"#!
!
!"#!
!
!"#!
!
!!"
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
!!!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾! 𝑅𝑅!
𝑅𝑅!"𝑟𝑟 +
𝑅𝑅!" +
− 𝛾𝛾𝑟𝑟𝑅𝑅−
𝑅𝑅𝛾𝛾! ! 𝑅𝑅
! 𝐼𝐼 + 𝐼𝐼
𝐼𝐼
+
𝐼𝐼
+
!
!"#!
!"#!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! !
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
1
!
∗
solution
the+optimal stack current
of a
𝐼𝐼!"
= (23) reveals
𝐼𝐼 + 𝐼𝐼!"#!
𝐼𝐼 consists
− 𝐼𝐼!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! !
𝑟𝑟 − 𝛾𝛾 !
!
𝑟𝑟𝑟𝑟 −
11
∗
− 𝛾𝛾𝛾𝛾 𝑅𝑅
𝑅𝑅!!
𝐼𝐼𝐼𝐼!"
𝐼𝐼𝐼𝐼! +
𝐼𝐼𝐼𝐼!"#! +
𝐼𝐼 − 𝐼𝐼
∗ =
=
+
+
𝑅𝑅
+ 𝑟𝑟 − 𝛾𝛾 !!𝑅𝑅! 𝐼𝐼!! − 𝐼𝐼!!
𝑟𝑟𝑟𝑟 −
!"
!"#!
𝑅𝑅!"
− 𝛾𝛾𝛾𝛾 !
!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
(23)
(23)
(18)
∗
𝐼𝐼𝐼𝐼!"
∗ = ! 𝐼𝐼! + 𝐼𝐼!"#! + ! 𝛷𝛷∆𝐼𝐼!
𝐼𝐼 + 𝐼𝐼!"#! + !!!
𝛷𝛷∆𝐼𝐼!
!" = !!!
!!! !
!!!
deviations in the current demand, results in the next expression for optimal control:
!
(18)
(18)
(19)
(19)
(20)
!
(24)
(24)
(24)
with
with
with
!
𝑟𝑟𝑟𝑟 −
− 𝛾𝛾𝛾𝛾 !𝑅𝑅
𝑅𝑅!!
𝛷𝛷
=
𝛷𝛷 = 𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 !!𝑅𝑅!
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 𝑅𝑅!
(19)
(20)
𝑟𝑟 − 𝛾𝛾 ! 𝜆𝜆 value.
𝑑𝑑𝑑𝑑 = This
𝑅𝑅!"reduces
(𝐼𝐼! + 𝐼𝐼!"#!
average
the) 𝑑𝑑𝑑𝑑
need for a-priori information on
In (20), λ and Id relate only by
!!
the driving cycle to only the future average current!!!
demand. The resulting optimal value for a constant
!! !
!!
!
In λ(20),
λ and I relate only by their average
need for a-priori information
(20)on the
𝑟𝑟 − 𝛾𝛾 value.
𝜆𝜆 value.
𝑑𝑑𝑑𝑑 =This
𝑅𝑅reduces
(𝐼𝐼! + the
𝐼𝐼!"#!
!"reduces
This
the) 𝑑𝑑𝑑𝑑
need for a-priori information on
Inbecomes:
(20), λ andd Id relate only by their!! average
!
(20) λ
driving cycle to only the future average
current
demand.
The
resulting
optimal
value
for
a
constant
𝑟𝑟
−
𝛾𝛾
𝜆𝜆
𝑑𝑑𝑑𝑑
=
𝑅𝑅
(𝐼𝐼
+
𝐼𝐼
)
𝑑𝑑𝑑𝑑
!!! !average current
!!!
!" ! The
!"#!
the driving cycle to only the future
demand.
resulting optimal value for a constant
1
becomes:
!!!
!!!
λ becomes:
=2
𝑅𝑅 (𝐼𝐼! +reduces
𝐼𝐼!"#! ) =the
0 need for a-priori information
(21) on
In (20), λ and Id relate only by their𝜆𝜆 average
𝑟𝑟 − value.
𝛾𝛾 ! !"This
λ and
Id relate
theiraverage
averagecurrent
value. This
reduces
need for
a-priori
information
on
In (20),
the
driving
cycle
to onlyonly
theby
future
demand.
The the
resulting
optimal
value
for a constant
1
*
the
driving
cycle
to
only
the
future
average
current
demand.
The
resulting
optimal
value
for
a
constant
Combining
this
solution
(21)
for
λ
with
(15)
results
in
the
optimal
control
variable
I
as
analytic
λ becomes:
𝜆𝜆 = 2
𝑅𝑅 (𝐼𝐼 + 𝐼𝐼!"#! ) = 0
(21)
FC
𝑟𝑟 − 𝛾𝛾 ! !" !
λ becomes:
solution
to the optimization problem:
1
Combining this solution (21) for λ 𝜆𝜆with
variable IFC* as analytic
= 2(15) 1results
𝑅𝑅 in(𝐼𝐼the
+ optimal
𝐼𝐼!"#!! ) =control
0
(21)
! !" !
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
!"
!!!
! 2
solution to the optimization
problem:
𝜆𝜆 =
𝑅𝑅
(𝐼𝐼
+
𝐼𝐼
)
=
0
(21)
∗
!" +
!
!"#!
(22)
𝐼𝐼!" =
𝐼𝐼! +
𝐼𝐼! + 𝐼𝐼!"#!
𝛾𝛾 𝐼𝐼!!"#!
! 𝑅𝑅 𝑟𝑟 −
! 𝑅𝑅
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
!"
!
!"
!
Combining this solution (21) for λ with (15) results in the optimal control variable IFC** as analytic solutiCombining this solution (21) for λ with (15) results in the optimal
control variable IFC as analytic
!
onCombining
to the optimization
problem:
𝑅𝑅 control variable I * as analytic
𝑟𝑟for
− λ𝛾𝛾 with
𝑅𝑅! (15) results in the optimal
!!! !"
this
solution
(21)
∗
solution
to
the
optimization
problem:
(22)
or
𝐼𝐼!" =
𝐼𝐼 + 𝐼𝐼!"#! +
𝐼𝐼 + 𝐼𝐼!"#!FC
𝑅𝑅!" problem:
+ 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! !
𝑅𝑅!" + 𝑟𝑟 − 𝛾𝛾 ! 𝑅𝑅! !
solution to the optimization
or
!"
(17)
(17)
result
in:
result
result in:
in:
!!
or
or
!
A closer observation of solution (23) reveals the optimal stack current consists of a constant part, reA
observation of
(23)
reveals
the
current consists
of
part,
A closer
closer
of solution
solution
(23)and
reveals
the optimal
optimal stack
stack
of aa constant
constant
part,
lated
to theobservation
average current
demand,
an instantaneous
part,current
relatedconsists
to the deviations
in the
derelated
to
the
average
current
demand,
and
an
instantaneous
part,
related
to
the
deviations
in
the
related
to
the
average
current
demand,
and
an
instantaneous
part,
related
to
the
deviations
in
the
manded current. This instantaneous part is split over fuel cell stack and battery, according their internal
demanded
current.
This
part is
over fuel
cell
stack
and
according their
demanded
current.
This instantaneous
instantaneous
is split
split
celland
stack
and battery,
battery,
their in
resistances
(23).
Expressing
this constant part
power
splitover
ratiofuel
as Φ,
introducing
ΔId according
as the deviations
Φ
,
and
introducing
ΔI
internal
resistances
(23).
Expressing
this
constant
power
split
ratio
as
d as the
Φ
,
and
introducing
ΔI
as
the
internal
resistances
(23).
Expressing
this
constant
power
split
ratio
as
d
the
current demand, results in the next expression for optimal control:
deviations in the current demand, results in the next expression for optimal control:
!
!!
Combining
Combining both
both constraints
constraints (9)
(9) and
and (10)
(10) with
with relation
relation (16)
(16) provides
provides an
an analytic
analytic solution
solution for
for λλ::
∗
(17)
𝐼𝐼! + 𝐼𝐼!"#! − 𝑟𝑟 − 𝛾𝛾 𝐼𝐼!" (𝜆𝜆) 𝑑𝑑𝑑𝑑 = 0
!!
!!!
!!
!"
5
(25)
(25) (25)
IMPLEMENTATION
IMPLEMENTATION
5.5 Implementation
The
constant
part
(24)
considered
the
point
operation
for
the
fuel
cell
stack,
from
which
the
The
constant
part
ofof
(24)
is is
considered
the
point
The
constant
part
of
(24)
is
considered
the
pointofof
ofoperation
operationfor
forthe
thefuel
fuelcell
cellstack,
stack,from
from which
which the
the fuel
fuel
cell
stack
current
changes
according
the
variations
in
the
current
demand
and
the
value
of
the
power
cell
stack
current
changes
according
the
variations
in
the
current
demand
and
the
value
of
the
power
cell stack current changes according the variations in the current demand and the value of the power
split
ratio.
When,
over
significant
time
horizon,
the
SOC
the
battery
decreases
increases,
the
split
ratio.
When,
ofof
the
oror
the
split
ratio.
When,over
overa aasignificant
significanttime
timehorizon,
horizon,the
theSOC
SOC
of
thebattery
batterydecreases
decreases
orincreases,
increases,
the
point
of
operation
has
been
too
low
or
too
high
respectively.
Essentially,
the
average
future
current
point
of
operation
has
been
too
low
or
too
high
respectively.
Essentially,
the
average
future
current
point of operation has been too low or too high respectively. Essentially, the average future current
demand
predicted
the
average
past
current
demand.
Therefore,
the
point
operation
of
the
fuel
demand
is is
predicted
byby
the
average
past
current
demand.
Therefore,
the
demand
is
predicted
by
the
average
past
current
demand.
Therefore,
thepoint
pointofof
ofoperation
operationof
ofthe
the fuel
fuel
cell
stack
can
be
considered
the
result
of
a
feedback
loop
on
the
SOC
of
the
battery.
This
results
in
cell
stack
can
be
considered
the
result
of
a
feedback
loop
on
the
SOC
of
the
battery.
This
results
in
the
cell stack can be considered the result of a feedback loop on the SOC of the battery. This results in
the
implementation
as
in
3.
comprises
aa slow
feedback
real-time
implementation
as indicated
in figure
3. This
implementation
comprises
a slow
feedback
loop,
the real-time
real-time
implementation
as indicated
indicated
in figure
figure
3. This
This implementation
implementation
comprises
slow
feedback
loop,
providing
robustness
on
the
SOC,
and
a
fast
feed
forward
path,
enabling
a
minimal
fuel
providing
robustness
on
the
SOC,
and
a
fast
feed
forward
path,
enabling
a
minimal
fuel
consumption.
loop, providing robustness on the SOC, and a fast feed forward path, enabling a minimal fuel
consumption.
consumption.
r 
Î d  I aux0
-
SOCref
+
ΔSOC
-
+
K
controller
Î d
1

r 
bias
Î FC 0
+
I d  I aux 0
+
feed
forward
+ Î FC Î *
FC Propulsion
system
+
min J
SOC
feedback
Figure 3. Fuel consumption minimizing EMS as combined feed forward path and feedback control loop.
(22)
(23)
(22)
or
A closer observation of
constant part,
(23)
or
related to the average current demand, and an instantaneous part, related to the deviations in the
ordemanded current. This instantaneous
− 𝛾𝛾 cell
𝑅𝑅 stack and battery, according their
1
part isthe
split
over𝑟𝑟 fuel
∗
A closer observation of solution
reveals
stack !current
of a constant part,
(23)
𝐼𝐼!"
= (23)
𝐼𝐼! + 𝐼𝐼!"#!
+optimal
𝐼𝐼 consists
− 𝐼𝐼!
! 𝑅𝑅 as! Φ, and
𝑟𝑟
−
𝛾𝛾
𝑅𝑅𝛾𝛾!ratio
1
𝑅𝑅
+
𝑟𝑟
−
𝑟𝑟
−
𝛾𝛾
introducing
ΔIdin
asthe
the
internal
resistances
(23).
Expressing
this
constant
power
split
∗
!"
!
related to the average current
demand,
and
an
instantaneous
part,
related
to
the
deviations
(23)
𝐼𝐼!" =
𝐼𝐼! + 𝐼𝐼!"#! +
𝐼𝐼! − 𝐼𝐼!
!
𝑅𝑅
+
𝑟𝑟
−
𝛾𝛾
𝑅𝑅
𝑟𝑟
−
𝛾𝛾
!"
! optimal control:
deviations
the current
demand, results
in the
next
expression
demanded in
current.
This instantaneous
part
is split
over
fuel cell for
stack
and battery, according their
A closer observation of solution (23) reveals the optimal stack current consists of a constant part,
Φ, and introducing
ΔId part,
as the
internal
resistances
(23).
Expressing
this
constant
power
split
ratio
as
A
closertoobservation
solutiondemand,
(23) reveals
theinstantaneous
optimal stack part,
current
consists
of deviations
a constant
related
the averageofcurrent
and an
related
to the
in the
deviations
in the
current
demand,
resultsand
in
the
next expression
for related
optimal to
control:
! is
related
to the
average
current
demand,
an
instantaneous
part,
the deviations
the
∗
demanded
current.
This
instantaneous
over +
fuel! cell
stack and battery,
accordingin
their
=part
𝐼𝐼 split
+
𝐼𝐼!"#!
𝛷𝛷∆𝐼𝐼
𝐼𝐼!"
(24)
!
!!! !
!!!
demanded
current.
This
instantaneous
part
is
split
over
fuel
cell
stack
and
battery,
according
their
internal resistances (23). Expressing this constant power split ratio as Φ, and introducing ΔId as the
16 internal resistances (23). Expressing this constant power split ratio as Φ, and introducing ΔI as the
d
deviations in the current demand, results
in
for optimal control:
! the next expression
!
∗
= !!!
𝐼𝐼! +next
𝐼𝐼!"#!
+ !!!𝛷𝛷∆𝐼𝐼for
𝐼𝐼!"
(24)
! optimal control:
deviations
in
the
current
demand,
results
in
the
expression
with
The rated maximum and minimum currents of both the fuel cell stack and battery are not considered
as bounds in the optimization. Still, it can be proven that the specified bound is the optimum choice
when such bound is exceeded. As consequence, the point of operation of the fuel cell stack has to be
corrected, but thanks to the feedback loop on the SOC, this issue is covered such that optimality is still
achieved.
17
6. Validation
The proposed EMS is validated through simulations and measurements. For simulations, validated models of an existing distribution truck are used (figure 4a). Measurements are taken on a 10 kW fuel cell
stack test facility (figure 4b).
The accuracy of the test facility and repeatability of experiments is such that conclusions on the fuel
consumption can be drawn with a significance of 0.1%. This high accuracy is obtained thanks to the very
predictable performance of the fuel cell system and the use of an SOC model running in parallel with
the actual battery.
It is concluded that AS provides a real-time strategy that performs slightly better compared to ECMS and
closely approaches the benchmark for the possible minimum in fuel consumption. The complexity of AS
is comparable to RE. Both methods are much less complex compared to ECMS and DP.
Figure 4. Fuel cell hybrid distribution truck (a) and fuel cell test facility (b).
For comparison, also other EMSs are evaluated, such as the Range Extender approach (RE), ECMS and
off-line Dynamic Programming (DP). RE refers to a constant fuel cell stack current. DP is based on full
available knowledge on the future driving cycle. Therefore, its results are considered a benchmark for
the smallest possible fuel consumption. Results for the proposed Analytic Solution are referred to as
AS. Results for simulations over three driving cycles [13] are presented in table 1. Real-time results from
experiments at the test facility are presented in table 2.
Table 1 Fuel consumption, simulated for a distribution truck with different driving cycles and EMSs.
RE
ECMS
AS
DP
NEDC Low Power
108.3%
103.2%
100.2%
100.0%
JE05
107.0%
102,5%
100.1%
100.0%
FTP75
110,2%
105,6%
101.5%
100.0%
Driving cycle
Table 2 Fuel consumption, measured at the test facility for two driving cycles and three EMSs.
RE
ECMS
AS
JE05
106.1%
100.3%
100.0%
FTP75
109.2%
100.8%
100.0%
Driving cycle
18
Figure 5. Fuel cell hybrid test and demonstration vehicle.
The Analytic Solution as EMS is demonstrated on-road using the fuel cell hybrid vehicle of figure 5. This
EMS in this vehicle operated over 1,5 years with different drivers with a 100% availability.
7. Conclusions
The proposed minimization of losses provides an analytic solution to the energy management problem
of a fuel cell hybrid propulsion system. The resulting EMS enables a minimum in fuel consumption, close
to the global minimum. The assumption that the future average current demand is predicted by the past
average current demand enables real-time implementation. In simulations and experiments, the proposed real-time EMS results in a fuel consumption within 2% of the minimum fuel consumption calculated
as benchmark off-line using dynamic programming.
The proposed EMS has some additional benefits. Thanks to the comparable first order behavior of fuel
cells and battery cells, the resulting analytic solution is simple and robust against parameter variations
and disturbances. Apart from these benefits, an analytical approach provides a more fundamental understanding of the energy management problem of fuel cell hybrid propulsion systems.
19
References
[1]Gao D., Jin Z., Lu Q., Energy management strategy based on fuzzy logic for a fuel cell hybrid bus,
J. of Power Sources, 2008
[2]Feroldi D., Serra M., Riera J., Energy Management Strategies based on efficiency map for Fuel Cell
Hybrid Vehicles, J. of Power Sources, 2009
[3]Kim M., Peng H., Power management and design optimization of fuel cell/battery hybrid vehicles,
J. of Power Sources, 2007
[4]Lin C., Kang J., Grizzle J.W., Peng H., Energy Management Strategy for a Parallel Hybrid Electric
Truck, proc. American Control Conf., 2001
[5]Johannesson L., Egardt B., Approximate Dynamic Programming Applied to Parallel Hybrid Powertrains, proc. Int. Fed. of Automatic Control, 2008
[6]Koot M., Kessel J.T.B.A., de Jager B., Heemels W.P.M.H., van den Bosch P.P.J., Steinbuch M., Energy
Management Strategies for Vehicular Electric Power Systems, IEEE trans. Vehicular Technology,
2005
[7]Kessels J.T.B.A., Koot M.W.T., van den Bosch P.P.J., Kok D.B., Online Energy Management for Hybrid
Electric Vehicles, IEEE trans. on Vehicular Technology, 2008
[8]Hofman T., Steinbuch M., van Druten R., Serrarens A., Rule-based energy management strategies
for hybrid vehicles, int. J. Electric and Hybrid Vehicles, 2007
[9]Tazelaar E., Veenhuizen P.A., van den Bosch P.P.J., Analytical solution of the energy management
for fuel cell hybrid propulsion systems, IEEE trans. Vehicular Technology, 2012
[10]Tazelaar E., Veenhuizen P.A., Jagerman J., Faassen T., Energy Management Strategies for Fuel Cell
Hybrid Vehicles; an Overview, proc. EVS27, Barcelona, Spain, 2013
[11] Kirk D.E., Optimal Control Theory; An Introduction Dover Publications, 1988
[12] Bryson A.E., Dynamic optimization, Addison Wesley Longman, 1999
[13] Dieselnet, www.dieselnet.nl December, 2010
[14]Tazelaar E., Energy Management and Sizing of Fuel Cell Hybrid Propulsion Systems, PhD thesis,
Eindhoven University of Technology, 2013
[15]Tazelaar E., Veenhuizen P.A., van den Bosch P.P.J., Grimminck M., Analytical solution and experimental validation of the energy management problem for fuel cell hybrid vehicles, proc. EEVC,
Brussels, Belgium, 2011
20
Distribution of the power for traction
for fixed gear electric propulsion systems
Edwin Tazelaar
HAN University of Applied Sciences, Arnhem, The Netherlands
Abstract
Most studies on power train design rely on deterministic driving cycles to define the vehicles longitudinal
speed. Especially simulations on hybrid propulsion systems use driving cycles to define the speed sequence of the vehicle and backwards calculate the power for traction. Disadvantages of this deterministic
approach are the limited value of one driving cycle to represent real-life conditions and the risk of ’cycle
beating’ in optimizations.
Observations suggest that the distribution of the power for traction is more easily characterized than
the distribution of the speed, as it tends to a bell-shaped curve. This study proposes to approximate this
bell-shaped distribution with a normal distribution for electric propulsion systems with a fixed gear ratio.
This proposal is motivated from simulations, chassis dynamometer experiments and real-world data.*)
1. Introduction
As hybrid power trains comprise a storage, it is not sufficient to size the power train components on
specifications as acceleration time and top speed only. Therefore, many design studies on hybrid power
trains use driving cycles -time sequences of speed samples (m/s)- as input to simulations that define
the required power for traction (W) [1, 2]. Such simulations are used for component sizing [2-6] and
the design of energy management strategies [7-12]. For different vehicle classes and purposes, general
accepted driving cycles are available. Examples are the NEDC for passenger cars, the FTP75 and JE05 for
light and heavy duty vehicles in an urban environment and the Braunschweig, NYbus and Beijing Bus
cycle for buses [13, 14]. As most of these cycles were initially developed for emission tests, initiatives as
the LA92/UCDS as successor of the FTP75 [13], the ARTEMIS project [15, 16] and others [17] intend to
provide cycles more suitable to modern requirements as a minimum fuel consumption. These driving
cycles provide useful deterministic requirement definitions for several applications, including hybridized propulsion systems. Still, the cycle chosen will never be driven in real life [18]. They fall short when
designing actual power trains. Therefore, sizing the system components and engineering the vehicles’
energy management system need a better and richer cycle definition for specific vehicle and traffic circumstances, less depending on a deterministic series of speed points [19, 20]. For energy management
systems, one approach to reduce this cycle dependency is to include an online cycle prediction. Such a
prediction can be based on statistics and historic data [21], on GPS and navigation data [21, 22] or on
dynamic traffic routing information [23, 24]. Prediction can help to increase the performance and robustness of the energy management system, but it does not support the sizing of the components in the
design phase. For sizing, a characterization less depending on a predefined speed sequence is needed.
Considering characterization for design, techniques are proposed as fuzzy logic and neural networks
[18, 25, 30], time series analysis [26] or statistics based methods as principle component analysis (PCA)
[27]. In [28], a method is presented that generates random driving cycles with statistical and stochastic
properties similar to a driving cycle provided as ’seed’ to the driving cycle generator. This reduces the
risk of ’cycle beating’ in design optimization. In a broader context, characterizations for other purposes
*)This paper contains parts and partly summarizes [33] and [36].
21
are presented, such as the characterization of driving styles [29, 30]. The methods and approaches discussed consider a driving cycle as a sequence of speed samples over time (m/s). Still, propulsion systems
have to provide the power for traction (W). Therefore, this study explores if a characterization of driving
cycles in terms of power for traction is an attractive alternative.
Objective of this document is to motivate that the statistical distribution of the power for traction for an
electric propelled vehicle, driven by a human, tends to one bell-shaped distribution. Such power distribution can cover a complete class of driving cycles. In addition, the paper proposes to approximate this
distribution with a normal distribution. The parameters of such distribution are linked to vehicle parameters and some key characteristics of a driving cycle representing the traffic environment. The paper
provides this relation using a general vehicle model.
2. Observations
2.1 First observations
Experiments and simulations with different vehicles, different driving cycles and different conditions
resulted in substantial data of speed and power for traction. This data has been obtained from a delivery van (Fiat Doblo), a mid-size distribution truck (Hytruck) [6] and an articulated trolley bus. Although
different, these vehicles have an electric propulsion system with one fixed gear in common.
Figure 1. Measured speed and power distribution delivery van.
From these experiments, three examples are presented, with their speed profile, and statistical distributions of speed and power for traction:
• Figure 1 shows a trip of the delivery van through a suburban traffic environment, where both
speed and power for traction are measured through the logging system on the vehicle. This trip
has a duration of 3165 seconds, an average speed of 43.1 km/h and a maximum speed of 82.0
km/h
• Figure 2 shows a driving cycle of the articulated trolley bus in an urban area, where the speed is
measured and the power for traction is derived through a vehicle model. The duration of this driving cycle is 4337 seconds, with an average speed of 25.8 km/h and a top speed of 81.9 km/h.
• Figure 3 presents simulation results where the JE05 standard is used as driving cycle, representing
a mix of urban and suburban traffic conditions as traffic environment [13]. The power distribution
is derived through a vehicle model of the midsize distribution truck, validated on component level
[6]. The JE05 driving cycle has a duration of 1829 seconds with an average speed of 27.3 km/h and
a top speed of 87.6 km/h.
In spite of the differences in vehicle class and the significant differences in speed distributions, the
distributions of the power for traction show some resemblance. All experiments provide bell-shaped
distributions for the power for traction.
For comparison, the normal distributions based on mean and variance of the considered data are included as curve in the graphs. But as the experiments cover a relative short time, resulting in an average
number of samples per bin less than 70, the significance of these observations is not sufficient to draw
final conclusions on the shape of the distributions. Therefore, experiments resulting in more data samples were initiated.
22
Figure 2. Measured speed and simulated power distribution articulated trolley bus.
23
Figure 3. Predefined speed and simulated power distribution mid-size truck.
Figure 4. Long-term speed and power distribution delivery van.
2.2 Long-term measurements
To cover the short observation time in the previous examples, an additional experiment was conducted,
where measurements of both speed and power for the delivery van were logged over several days of
operation. The resulting distributions are presented in figure 4 with the standstill/idling
times skipped from the data.
2.3 Chassis dynamometer results
To verify if such a bell-shape distribution also holds for predefined styled driving cycles as the NEDC, a
comparison is made between the NEDC as simulated driving cycle and the NEDC tested on a roller test
bench using the delivery van [31]. The NEDC Low Power version is used to reflect the capabilities of the
vehicle considered. Figure 5 presents the results for simulation and figure 6 presents the results derived
from the roller test bench.
The experiment includes rides in the city, the suburbs and the countryside. Where the distributions presented in the previous examples refer to flat terrain, the distributions of figure 4 include trips in some
more elevated terrain.
This experiment, including over six hours of driving, supports the suggestion that, over a longer time horizon, the distribution of the power for traction is bell-shaped, with as exception a spike at zero power.
Figure 5. Distributions for the NEDC Low Power, derived from simulations.
24
25
3. Motivation for a normal distribution
Both in measurements, roller test bench experiments and simulations and for different vehicles, the
distributions of the power for traction show bell-shaped curves, especially when driving cycles with a
significant duration are evaluated. When considering sizing the components of a propulsion system, a
much longer time is relevant: the lifetime of the vehicle. Extending the horizon of observation to the
lifetime of the vehicle, it is stated that the distribution of the power for traction is bell-shaped, with a
peak around zero due to idling. Steady power consumption by auxiliaries or electric heating would shift
this peak to non-zero values. With respect to the shape of the distribution, it is postulated that:
Over the lifetime of the vehicle, the distribution of the power for traction
is sufficiently accurate approximated by a normal distribution.
A peak at zero to represent idling might be included when convenient. Coasting will not introduce a
peak as will be explained later.
Figure 6. Distributions for the NEDC Low Power, derived from chassis dynamometer results.
Clearly, the simulated case does not result in a bell-shaped distribution, as the number of samples is limited and as the NEDC Low Power driving cycle is artificially constructed. The measured speed distribution
resembles the original NEDC Low Power cycle: the dominant velocities are still clearly visible. However, the
power distribution is much more blurred, suggesting a tendency towards a more bell-shaped distribution.
To evaluate the difference between a real-world driving cycle and an experiment on the roller test
bench, the measured driving cycle of figure 1 is replayed on the chassis dynamometer. The results are
presented in figure 7. Except for a spike around 8 kW, both speed and power distribution resemble the
results of figure 1. This indicates the roller test bench is useful to represent real-life conditions with respect to speed and power distributions.
To provide some support for the hypothesis for a normal distributed power for traction, a first qualitative motivation is presented in this chapter. This motivation is based on a combination of driver behavior
and vehicle properties, as indicated in figure 8.
traffic
disturbances
human
driver
vref
+
driver
-
controller
P
vehicle
v
process
information
Figure 8. The power for traction as the result of a control loop with the driver as controller.
Figure 8 presents a model of the driver-vehicle interaction with respect to the vehicle speed, based on
[32]. Using the information of the surrounding traffic environment, the driver continuously adapts the
virtual setpoint vref for the desired speed of his vehicle. Observing the actual speed v, of the own vehicle, the driver acts as controller by changing the power for traction P through the accelerator pedal, to
reduce the difference between actual speed v and desired speed vref. At it is the same driver that has to
process all information, both the setpoint and the control loop update at a comparable rate.
The vehicle itself is a dynamic (nonlinear) process with a low pass behavior. To obtain the desired output, the driver acts as controller. As the rate of the driver with which he can process information is limited in terms of bandwidth, also the overall response from desired speed vref to the power for traction
P is low pass. This is supported by literature, proposing to approximate human control loops with a first
order transfer function and delay [32, 34, 35].
Figure 7. Distributions for real-world data, derived from chassis dynamometer results.
26
The observation that the power for traction is the result of a low pass dynamic system supports the
hypothesis of a normal distribution, as low pass systems provide normal distributed outputs on arbitrary
distributed random inputs.
27
Still, the motivation for a normal distribution is an observation supported by arguments, rather than a
proof. The vehicle itself is a nonlinear dynamic process. This hinders a mathematical proof: Unlike linear
input-output relations, nonlinear properties result in higher order harmonics at the output. Therefore,
an ultimate low pass behavior cannot be guaranteed.
Variables as wind speed and road inclination can be considered disturbances in the control loop. Over
the lifetime of the vehicle, also the number of passengers and amount of payload can be considered
disturbances. As several of these disturbances are uncorrelated, over time, based on the Central Limit
Theorem, these disturbances further support a normal distribution of the power for traction.
4. Discussion
To further investigate the validity of the normal distribution hypothesis, measured data from electric
propelled vehicles over longer periods of operation with a sufficient high sample rate should be available. As such data is not available yet, the hypothesis should be considered a reasonable assumption.
The motivation for a normal distribution relies on the feedback loop of figure 8. When this control loop
is interrupted, the motivation for a normal distribution partly fails. This is the case during coasting (the
driver releases the accelerator pedal and just accepts the resulting speed change), during gear shifting
(again the accelerator pedal is released and for a short moment the power for traction reduces to zero
or possibly a constant power level). Also when the driver shifts to mechanical braking, the loop is interrupted. First generation electric vehicles tend to implement braking to resemble the brake behavior of
gasoline cars, but we observe a shift to maximum regenerative braking on the electric motor, or driver
adjustable amount of regenerative braking. This improves the electric efficiency of the car and therefore
its driving range. Examples are the BMW i3 commercial vehicle (maximum regeneration) and the Skoda
Octavia green E line concept car (adjustable regeneration).
To evaluate the impact of gear shifting and mechanical braking, experiments are carried out with an
electric concept vehicle with a 5-shift gearbox. In addition, the inverter is reprogrammed such that the
accelerator pedal imitates the behavior of an ordinary gasoline vehicle: When released, only a small
amount of power is regenerated from the kinetic energy of the vehicle, mimicking coasting. In addition,
the drivers could use the 5-shift gearbox as in a conventional vehicle. With these adjustments, a commuter trip of approximately 40 minutes is made in an urban environment.
Figure 9. Measured distributions under gasoline vehicle imitating conditions.
5. Conclusions
The objective of this study was to evaluate if characterizing driving cycles by their power for traction is
acceptable and beneficial. Experiments with different electric vehicles on the road, on a chassis dynamometer and in simulations, show that the distributions of the power for traction tend to be bell-shaped. Next, it is motivated this shape is acceptable approximated with a normal distribution.
A normal distribution is defined by its mean and variance. This mean and variance are directly linked
to the vehicle parameters and key properties of the considered driving cycle. As a result, the power
distribution not only represents the considered driving cycle, but more general the vehicle in a traffic
environment for which the considered driving cycle is one observation. Therefore, representing the requirements for a vehicle in terms of a power distribution is more convenient than using one or a limited
set of deterministic driving cycles.
The results are presented in figure 9. Compared to the previous bell-shaped power distributions, this
figure shows an additional peak at the negative “accelerator pedal released” power, and the resemblance with a normal distribution is diminished. Apparently, the driver’s behavior is affected by the programmed behavior of the accelerator pedal. This was confirmed by the test drivers. These results support the
control loop as a model for the interaction between driver and vehicle and the closed loop assumption
where the driver continuously controls the power for traction. This is also in line with the simulation
results of figure 5, where the power for traction is the result of a single backwards relation, instead of a
control loop.
28
29
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31
Rapid control prototyping platform for electric
and hybrid vehicle drive lines
2. Model based development V-cycle
Jan Benders
When looking at model based development techniques for the development of control systems - or E/E
systems in general - a simplified V-cycle can be derived, allocating the different primary use cases of model based development in the left and right side of the V-cycle, covering respectively specification and
design on the left and verification, validation and optimization on the right (Figure 1).
Note that this V-cycle is not discriminating in system, sub-systems and components levels. The mentioned steps can be applied to all integration levels.
HAN University of Applied Sciences, Arnhem, The Netherlands
Abstract
This article describes the development of a Battery Hybrid Electric Vehicle (BHEV) RCP Platform in the
light of the RAAK-PRO project Electric Power Train (EPT). During the first phase of EPT, an already existing platform initiative was further developed to become a safe, universal RCP platform for BHEV applications, ultimately usable for series production developments. This development was accompanied by
the development of a ‘Functional Safety Handbook’ in order to comply with ISO26262 related procedures
and methods when applying the platform in BHEV applications. As the platform development proceeded, it became clear that this technology became useful in a wider variety of applications as several
HAN partners and clients expressed interest in the solution. As a result, new development requirements
did arise, surpassing those of the EPT project, not only because of the wider application range but also
because of the required development investment. Therefore, further development of the platform was
moved into a separate project, called Fast & Curious. Fast & Curious focuses on a universal RCP platform
for SME’s and HAN, in both R&D and educational settings. Beyond Fast & Curious, the project SMARTcode is now aiming series product development of control systems, including the development of a recommended practice according to ISO26262. With Fast & Curious and SMARTcode the original goals of EPT
have been extended towards a wider application arena and a larger user base in the form of a growing
community of companies and institutes, while still acknowledging the validity of the original development goals.
1. Introduction
Rapid Control Prototyping - or RCP - is a relevant development step on the left side of the V-cycle requiring the correct software tools in order to automatically generate code from a model. This code should
be generated into a widely applicable, re-usable hardware platform. Market solutions exist, but are
highly expensive and therefore not feasible in the HAN R&D environment where educational usage is
essential and SME’s form a significant part of the customer base.
For this reason, HAN initiated the in-house development of an RCP solution preceding the EPT project.
This development was further strengthened by defining it as one of the technical focus areas of the
EPT project. During the EPT project, goals were formulated to realize a RCP solution in the form of a
“BHEV RCP Platform” with the ultimate future goal to extend the platforms’ usability for series production usage. Series products need to be certified; therefore the development process of the products
should comply with standards and regulations. In this respect, the ISO26262 standard [1] concerning
functional safety in automotive E/E systems is one of the key guidelines to follow. Electric power trains
can generate high torque instantly, potentially leading to danger, and this could occur due to a failure in
the BHEV RCP Platform. Because of the complexity of the required processes, procedures and methods
in ISO26262, the initiative was born to develop a handbook for applying ISO26262 in the scope of the
electric power train application area targeted at SMEs.
32
The standardized V-cycle as defined in parallel in Germany and the USA covers system level requirements and design, followed by decomposition into sub-system and components on the left side, ending
up in component implementation down at the bottom. On the right side a verification and validation
takes place, with an upwards integration or re-composition from sub-systems up to system level.
Figure 1. ‘Model Based Development V-cycle’
On the top left simulation is used for the first design and development. Simulation allows for setting up
a plant model in conjunction with a controller algorithm. This enables deeper understanding of system
and controller behaviour and allows for refining control strategies in a very early stage. Due to the gained knowledge in detailed system behaviour, this step can be - and normally is - used to refine requirements on all integration levels.
Since models are the key elements in model based development, the automatic generation of code from
these models can significantly enhance the efficiency and effectiveness of the development process. It
enables the 3 central steps in the abovementioned V-cycle:
-
Rapid Control Prototyping, where code from the control algorithm model is automatically generated for a universal RCP control unit;
- Implementation, where code is again automatically generated from the control algorithm model.
In this case the code is generated for the series production system so either the model has to be
extended with aspects like robustness and diagnostics, or the generated code has to be combined
with other code comprising these necessary aspects for series products;
-
Hardware In the Loop (HIL) testing, where code is generated from plant models and executed on a
computer system capable of running the models in real-time: a HIL simulator. With the correct input
and output interfacing, a HIL simulator can electrically simulate a system to the level that control
system can be connected to the HIL simulator as it is connected to a real world system. This enables
reproducible, 24/7 conductible automated testing and validation of the control system behaviour.
33
The BHEV RCP Platform is primarily intended for RCP as its name implies, but ultimately the solution is
also intended for usage in series products, covering the implementation phase as well when this solution becomes available. Obviously, much of the automatic code generation technology used for RCP can
be re-used for final implementation. To a certain extend this is also valid for the code generation required for autocoding the plant models for a HIL simulator. In some cases an RCP platform may actually
serve as a simple HIL simulator.
The upper right part of the cycle mentions function optimization, which seems to be separated from the
RCP stage, but as the following paragraph describes, this phase is also partly covered by the BHEV RCP
Platform development.
3. BHEV RCP platform
The BHEV RCP Platform comprises 2 main developments:
-Development of a so called ‘embedded target’ for generating code from MATLAB/Simulink for a
Rexroth RC30 series automotive specified controller (ECU). This target originally got the name ‘RC30
Target’ and is now being part of the HANcoder library of embedded targets, ex-tending its full name
to ‘HANcoder RC30 Target’;
-Development of an application engineering tool, named HANtune, to serve as a real time dashboard for the algorithm running in the RC30 ECU.
In the light of EPT, support for a controller with integrated safety functionality was realized in order to
prepare future support for series production developments requiring these aspects. More information
on the supported hardware in HANcoder RC30 Target can be found at www.han.nl/rc30target.
The attractive aspect of the RC30 series controllers is the series production quality and service, combined with a significant amount of inputs and outputs providing a wide variety of functionality. This enables
RCP and series product development based on the same hardware family, in line with the set targets for
the BHEV RCP Platform.
When the generated code is executed in an RC30 controller, a wish for monitoring and optimizing the
algorithms’ behaviour emerges. This is enabled by HANtune shown on the right side of the picture. Via
the industry standard XCP protocol, HANtune connects to the running algorithm in the RC30 controller. When connected, the user can visualize, log and optimize signals and parameters in real-time. This
provides an efficient workflow for RCP activities. The logged data can be transferred back to MATLAB/
Simulink® for simulation purposes as well. More information on HANtune can be found at www.han.nl/
hantune.
During the EPT project, several students worked together with employees on the continuous development of HANcoder RC30 Target and HANtune, upgrading them to an attractive solution. On the educational side this resulted in combining HANtune with another code generation solution based on a lowcost Freescale HCS12 microcontroller already in use at minor ‘Autotronica’. By combining these tools,
students got access to model based development tools resulting in a significant increase of effectiveness
in the minor projects. Using these tools, students are able to design, implement and test algorithms
with higher complexity in shorter time frames compared to the original process using manual coding.
Students can put more focus on functionality instead of programming language and microcontroller behaviour when using the tools. Besides, true model based development scenarios became reality in our
educational environment as described below in the application examples.
4. Extending the horizon
Figure 2. HANcoder and HANtune
On the left, the Simulink environment is shown with the library browser containing the available functional blocks for building models on the far left. HANcoder RC30 Target adds a library of blocks for connecting a Simulink algorithm to inputs and outputs of an RC30 controller. Additionally, some blocks for
system management (e.g. hardware configuration) and memory management are provided.
When the necessary blocks of HANcoder RC30 Target are properly configured and connected to the
model, code can be generate from the model fully automatically. The generated code can directly be
programmed into the controller, shown in the middle. HANcoder RC30 Target supports different versions of the RC30 family. The differences in hardware versions are twofold:
- The amount of relatively expensive power outputs is varied for unit price reasons;
-More recent hardware versions contain additional safety aspects for supporting safety relevant
application development.
34
In parallel to the educational integration, several parties became interested the HANcoder/HANtune
solution as well, resulting in co-development [2] by one of these parties. As a consequence the application horizon extended beyond that of EPT, leading to a proposal for a separate project focusing on tool
development in a community environment for RCP purposes: the RAAK-MKB project Fast & Curious [3].
During Fast & Curious, many project partners started adopting the HANcoder/HANtune solution in their
development process, specifically for prototyping purposes. The usage was not limited to Fast & Curious partners. Other parties started using the tools as well [4, 5, 7]. During Fast & Curious, several SME
partners expressed a wish for integration of RCP and series production developments. Consequently, a
new RAAK-MKB project called SMARTcode [6] focusing on model based development for series products
started in October 2014.
5. Safety handbook
As mentioned in the introduction, in early EPT stage the idea arose to develop a pragmatically oriented
handbook for applying the functional safety related procedures and methods as ISO26262 proposes in
electric power train control systems. Since ISO26262 is very comprehensive and leaves interpretation
and choice of methods open to some extent, a handbook can be of practical assistance, especially for
SMEs that do not have the necessary competences and experience available.
35
ISO26262 covers the whole product lifecycle. The handbook activities in the light of EPT focused mainly on the concept phase. A good handbook uses real life examples, so the RAAK International project
PLUTO project served one of the main cases for the handbook. Various bachelor and master students
worked on the handbook covering various aspects. Some examples of the activities:
-Development of a good format for the handbook with the appropriate interactivity, including a
short explanation of background and content of ISO262626;
-Design of generically usable torque manager for electric power trains using model based development techniques and tools;
-Assessment of the model based verification and validation tools, such as Simulink Design Verifier,
being able to automatically generate test cases for a given algorithm model to asses model design
errors using formal methods;
-PLUTO control algorithm development was extended with HARA (Hazard Analysis and Risk Assessment) in order to adapt the controller design with respect to functional safety, providing a real life
HARA example and good practice input for safe controller design on the model level.
Figure 4. Hydrogen/electric powered HyDoblo vehicle
The current status of the handbook is an interactive Word/PDF document describing the largest part
of the concept phase as defined by ISO26262, including a flowchart approach assisting in proper process execution, see Figure 3. This handbook will be further extended towards an SME oriented recommended practice in the light of the SMARTcode project.
Figure 3. Interactive flowchart in the safety handbook
6. Application examples
HANcoder and HANtune have become the standard tools in control developments within the Automotive and Engineering departments of HAN. Many community members of the projects Fast & Curious
and SMARTcode use this toolset in control system developments. Figures 4 and 5 show some example
applications that have been developed in a model based way using the tools.
The HyDoblo contains an electric power train combined with a fuel cell system, providing both fuelling
options from the grid and from compressed hydrogen. Top level power train control and complete fuel
cell system control is handled by a Rexroth RC36-20/30 controller, programmed in Simulink, using HANcoder.
Figure 5. PLUTO hybrid power train test bed
The PLUTO project targets at PLUg-in hybrid power train TOols. Within PLUTO, a plug-in hybrid power
train test bed was developed, including all the necessary control systems. PLUTO uses automatically
generated code from HANcoder in several ways:
-The PCU (Power train Control Unit) executing high level power train management is implemented
in a Rexroth RC28-14/30 controller by the HAN;
-The TCU (Transmission Control Unit), controlling the used twin speed power shift gearbox is implemented in a Rexroth RC28-14/30 controller by Punch Powertrain;
-
DC bus safety and control is implemented in a Rexroth RC28-14/30 controller by HAN.
The algorithms in all controllers are defined in MATLAB/Simulink and autocoded in the Rexroth hardware using HANcoder RC30 Target. The resulting software has been tuned using HANtune.
36
37
During the control system development in PLUTO plant models have been developed for power train
components and vehicle behaviour. These models have been used in various stages of the development,
when no real power train hardware was yet available:
This type of concurrent engineering is one of the powerful aspects of model based development in a
real life project, utilized by the PLUTO control team in this way (see Figure 6).
various projects. Because of a growing interest beyond the scope of EPT, the RCP platform development
took a next step in the projects Fast & Curious and SMARTcode, resulting in a community around model
based development. By migrating the development into this community environment, the original
targets of the BHEV RCP Platform have been surpassed in many aspects. However, for the same reason
some original development targets, like the universal safety controller, have not yet been fully realized.
The expectation is that the market as represented by the current group of community members will
eventually drive the development towards similar solutions if this is really a market wish. That immediately shows the relevance of the new development setting: further, market driven development, automatically optimizing application potential.
7. Model based development community
References
The EPT developments of the BHEV RCP Platform induced the projects Fast & Curious and SMART-code,
resulting in a community centred around model based development.
[1]ISO 26262-1:2011 - Road vehicles - Functional safety -, International Organization for Standardization, 2011
[2]J.Benders, A.Serrarens, Han verlaagt drempel modelgebaseerd ontwikkelen in automotive, Bits &
Chips 4, 2012
[3]HAN University of Applied Sciences (2013), Project Fast & Curious, retrieved on 2nd of February
2015 from http://www.han.nl/onderzoek/werkveld/projecten/fast-and-curious/
[4]Robert Cloudt, Diagnostics Development for Cost-effective Temperature Sensor based Particulate
Matter OBD Method, SAE paper 2014-01-1550, 2014
[5]Twan Heetkamp Trailers (2013), New Cool, retrieved on 2nd of February 2015 from http://www.
thtrailers.com/nl/new-cool
[6]HAN University of Applied Sciences, SMARTcode project for model based series production
development, retrieved on 2nd of February 2015 from http://www.han.nl/international/english/
about-han/news/smartcode-series-producti-1/
[7] J.Benders, C.J.M.Heemskerk, R.de Zaaijer, Modelgebaseerd ontwikkelen in het mkb, Mechatronica
& Machinebouw 2, 2015
-
-
At simulation level for early development of algorithms;
In HIL simulation to test and optimize real-time behaviour of the control algorithms.
Figure 6. HAN model based development community
The community not only focuses on automotive applications. Extension towards industrial applications
has been implemented. Within the community a Change Control Board, existing of commercial partners
and HAN, is forcing priorities for further development in order to ensure a ‘market push’ oriented development. The combined industrial and automotive orientation of the community has shown its potential
in bilateral exchange of experience and requirements for future developments. One of the interesting
results is the realization of the new STM32 Target being usable in both application areas. This target is
currently the most used solution in the HANcoder library.
8. Conclusions
The BHEV RCP Platform activities in EPT have fit the need for adequate RCP solutions within the project scope and beyond. Development of the platform as intended was initially executed during phase 1
of the project, resulting in an effective first platform solution. This solution was successfully applied in
38
39
A prototype emission free cooling trailer
Bram Veenhuizen1; Menno Merts1, Karthik Venkatapathi1, Erik Vermeer2
module, was designed and tested by Bahaj et al. in 2000 [2]. A comprehensive feasibility study on solar
powered refrigeration for transport application has been reported by Bergeron [3]. This paper concludes that “…the economic justification for wide spread use of solar is moderate, but not compelling.
However, as the price of diesel increases and the price of solar modules and vacuum panels decreases,
the economic case will improve. These are expected trends. In addition, any new regulations impacting
diesel emissions will likely favour solar”.
1 HAN University of Applied Science, Arnhem, the Netherlands
2 TMC Mechatronics, Eindhoven, the Netherlands
The main layout of the proposed system and its main characteristics will be presented. The control system will be described, including the energy management system. The system was realized and preliminary test results will be given.
KEYWORDS – cooling trailer, energy management, solar power, brake energy recuperation.
System description
Abstract
A schematic overview of the system is represented in Figure 1. The subsystems will be described in the
following sections:
Cooling trailers need a substantial amount of power to drive the cooling compressor. This power is produced by linking the compressor to the engine power take-off or by a reefer unit.
An interesting option is to design an autonomous cooling trailer that powers the compressor from brake
energy, solar power and from the grid. This paper describes the design and test of such a system, which
also comprised a lead acid type battery for storing electric energy. The objective of this study was to
design and realize a fully functional prototype trailer and also to design an energy management strategy
for this system. The paper presents some preliminary test results, showing its feasibility.
The roof of the trailer is covered with about 35 [m2] PV solar array, generating approximately 4.5kWp
power. The array is coupled to an inverter and a battery, which acts as a buffer for storing electrical
energy.
Battery:
Introduction
Cooling Unit
Fuel consumption of road going vehicles is an increasing concern for the entire automotive industry.
Commercial vehicle manufacturers spend a lot of their efforts in improving the fuel economy of their
vehicles, not only to comply with government regulations, but also to preserve the competitiveness of
their valued customers. Trailer manufacturers also contribute to this by making their products lighter.
Nowadays 10 to 15% of all goods are transported in conditioned trailers. Most conventional cooling systems on trailers, also referred to as reefer systems, are powered by separate diesel engines. An obvious
advantage of these systems is the instantaneous availability of ample cooling power, independent of the
towing truck. Disadvantages are: high
weight, additional engine maintenance, relatively low efficiency and high
exhaust and noise emissions. In other
systems, the electrically driven compressor is powered via a hydraulic drive
Solar Panels
from the truck engine power take-off.
Cooling Buffer
An overview of the approaches to reduce energy consumption and environmental impacts of refrigerated food
transport is given by Tassou et al. [1].
Inverter
This paper describes the design of a
Wheel driven
novel system, comprising a regenerative
generator
braking module, mounted on one of
Battery
the trailer axles, a solar panel module,
380 VAC socket power supply
a trailer box and a Li-ion battery for
energy buffering. A comparable system,
but without the regenerative braking
Figure 1. Schematic overview of the Prototype Emission Free Cooling Trailer
40
Solar panels:
1
This paper has been presented at the FISITA world congress, Maastricht, 2014
For first tests a 48 [V], 30 [kWh] lead acid battery was mounted. This battery serves a number of purposes: storing electrical energy from the solar array and regenerative braking and powering the cooling
machine.
Regenerative braking axle:
A special axle (see Figure 2) was developed for the
prototype trailer, comprising a drive shaft connected
to one of the trailer wheels, and a hydraulic pump
with variable displacement, which in turn is connected to a hydraulic motor, driving a generator. This
solution allows for a continuously variable control
of the generator speed, independent of the wheel
speed. When the braking light of the trailer is lit, the
axle can be used to generate braking power which
can be recuperated in the battery, instead of wasted
in heat.
Figure 2. Picture of regenerative braking axle.
Cooling unit:
Cooling power is generated by an electrically driven TRS Alaska cooling unit.
Inverters:
Three Victron Energy 48/10K Quattro inverters were used to convert the battery DC power into 380Vac50hz. This power can then be used to drive the compressor of the cooling unit. Also the power from the
41
regenerative braking axle is connected to these inverters, see also Figure 3.
An additional input to the inverters allows for charging of the battery from a mains supply.
Cooling buffer:
The cooling buffer was not used during the tests described in this paper.
The control system
The Rexroth RC28 control unit forms the heart of the control unit (see Figure 3). This industrial controller has a powerful processor, and is completely programmed by Simulink code generation. Monitoring
and calibration of the software can be done by using HAN-Tune. Basically the controller has three tasks,
communication, energy management and component protection.
RC2-21
Although with current power levels not a realistic threat, the component protection software has to
prevent blocking and thus sliding of the trailers’ wheel, caused by a too large power take off. Comparable to ABS functionality, the controller is monitoring the maximum deceleration of the trailer wheel.
At deceleration above a threshold based on the friction between tire and road surface, the generator is
switched off.
Power flow
Analogue signals
CAN bus signals
Can bus 3
Can bus 1
Brake light
RC28
Operation
Can bus 2
TRS
hydropower
kWh
sensor
Victron
AC IN 1 Quattro's
kWh
sensor
AC OUT
AC IN 2
Battery
kWh
sensor
kWh
sensor
Alaska
TRS
Cooling
Unit
PV cellen
Mains
220/380 Vac

Temperature [ C]
Figure 3. System overview, showing power flow, analogue and CAN-bus signals.
The control unit obtains information from the inverters, the hydraulic system and external sensors. As
output it sends the necessary control signals for energy generation and hydraulics. The control unit also
takes care of an external CAN logging device, which is able to store desired information on a flash card.
Also a display is controlled, which can inform the truck driver about the actual status and errors in the
system. Exchange of information between the systems is done via CAN.
Energy management.
The main task of the energy management system is to ensure cooling of the payload. Temperature
control is done by the cooling system itself, but the trailer control unit has to make sure there is always
energy available for cooling. Multiple energy sources are available and the system has to select the preferred one(s), balancing the availability and price of energy.
The energy that is always for free comes from the solar array. So when it is available it is used to charge the batteries, until the desired maximum SOC is reached. Additional power is available from the
42
a
20
15
50
0
20
0
0.5
b
1
1.5
2
Time [h]
Trailer temperature
2.5
-20
0
c
1
1.5
Time [h]
Battery
2
AC OUT
Battery
AC IN1
0
-5
2.5
50
0
d
Solar
5
-10
0.5
Power
10
0
100
SOC [%]
Communication
100
Trailer speed
Power [kW]
Can bus 4
CAN
Logging
device (16GB)
With the hydraulic transmission from wheel to generator it is possible to run the generator at its required speed down to very low vehicle speeds. This leads to increasing torque in the power take off. To
ensure the integrity of all mechanics, the generator is switched off below a certain vehicle speed, and
thus above a certain torque limit.
6
Status
Bosch sensor
Component protection.
Vwheel [km/h]
Trailer/Truck
CAN bus
generator, connected to the regenerative braking axle. Energy from the generator is for free when it is
created during deceleration or braking of the trailer, so this is preferred. When still not enough energy
is available, for example during a long highway drive during the night, the control system can decide to
enable the generator also during constant driving. The truck engine has to produce extra power for this.
Because of the high efficiency and low emissions of the truck engine this still is a sensible way of energy
generation. When the trailer is connected to the grid this energy is used for both cooling and charging of
the batteries.
0
0.5
1
1.5
Time [h]
2
2.5
Software Status
e
Standstill
V > 30 [km/h]
Charge demand
Sw itch ON
4
2
0
0.5
1
1.5
Time [h]
2
2.5
0
0
0.5
1
1.5
Time [h]
2
2.5
Figure 4. Test results for Test 1.
43
15
0
20
0
b
1
2
Power [kW]
50
3
Time [h]
Trailer temperature
-20
0
c
Time [h]
Battery
0
2
AC OUT
AC IN1
0
Time [h]
2
3
Solar
AC OUT
Battery
15
5
1
Energy
20
10
AC IN1
e
0
1
Time [h]
2
3
-5
0
1
Time [h]
2
3
Several tests have been carried out with prototype cooling trailer and results of two experiments are
reported in this section.
Test 1
Vwheel [km/h]
100
Trailer speed
a
50
0
8000
20
Power [kW]
Test 1 was done by driving the trailer at
a constant speed and was intended to
test the functional performance of the
regenerative braking axle, mainly. The
vehicle was driving 90 [km/hr] most of
the time, see Figure 4. The solar array
was not operating during this test but
the axle was used frequently, as can be
seen in Figure 4d. Here it can be seen
that the axle (signal AC IN1) during this
test produces about 15 [kW] when it
is activated. When the cooling unit is
running, it consumes about 5 [kW] (signal AC OUT), leaving about 10 [kW] to
charge the battery. From Figure 4b it can
be concluded that the setpoint tempe-
8050
8100
8150 8200 8250
Time [sec]
Power
8300
8150 8200 8250
Time [sec]
8300
8350
8400
0
-10
8000
Solar
AC OUT
Battery
AC IN1
8050
8100
As shown in the sections above, the sizing of the various components (battery capacity, regenerative
braking axle power, solar array power) may need further optimization. The system replaces a conventional reefer system, powered by a diesel engine. These systems represent a weight of approximately 750
[kg]. The weight of the proposed system is primarily determined by the battery weight, which amounts
also to approximately 500 [kg]. A further reduction in weight may be reached by optimizing battery
capacity and by using Lithium based battery technology.
A sophisticated energy management system may further reduce the fuel consumption. An example is
the use of a planning system. If the next truck-stop duration will be sufficiently long and a grid connection is available, it may be advantageous to arrive there with a low battery SOC, since energy from the
grid comes at lower cost than energy generated by the truck engine.
A system described in this paper comes with additional advantages. The truck driver may use the
available energy in the battery for hotel load applications. This may also lead to additional fuel savings
and emission reductions, since the truck engine is not needed for this. The required inverter is already
present, avoiding extra cost.
Conclusions
In this paper, a Prototype Emission Free Cooling Trailer was presented. The system design was described, as well as the trailer control system. Experimental test results were presented, showing that the
system is feasible from an energy perspective. Further work needs to be done in order to optimize
component sizing and the energy management.
b
10
Test 2 showed frequent stops and two periods of system shutdown, starting at t=0.92 [h] and at 2.05
[h]. During these periods, also the cooling unit was shut down, and the temperature can be seen to rise
quickly (see Figure 5b). When the vehicle speed becomes lower than 30 [km/h] the braking axle cannot
supply energy to the cooling unit and the required power is obtained from the battery. Figure 6 shows
a delay of over 30 [sec] after the trailer reaches 30 [km/h] before the regenerative braking axle starts
producing power. This delay is mainly caused by the inverters. Nevertheless, also for this test the axle
is able to increase the SOC of the battery, indicating that the power rating of the axle is sufficient for its
purpose. The solar array, however, supplies just about 0.8 [kW] on average during Test 2, while the average power consumed by the cooling unit amounts to 4.7 [kW]. Further analysis is required to investigate
whether the installation of the solar array pays off under western European climate conditions.
Discussion
0
Figure 5. Test results for Test 2.
44
Test 2
Battery
25
3
From this measurement it can be seen that the regenerative braking axle alone is quite capable to supply the power needed to cool the trailer and to sustain or increase the battery SOC (Figure 4d).
Solar
-5
50
0
d
rature is reached at about 1.3 [h]. This causes the subsequent on-off behaviour of the cooling unit, also
witnessed in Figure 4d. The Charge demand signal of the battery is set to zero when the battery SOC
reaches 80 [%] and is set to 1 when the SOC drops below 70 [%] (see Figure 4e). For this reason the generator stops supplying power at t~1.8 [h] and the battery is used from that time onwards to cool the trailer.
5
-10
1
Power
10
0
100
SOC [%]
a
20
Energy [kWh]
Vwheel [km/h]

Temperature [ C]
100
Trailer speed
8350
8400
Figure 6. Test 2 data, showing delay in power generation
from the regenerative braking axle.
Acknowledgements
All data presented in this paper have been supplied by the companies TRS transportkoeling bv, Twan
Heetkamp trailers bv, both partners in the ecotrailer project. Their contribution is gratefully acknowledged. Acknowledgements also to TMC for their support.
45
References
[1] S. Tassou, Y. Ge and G. De-Lille, “Food transport refrigeration – Approaches to reduce energy consumption and environmental impacts of road transport,” Applied Thermal Engineering, vol. 29, p.
1467–1477, 2009.
[2] A. Bahaj, “Photovoltaic power for refrigeration of transported perishable goods,” in Photovoltaic
Specialists Conference,, 2000.
[3] D. Bergeron, “Solar Powered Refrigeration for Transport Application -- A Feasibility Study,” Sandia
National Laboratories, Albuquerque, 2001.
Design and test of a battery pack simulator
Stefan van Sterkenburg1, Ton Fleuren1, Bram Veenhuizen1, Jasper Groenewegen2
1 HAN University of Applied Science, Arnhem, The Netherlands
2 DNV GL, Arnhem, The Netherlands
Abstract
This paper describes the design of a battery pack simulator (BPS). The BPS is an electronic device that
emulates all cell voltages of a battery pack that consists of at most 248 cells in order to test battery management systems (BMS). Cell characteristics and environmental conditions of individual cells, such as
capacity, ambient temperature and internal resistance can be adjusted in order to examine the unbalance in a battery pack on the functioning of the BMS. The BPS implements a real time simulation model
that consists of interacting electric and thermal submodels. The BPS uses xPC Target and Simulink as
real-time software environment. A hardware interface converts all simulated cell voltages and temperatures to real voltages that can be connected to the voltage and temperature sensor inputs of a BMS. We
have tested the BPS by comparing the emulated cell voltages of it to the cell voltages of a real battery
pack that consists of LiFePO4 cells. In order to do so, a thermal cell model and third order equivalent
network model has been derived by analysing EIS1-measurements and pulse current measurements.
The found network is implemented in the BPS. Test results show that the maximum deviation between
the real and emulated cell voltages is at most 0.5%. We have connected our BPS to a Lithiumate Pro
BMS and evaluated the capability to test it with our BPS. We found that basis functionality, like SoC estimation and over and under voltage detection could be tested well. At this moment, the BPS is not suited
to test the balancing feature of BMS. Further development of the BPS is needed to enable the testing of
this feature of BMS as well2.
1
46
2
Electrochemical Impedance Spectroscopy
This paper is a revised version of the paper “Design and test of a battery pack simulator”, presented in EVS 27, 2013.
47
1. Introduction
The testing and evaluation of battery management systems (BMS) is in practice an awkward and time
consuming activity. A complete test includes temperature tests, tests of unbalances in battery packs,
tests of unknown initial state of charge values, and so on [1]. Also, it is often difficult to test the BMS in a
laboratory with the same load and environmental conditions that a battery pack might face in practice.
In order to facilitate the testing of BMS we have designed a battery pack simulator (BPS). The BPS is an
electronic device that emulates all cell voltages of a battery pack under various adjustable battery conditions. The BPS can be used also for testing voltage management systems of fuel cell stacks.
The BPS calculates the voltages and temperatures of all cells in a battery pack that consists of at most
248 cells. The user can define his own battery model, so batteries with different cell chemistries can
be emulated. Cell characteristics and environmental conditions of individual cells, such as capacity or
ambient temperature can be adjusted in order to test the impact of unbalance in a battery pack on the
functioning of the BMS. The BPS implements a real time simulation model that comprises interacting
electric and thermal submodels. A hardware interface converts all simulated cell voltages and temperatures to real voltages that can be connected to the voltage and temperature sensor inputs of a BMS.
This paper explains the architecture of the battery simulator, the hardware interface, the modeling of
a battery pack that consists of LiFePO4 cells and the basic testing of an Elithion Lithiumate PRO BMS by
using our BPS.
2. Architecture of battery pack simulator
The hardware of the BPS consists of four main components (Fig.1): a host PC, an xPC target, a FGPA
board and a set of at most 64 isolation amplifiers.
1. The subsystem ‘UDP data from host’ handles the model input that is sent from the host to the xPC
target. The model input consists of the battery current and adjustable cell parameters and variables.
The model in Fig.2 defines 5 adjustable cell parameters, but this can be expanded according to the
user’s needs. Port numbers of the UDP packets are used to identify the adjustable variables and parameters in order to support flexible and easy configurable model implementations.
2. The subsystem ‘UDP data to host’ handles the model output that is sent from the xPC to the host. The
model output consists of terminal voltages, temperatures and state of charge of all cells.
3. The subsystem ‘Battery pack model’ defines the cell and battery pack model and is discussed in more
details later in this paper.
4. The subsystem ‘FPGA board’ provides the model output that is sent to the FPGA board. The model output to the FPGA board holds 512 variables. In the configuration, shown in Fig.1, the output
consists of 248 cell voltages, 248 cell temperatures, the battery pack voltage, a current sensor output. Zero value placeholders are used if the battery pack consists of less than 248 cells. All simulated
output variables are scaled to 16-bit variables that can be processed by the digital to analog converters (dac’s) in the isolation amplifiers. An offset calibration feature is implemented to correct offset
error(s) that may arise in the analog signal processing circuit. Raw datagrams are used for the communication between the xPC and FPGA since it is a fast and rather simple protocol.
The FPGA board is the interface between the xPC target and the isolation amplifiers. The FPGA is responsible for receiving and decoding the Raw datagram packets sent by the xPC target. The FPGA decomposes the 512 output variables into 64 sets of 8 variables that are transmitted to isolation amplifiers
circuits by 64 SPI busses. All 64 SPI busses share the same clock- and latch signal in order to limit the
number of signals and to provide synchronous sampling. Each SPI bus has its own data signal. The SPI
busses are connected to isolation amplifiers. Each isolation amplifier circuit provides eight analog voltages that can be configured to represent cell voltages, temperature sensor signals or a combination of
both. The isolation amplifiers provide the analog voltages that can be connected to the BMS cell voltage
inputs or temperature or current sensor inputs. Its working is discussed in the next section.
3. Isolation amplifier circuit
Figure1: Block diagram of the battery pack simulator
The Host PC is the interface between the user and the model. The host PC enables the user to start or
stop a simulation, watch simulation results and to enter the model input.
The battery model runs on an xPC target and consists of four subsystems (Fig.2):
Fig.3 shows the functional block scheme of the isolation amplifier circuit. The SPI-bus operates at a bit
rate of 500 [kbit/s]. The chosen bit rate limits the maximum sample rate to at most 3.9 [kSamples/s]
(=[500kbits/s] / (8 words * 16 bits)).
The SPI signal are galvanic isolated from the FPGA board by optocouplers. The secondary sides of the
optocouplers are connected to 8 daisy-chained 12-bit dac’s. The dac’s are grouped in two sets of four
isolation amplifiers that share a common power supply. An analog processing circuit converts the voltages of the dac’s to output voltages. The design of the analog circuit allows the following configurations:
- Configuration 1 is applicable to emulate the cell voltages of eight subsequent cells when the BMS
under test uses multiple cell monitor units. Then, it applies: Uout1 = Uadc1, Uout2 = Uout1 + Uadc2, Uout3 =
Uout2 + Uadc3 and so on. The gnd2 connection of the upper analog process circuit must be connected to
Uout4 of the lower analog process circuit.
- Configuration 2 is applicable to emulate the temperature sensor voltages of eight subsequent temperature sensors when the monitor unit of the BMS under test uses analog temperature sensors with a
common ground. In that case, it applies:
Uout1 = Uadc1, Uout2 = Uadc2, Uout3 = Uadc3 and so on.
Figure 2: Block scheme of the model that runs on the xPC target
48
49
Figure4: Battery pack that
consists
of 12 cells.
Figure4:
Battery
pack that consists of 12 cells.
In this paper we describe
the modeling
a battery
that consists
of twelve
Sinopoly
In this
paper we ofdescribe
thepack
modeling
of a battery
pack 100
that [Ah]
consists
of twelv
In this paper we describe the modeling of a battery pack that consists of twelve 100 [Ah] Sinopoly
were4 placed
in one
row (see
LiFePO4 cells. The 12 cells
cells.
12 cells
wereFig.4.).
placed in one row (see Fig.4.).
LiFePO4 cells. The 12 cells
were placedLiFePO
in one row
(seeThe
Fig.4.).
4.1
Thermal submodel
4.1
Thermal submodel
4.1
Thermal submodel
The thermal model of the battery pack is based on the thermal heat conductance between the cells
The thermal
the
battery
pack
based
theequation
thermal
conductance
between
cells
The
thermal
model
ofWe
theon
battery
pack is
based
on thethe
thermal
heatthe
conductan
mutually and from
the cellsmodel
to the of
environment
[2],
[3],is [4].
used
1heat
to
determine
cell
to theand
environment
[2], [3],
[4]. environment
We used equation
to determine
the cell 1
mutually
from the cells
to the
[2], [3],1[4].
We used equation
temperatures.mutually and from the cells
temperatures.
𝑇𝑇! = 𝑇𝑇!!
Figure 1. Functional block scheme of the isolation amplifiers
The gnd2 connection of the upper analog process circuit must be connected to gnd1 of the lower analog
process circuit.
- Configuration 3 is applicable to emulate the cell voltages when the BMS under test uses single cell monitor units. In that case, the outputs emulate alternately cell voltage and temperature sensor signals of
four subsequent cells. it applies:
Uout1= Uadc1 (voltage ofcell 1),
Uout2 = Uadc2 (temperature sensor voltage cell 1)
Uout3 = Uout1 + Uadc3 (voltage of cell 2)
Uout4 = Uout1 + Uadc4 (temperature sensor voltage cell 2)
Uout5 = Uout3 + Uadc5 (voltage cell 3)
Uout6 = Uout3 + Uadc6 (temperature sensor voltage cell 3)
Uout7 = Uout5 + Uadc3 (cell voltage cell 4)
Uout4 = Uout5 + Uadc8 (temperature sensor voltage cell 4)
The gnd2 connection of the upper analog process circuit must be connected to Uout3 of the lower analog
process circuit.
The maximum output current of the emulated voltages is limited to 30 [mA]. This is an important restriction that makes the BPS not suited to test the balancing feature of a BMS.
1
+
𝐶𝐶!!
temperatures.
𝑃𝑃! +
!"#$$%
!"#$$%
1
!!
!"⋅ 𝑇𝑇 −
! 𝑇𝑇
!!
+ 𝑃𝑃! +𝜅𝜅𝜅𝜅!"
+ 𝜅𝜅!"
⋅ 𝑇𝑇! − 𝑇𝑇! 𝑑𝑑𝑑𝑑 !! −
!𝑇𝑇! 𝑑𝑑𝑑𝑑 (1)
! ⋅ 𝑇𝑇
𝐶𝐶!! !!!,!!!
! 𝑇𝑇
𝜅𝜅!!"𝑇𝑇!⋅ =𝑇𝑇 !!𝑇𝑇!−
+!
!!!,!!!
0
ca
0
where: Ti0 = initial
temperature
cell i,where:
Cth = heat
cell,capacity
Pi =of
heat
in capacity
cell
i, κica of
= cell,
temperature
of
i, Cthtemperature
= of
heat
ofdissipation
cell,
heat
dissipation
in cell
κi =diss
where:
Ti = initialof
=capacity
initial
cell
i, CthP=i =heat
Pi =i, heat
Ti cell
a
cc
a conducheat conduction
coefficient
fromcoefficient
cell i heat
to ambient,
Tia i=to
ambient
temperature
i, κijcc= heat
ambient
temperature
cell i, κijtemperature
= heat
heat
conduction
from
cell
ambient,
Ti = cell
ambient
conduction
coefficient
from
ioftocell
ambient,
Ti =of
tion coefficientconduction
from cell icoefficient
to cell j. from
cell
i
to
cell
j.
conduction coefficient from cell i to cell j.
The heat production
of cell
i consists
a part that arises
from
the entropy
change
of the change
reactionofand
a
The heat
production
of of
cell
of a part
from
entropy
the
Thei consists
heat production
ofthat
cellarises
i consists
ofthe
a part
that arises from
thereaction
entropyand
chang
part caused byathe
overpotential
voltage.
In
our
model
it
is
calculated
as
[5]:
part caused by the overpotential voltage. In our model it is calculated as [5]:
a part caused by the overpotential voltage. In our model it is calculated as [5]:
𝑃𝑃! = 𝐼𝐼 ⋅ 𝑈𝑈!"#,! − 𝑈𝑈!,!
𝑃𝑃! = 𝐼𝐼 ⋅ 𝑈𝑈!"#,! − 𝑈𝑈!,!
Uocv = open
terminal
cell i and
Ut,i =voltage
terminalofvoltage
of cell
Uocv
= open
terminal
cell i and
Ut,i =i. terminal voltage of cell i.
Uocv = open terminal
voltage
of cellvoltage
i and
Uof
t,i = terminal voltage of cell i.
Wethe
didreversible
not take heat
the reversible
effectthe
into
account.heat
Weheat
measured
the
of a cell
We into
didheat
not
take
reversible
effect
into account.
We
measured
theby
heat
We did not take
effect
account.
We
measured
the
capacity
of heat
a cellcapacity
by memeans
of
a
calorimeter
and
found
a
value
of:
of calorimeter and found a value of:
ans of a calorimeter and found a valuemeans
of:
Cheat = 4090 [J/K].
Cheat = 4090 [J/K].
Cheat = 4090 [J/K].
-1
-1 -1
-1 [6].
-1
This corresponds
a specific heat
of
1.3capacity
[J·g-1·Kto
].of
15%
than
reported
⋅Kmore
].capacity
This
is 15%
more
than
by [6].
Thistocorresponds
to acapacity
specific
heat
1.3 is
[J⋅g
⋅K ].reported
This is 15%
more tha
This corresponds
aThis
specific
heat
of 1.3
[J⋅gby
The heat conduction
coefficients
have
been determined
bydetermined
measuring
the
stationary
temperature
rise
The heat
conduction
coefficients
been
measuring
the stationary
temperature
The heathave
conduction
coefficientsby
have
been determined
by measuring
therise
station
of all cells when
the
pack
is
loaded
by
an
80
[A]
alternating
charge
/
discharge
current
(see
Fig.5).
of all cells when the pack is loaded by an 80 [A] alternating charge / discharge current (see Fig.5).
of all cells when the pack is loaded by an 80 [A] alternating charge / discharge cur
4. Battery pack model
In this section we discuss the battery pack model subsystem (see Fig.2). This subsystem calculates the
cell voltages and temperatures of all cells of a battery pack. In principle, the user may define any cell
modeling method such as analytical, electrochemical or electric circuit model techniques that can be
implemented in Simulink. In this paper we focus on the cell modeling by means of electric circuit models
because of its relative simplicity. A simple model is preferable because the model must be calculated
real time for each cell.
Figure4: Battery pack that consists of 12 cells.
50
Figure5: Symmetrical charge/discharge current that is used to heat up the battery cells.
51
Figure5: Symmetrical charge/discharge current that is used to heat up the battery cells.
Because
of the symmetry
of the current,
the average
internal
power
lossbattery
in the cell
Figure5:
Symmetrical
charge/discharge
current
that is used
to heat
up the
cells.can be calculated
Because of the symmetry of
as:the current, the average internal power loss in the cell can be calculated as:
Because of the symmetry of the current, the average internal power loss in the cell can be calculated
!
1
as:
𝑈𝑈 ⋅ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐼𝐼 ⋅ 𝑑𝑑𝑑𝑑 (2)
𝑃𝑃! = 80 𝐴𝐴 ⋅ ⋅
𝑇𝑇 ! !
!
1
= 80 U
𝐴𝐴 =⋅ terminal
⋅
𝑈𝑈 ⋅ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐼𝐼 ⋅ 𝑑𝑑𝑑𝑑 (2)
𝑃𝑃!where:
i
𝑇𝑇 ! ! cell voltage of cell i, T = period time of current.
4.2.1 Open terminal voltage measurements
We measured the terminal voltage during a C/50 charge and discharge current at temperatures of
T=0[°C], T=25[°C] and T=40[°C]. Fig.6 shows the measurement results. The graphs show a clear hysteresis effect that increases at low temperatures. This effect has also been reported in literature [7], [8]. In
our model we didn’t include the hysteresis effect. We used the average value of the terminal voltage of
the charge and discharge curve as the open circuit voltage as a function of the SoC.
where: Ui = terminal cell voltage of cell i, T = period time of current.
Table 1 lists the average power losses and temperature rise of each cell.
cell voltagerise
of of
celleach
i, T =
period time of current.
where:losses
Ui = terminal
Table 1 lists the average power
and temperature
cell.
Table 1 lists the average power losses and temperature rise of each cell.
Table1: Power losses and temperature rises of the cells due to an alternating 80[A] charge/discharge
Table1: Power losses and temperature rises of the cells due to an alternating 80[A] charge/discharge
current.
current.
Table1:
cell nr. Power
1 losses
2 and3 temperature
4
5rises of
6 the cells due to an alternating 80[A] charge/discharge
2
3
4
5
6
current.
P [W]
4.1
4.3
4.4
4.1
4.5
4.5
cell nr.
1
Pi [W]
4.1
ΔTi [°C]
9.4 Pi [W] 11.5
4.1
cell nr.
7
Pi [W]
4.3
ΔTi [°C]
i
4.3
4.4
4.1
4.5
ΔTnr.
cell
i [°C] 1 9.4 2 11.5 3 12.5 4 13.2 5 13.1 6 13.4
4.3 12.5
4.4
4.1 13.2
4.5
4.5 13.1
cell
nr. 9.4
7
8
9
10 13.1
11 13.4
12
11.5
12.5
13.2
ΔT
i [°C]
Pi [W] 8 4.3
4.5 9 4.2
4.7 104.3
4.7 11
13.3 11
11.6 129.7
ΔTnr.
cell
i [°C] 7 13.4 8 13.2 9 13.7 10
4.5
4.2
4.7
4.3
4.5
13.4
12
4.7
Pi [W]
4.3
4.5
4.2
4.7
4.3
4.7
13.4
13.2
13.7
13.3
11.6
9.7
Based
on
the 13.2
construction
of the 11.6
battery9.7
pack, we assume three independent heat conduction
13.4
13.7 13.3
ΔT
i [°C]
coefficients. These are:
ca
Based on the construction
battery
pack, we assume
three
independent
heat
conduction
coeffiκof
: the
heat
coefficient
of cell
1 pack,
and 12we
(the
cells
on three
the edge
of the pack)
to ambient
1 the
Based
on
the conduction
construction
of the
battery
assume
independent
heat
conduction
ca
cients. These are:
κ2 : the heat
conduction
coefficients.
These
are: coefficient of all cells but cell 1 and 12 to ambient
κ1ca: the heat conduction coefficient
of cell
1 and 12coefficient
(the cells on the edge
of
the pack)
to ambient
cacc
theheat
heat
conduction
two12
adjacent
cells
or i=j-1).
κ1κij : :the
conduction
coefficient ofbetween
cell 1 and
(the cells
on (i=j+1
the edge
of the pack) to ambient
ca
κ2 : the heat conduction coefficient
of all cells but cell 1 and 12 to ambient
ca
heat
conduction
coefficients
cells1 are
to be 0.
κ2The
: the
heat
conduction
coefficientofofnon-adjacent
all cells but cell
andassumed
12 to ambient
κijcc: the heat conduction coefficient
between two adjacent cells (i=j+1 or i=j-1).
cc
κij : the heat conduction coefficient between two adjacent cells (i=j+1 or i=j-1).
The heat conduction coefficients
of non-adjacent cells are assumed to
0.
ca be ca
cc
Figure 6: The graphs on the left show the charge and discharge curve at T=0[°C], T=25[°C], and T=40[°C] at a charge and discharge
rate of C/50.The graph on the right shows the average terminal voltage of charge and discharge curve as a function of the SoC.
4.2.2 EIS-plot measurements
We measured the cell impedance as a function of the SoC at a temperature of 0 [°C], 25 [°C] and [40°C].
The measurements were carried out by means of an IVIUM STAT impedance analyzer in a frequency
range of 50 [mHz] to 8 [kHz] while discharging the cell at a current rate of C/10. Fig.7 shows the EIS-plots
that are made out of the measurement results. We have limited the frequency range of the EIS-plots
from 50 [mHz] to 659 [Hz], because at higher frequencies the self inductance of the cell was getting the
dominant impedance and we did not include this in our model.
andare
κij assumed
were determined
Theheat
three
heat conduction
coefficients
κ1 , κ2 cells
The
conduction
coefficients
of non-adjacent
to be 0. by least squares fitting of
the data of tableca 1 and
the
equation
3.
ca
cc
The three heat conduction coefficients κ1 , κ2 and κij were determined by least squares fitting of the
!"#$$%
ca
ca
cc
The three
coefficients κ1 , κ2 and κij were determined by least squares fitting of
data of table 1 and the equation
3. heat! conduction
!"
!!
𝜅𝜅 ⋅ Δ𝑇𝑇! − Δ𝑇𝑇! (3)
𝑃𝑃 = 𝜅𝜅 ⋅ Δ𝑇𝑇! + the! data! of table
1 and the!"equation
3.
!!!,!!!
!"#$$%
!!
𝜅𝜅!"
⋅ Δ𝑇𝑇! − Δ𝑇𝑇! (3)
𝑃𝑃! = 𝜅𝜅!!" ⋅ Δ𝑇𝑇!! + We found the following
values:
!!!,!!!
We found the following values:
We found the following values:
κ1ca = 0.47 [W/K]
κ2ca = 0.34 [W/K]
κijcc = 0.06 [W/K]
4.2 Electric submodel
In order to determine the electric model, we have executed three kinds of measurements:
1. Open terminal voltage measurements
2. EIS-plot measurements
3. Current pulse measurements
52
Figure 7: EIS-plots of the cell impedance at three temperatures.
The graphs show the impedances in the frequency range of 0.05
[Hz] ≤ f ≤659 [Hz].
53
25
0.24
21
5.0
40
0.04
88
3.5
We have investigated the dependency of Rs and Cs on chargin
We did not measure a clear dependency. Therefore, we assum
4.2.3 Current pulse measurement
We have measured the pulse response of 12 cells in series that are loaded by the pulse shaped current
profile that is shown in Fig.8. We have measured the pulse response for charge and discharge current
pulses. The measurement has been performed at room temperature (22 [°C]).
charging and discharging.
Table2: Values of Rs, Cs and τs at three temperatures. The time constant τs is calculated as: τs= Rs · Cs
The values of Rla, Rlb, Cla and Clb were determined from the
T [°C]
Rs [mΩ]
Cs [F]
τs [ms]
0
1.4
6.4
9.0
25
0.24
21
5.0
40
0.04
88
3.5
25
0.24
21
40
0.04
88
25
5.0
40
0.24
21
sample time of the pulse current measurements was 0.1 [s].
the time constant of the Rs//Cs circuit, so the transient beha
determined by the Rla//Clb and Rlb//Clb circuits.
By curve fitting of the cell voltage during the rest period interva
a function of the state of charge.
5.0
0.04
88
3.5
!!
!!
!! (4)
𝑈𝑈!"## 𝑡𝑡 =of 𝑈𝑈R!s+and
𝐼𝐼!"#$Cs∙ on
𝑅𝑅!"charging
∙ 𝑒𝑒 !! + 𝑅𝑅and
3.5 have investigated the dependency
!" ∙ 𝑒𝑒discharging
We
at room tem
We have investigated the dependency of Rs and Cs on charging and discharging at room temperature.
We did
have
investigated
dependency
Rs and aCclear
and
at
room
temperature.
We
did not of
measure
dependency.
Therefore,
assumed
thatboth
the values above app
s on
We
not
measure a the
clear
dependency.
Therefore,
wecharging
assumed
thatdischarging
the values we
above
apply
for
We did not
measure
a clear charging
dependency.
Therefore, we assumed
that
the values above apply for both
and discharging.
Where: U
charging
and
discharging.
cell(t) = cell voltage as function of time, U0 = station
charging and discharging.
pulse (-50 [A] for discharge pulses and 50 [A] for charge pul
The
values
of
R
,
R
,
C
and
Clb were
determined
from the[10].
pulse
la
lb from
la the
The values of Rla, Rlb, Cla and Clb were determined
current
measurements
Thecurrent
sam- measurements
Rpulse
la//Cla circuit, τlb = Rlb⋅Clb = time constant of the Rlb//Clb circuit
Thetime
values
of R
, Cla and
Clb were
determined
from
theismeasurements
pulse
[10].
The
sample
time of
the
current
was
[s]. This
isthe
more
la, Rlb
ple
of the
pulse
current
measurements
waspulse
0.1 [s].
This
more current
than 10 measurements
times0.1
higher
than
timethan 10 times h
sample time
of R
the
pulse
current
measurements
wasR0.1
[s].
This is
more
than
10
times
higher
than
the
constant behavior
of the
//CWe
so
the
transient
behavior
of
the
measured
constant
of the
sotime
the
transient
the
measured
pulse
response
is determined
sof
s circuit,
s//C
s circuit,
didn’t
measure
the
temperature
dependency
of R pulse
, R , re
C
la
lb
thethe
time
//Cs circuit, by
so the
the Rtransient
behavior
of the
measured pulse
is polarization of
by
Rlaconstant
//Clb and of
Rlbthe
//ClbRcircuits.
Rlb//Clb circuits.
sdetermined
la//Clb and
temperature
dependency
of the response
concentration
By
curve fitting
of the
celllb voltage
during
theofrest
intervals
to equation
4, we intervals
found thetovalues
as 4, we found the
and
circuits.
determined
by the
Rla//C
By Rcurve
the period
cell voltage
during
the rest
period
equation
lb//C
lbfitting
[9] for
LiFePO
4 cells. We used the empirical relations describe
aByfunction
of theofstate
of charge.
curve fitting
the cell
voltage
during
intervals to equation 4, we found the values as
a function
of the rest
stateperiod
of charge.
a function of the state of charge.
Figure 8 left: Current and voltage of cell 1 during a discharge pulse current profile. Each pulse lasts 600 [s] and is followed by a
600 [s] rest period. Figure right: Current and voltage of cell 1 during a charge pulse current profile. Each pulse lasts 600 [s] and
is followed by a 600 [s] rest period.
4.2.4 Battery model
The base of our battery model is the practical circuit-based model proposed in [9]. This model reduces
a typical Randell circuit that applies for LiFePO4 cells to a second order impedance circuit. We adapted
this circuit to a third order circuit as shown in Fig.9. In Fig.9, Rb represents the bulk resistance, the parallel circuit of Rs and Cs models the activation polarization and the parallel circuits of Rla//Cla and Rlb//Clb
model the concentration polarization.
𝑈𝑈!"##
𝑈𝑈!!!"## 𝑡𝑡 = 𝑈𝑈!!! + 𝐼𝐼!"#$ ∙ 𝑅𝑅!" ∙
𝑡𝑡 = 𝑈𝑈! + 𝐼𝐼!"#$ ∙ 𝑅𝑅!" ∙ 𝑒𝑒 !! + 𝑅𝑅!" ∙ 𝑒𝑒 !! (4)
We determined the values of Cs and Rs from the half circles of the EIS-plots. The EIS-plots of Fig.7 show
that Cs and Rs have a hardly noticeable dependency on the SoC. Therefore, we neglected this dependency in our model. However, the EIS-plots of Fig.7 do show a dependency on the temperature. Table 2
shows the values of Cs and Rs that we found via curve fitting of the EIS-plots.
!!
!! (4)
!" ∙ 𝑒𝑒 𝑇𝑇)
= 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅+!"#𝑅𝑅(𝑠𝑠𝑠𝑠𝑠𝑠,
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
voltage,
Where: Ucell(t) = cell voltage as function of time, U0 = stationary
𝑇𝑇) Ipuls = magnitude
𝑅𝑅 (𝑠𝑠𝑠𝑠𝑠𝑠,
Rla//Cla circuit, τlb = Rlb⋅Clb = time constant of the Rlb//Clb circuit.
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
We didn’t measure the temperature
dependency of Rla, Rlb, C
and C
𝐶𝐶!"
22°𝐶𝐶)
la (𝑠𝑠𝑠𝑠𝑠𝑠,
lb. Instead, we assume
We didn’t measure the temperature dependency of Rla, Rlb, Cla and Clb. Instead, we assumed
that the
We didn’t measure the temperature
dependency
of Rlaof, R
we assumed
that the
temperature
dependency
the
concentration
polarization
of our cells
is the same as is me
lb, C
la and Clb. Instead,
temperature dependency of the concentration polarization of our cells is the same as is measured in [9]
temperature dependency of[9]
theforconcentration
polarization
of our
cells isrelations
the same
as is measured
LiFePO4 cells.
We used the
empirical
described
in(𝑠𝑠𝑠𝑠𝑠𝑠,
[9] and
𝑇𝑇) inscaled them as fol
𝐶𝐶!"
for LiFePO4 cells. We used the empirical relations
described
in(𝑠𝑠𝑠𝑠𝑠𝑠,
[9] and
scaled
them
as follows:
𝑇𝑇) =
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠,
22°𝐶𝐶)
∙
𝐶𝐶!"#
[9] for LiFePO4 cells. We used the empirical relations described in [9] and scaled them as𝐶𝐶follows:
!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
Charging:
Discharging:
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
Discharging:
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
Charging:
is applied in our battery pack simulator.
Charging:
!"
Where:
(t) = cell voltagepulse
as
ofoffor
time,
UU0 0= =stationary
voltage,
= magnitude
current
𝑇𝑇)50= [A]
𝑅𝑅Ipuls
(𝑠𝑠𝑠𝑠𝑠𝑠,
22°𝐶𝐶)pulses),
∙ of of
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠,
asfunction
function
time,
stationary
voltage,
Ipuls
=
magnitude
Where: U
Ucell
!"#
(-50 [A]
discharge
pulses
and
for
charge
τla22°𝐶𝐶)
= Rla⋅Cla = time const
cell(t) = cell voltage
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠,current
pulse
(-50
[A]
for
discharge
pulses
and
50
[A]
for
charge
pulses),
τ
=
R
·C
=
time
constant
of
the
la
la
la
pulse (-50 [A] for dischargeRpulses
and
50
[A]
for
charge
pulses),
τ
=
R
⋅C
=
time
constant
of
the
la Rlb
la//Clb circuit.
la//Cla circuit, τlb = Rlb⋅Clb = time constantlaof the
Rla//Cla circuit, τlb = Rlb·Clb = time constant of the Rlb//Clb circuit.
Charging: Figure 9: Third order impedance circuit that
!!
𝑒𝑒 !!
Discharging:
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!"circuit
(𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
Where: Rlc(soc,T), Rld(soc,T), Clc(soc,T)
and𝑇𝑇)Cld(soc,T) are the
parameter equations given in [9].
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠,
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
In the equations above, the subscript
is used for charging and d for discharging.
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠,c 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
55
𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝑅𝑅!"# (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
𝑅𝑅!" (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶)
𝐶𝐶!" (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇)
𝐶𝐶 (𝑠𝑠𝑠𝑠𝑠𝑠, 𝑇𝑇) = 𝐶𝐶 (𝑠𝑠𝑠𝑠𝑠𝑠, 22°𝐶𝐶) ∙
Discharging:
54
Where: Rlc(soc,T), Rld(soc,T), Clc(soc,T) and Cld(soc,T) are the circuit parameter equations given in [9].
In the equations
above,
the subscript
c isand
used
charging
andcircuit
d for discharging.
Where:
Rlc(soc,T),
Rld(soc,T),
Clc(soc,T)
Cldfor
(soc,T)
are the
parameter equations given in [9].
In the equations above, the subscript c is used for charging and d for discharging.
Fig.10 shows the values of Rla, Rlb, Cla and Clb for charging and discharging.
Fig.10 shows the values of Rla, Rlb, Cla and Clb for charging and discharging.
Fig.10 shows the values of Rla, Rlb, Cla and Clb for charging and discharging.
5. Model validation
In this section we discuss the model validation. We do this by comparing the cell voltages and temperatures of the battery pack shown in Fig.4 to the simulated results of a model of the same pack. As
the validation current profile, we used the measured current of the motor controller of an electric Fiat
Doblo while driving the low power NEDC [11]. We have loaded the battery pack twice with the measured current. We introduced a 300 [s] rest period at the end of the first and second current profile. Fig.11
shows the validation current. The validation current varies between -203 [A] and 367 [A] and has a standard deviation of 94 [A]. The sample time of the validation measurement is 0.1 [s].
Figure11: Validation current profile
Figure 10: Values of R , R , C and C for charging and discharging. The solid lines are the graphs
la
lb
la
lb
Figure 10: Values of Rla, Rlb, Cla and Clb for charging and discharging.
The
solid
lines are
the graphs for discharging; the dashed
lines are the graphs for charging.
for
discharging;
theofdashed
are C
the
graphs
for charging.
Cla and
charging
and discharging. The solid lines are the graphs
Figure
10: Values
Rla, Rlb,lines
lb for
for discharging; the dashed lines are the graphs for charging.
The measurement started with a fully charged battery pack at 21 [°C]. At the end of the measurement,
The bulk resistor Rb is determined from the pulse current measurements. Because the sample time isthe SoC was reduced to 49%.
much
larger
thancurrent
constant
of the
Rs//Cscurrent
andsample
much
smaller
the timethe
constants
of the
determined
from
pulse
measurements.
sample time
isFig.12 left shows the averaged measured and simulated cell voltage as a function of the time. We
The
bulk
resistor
Rthe
The bulk resistor Rb is determined
from
the
pulse
measurements.
Because
the
time
is thanBecause
b is time
R
//C
and
R
//C
circuits,
it
applies:
much larger than the time constant
of
the
Rs//Cs
and
much
smaller
than
the
time
constants
of
the
averaged the cell voltages of all cells. Fig.12 right shows the difference of the averaged measured and
la
lalarger than
lb
lbthe time constant of the Rs//Cs and much smaller than the time constants of the
much
simulated cell voltages.
Rla//Cla and Rlb//Clb circuits, it applies:
Rla//Cla and Rlb//Clb circuits, it applies:
∆U
𝑅𝑅! + R ! = −
∆I
∆U
𝑅𝑅! + R ! = −
∆I
Where: ΔU and ΔI are the voltage and current change, measured at the end of the pulse and start of
Where: ΔU and ΔI are the voltagethe
and
current
change,
at and
the end
of the
pulse and
start ofatthe
rest
period.
Where:
ΔU
and
ΔI aremeasured
the voltage
current
change,
measured
the end of the pulse and start of
rest period.
the rest period.
At T=22 [°C] we found 0.8 [mΩ] for the sum of Rb and Rs. For the bulk-resistance we then find:
At T=22 [°C] we found 0.8 [mΩ] for
the sum of Rb and Rs. For the bulk-resistance we then find:
At T=22 [°C] we found 0.8 [mΩ] for the sum of R and R . For the bulk-resistance we then find:
b
Rb = 0.5 [mΩ]
s
Rb = 0.5 [mΩ]
Rb = 0.5 [mΩ]
The value of Rb found above is about 0.6 [mΩ] less than the value that can be derived from the EIS-
The value of Rb found above is about 0.6 [mΩ] less than the value that can be derived from the EISplots.value
Thisofdifference
causedabout
by the
resistance
the that
connections
to the from
IVIUM
above
0.6contact
[mΩ] less
than the of
value
can be derived
theSTAT
EISRb found is
plots. This difference is caused byThe
the contact
resistance
of theisconnections
to the IVIUM
STAT impedanimpedance
analyzer
that
is
used
to
measure
the
EIS-curves.
plots.
difference is caused by the contact resistance of the connections to the IVIUM STATFigure 12 left: Simulated and measured cell voltage. The simulated and measured voltages are averaged over all 12 cells. The
ce analyzer that is used to measure
theThis
EIS-curves.
impedance analyzer that is used to measure the EIS-curves.
Finally, the
Finally, the SoC in our model is calculated
by:SoC in our model is calculated by:
graph on the right shows the difference of the averaged simulated and measured cell voltage.
Finally, the SoC in our model is calculated by:
𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆𝑆𝑆! − 𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆𝑆𝑆! − !
𝐼𝐼 ⋅
!
!
𝐼𝐼𝐶𝐶⋅
! !
𝐶𝐶!
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Where: SoCi is the initial state of charge of cell i and Ci is the capacity of cell i.
56
Fig.12 shows a gradually decreasing difference between the simulated and measured cell voltage of +20
[mV] at the start of the test to -20 [mV] at the end of the test. This might be explained by the hysteresis effect that is not taken into account by our model. Also the modeling of the activation polarization
effects by only two parallel circuits of R and C might explain this difference.
57
The absolute difference of the simulated and measured voltages is in the entire time range below the
50 [mV]. The standard deviation of the difference of simulated and measured voltage is 14 [mV]. This
can be translated to a difference in the overall resistance of 0.15[mΩ], which is about 10% of the sum of
all resistances in the model.
Fig.13 shows the measured and simulated temperature of one cell at the edge of the pack and a cell in
the middle of the pack. Also, the measured ambient temperature is shown.
6.1
Accuracy of emulated voltages
The accuracy of the emulated voltages of the BPS is determined by comparing the measured emulated
cell voltages to the model output. We use the current profile shown in Fig.11 for this test. The sample
time of the test is 0.1 [s]. Fig.14 shows the emulated and simulated cell voltage of cell 1. The measured
and simulated cell voltages of the other cells are comparable. The difference between both signals becomes only visible when zoomed in to a small area of the graph.
Figure14: Emulated and simulated voltage of cell 1
when the battery pack is loaded with the validation
current profile of Fig11.
Figure13: Simulated and measured temperature of cell 1 at the
edge of the battery pack and cell 6 in the middle of the battery
pack. The black line is the measured ambient temperature.
Fig.13 shows that the difference between the simulated and measured temperatures gradually increases to about 2 [°C] at the end of the test. This difference can be explained by the raise of the ambient
temperature during the measurement, while the ambient temperature in the model has been constant.
We also observe more high frequency components in the response of the measurements. We explain
this by the heat dissipation at the poles caused by the contact resistance. The temperature sensors are
mounted on the poles of the cells, so the measured temperatures are relatively strongly influenced by
the local heat dissipation on and around the poles. In the model, the poles and connection strips are
part of the heat capacity of the cell. No temperature gradients exist in the cell model which results in a
flatter temperature graph.
The standard deviation of the difference of the emulated and simulated signal is 1.1 [mV]. This corresponds to the resolution of the digital to analog converter that is used in the isolation amplifier circuit.
6.2
Transient behaviour of emulated voltages
We determined the transient behaviour of the isolation amplifiers by measuring the rise and fall time on
a step response. For this test, we emulated a current sensor with a sensitivity of 100 [A/V]. We examined the current sensor output voltage on an oscilloscope when a block shaped current is simulated that
alternates between -200 [A] and +200 [A] at a frequency of 10 [Hz]. Fig.15 shows the rising and falling
edge of the emulated current sensor voltage.
6. Battery pack simulator tests
In order to evaluate the BPS, we have connected it to an Elithion Lithiumate Pro BMS. The BPS was
loaded with the twelve cells battery pack model that is discussed in the previous sections. We tested the
following features:
- The accuracy of the emulated cell voltages of the BPS compared to the model output.
- The transient behaviour of the emulated voltages.
- The SoC estimation of the BMS when it is connected to the BPS.
- The correct working of the under and over voltage detection of the BMS.
- The maximum sample rate of the BPS
As explained before, we cannot test the balancing functionality of the BMS because of the limited output current of the isolation amplifiers. Also, we did not connect the temperature sensors of the cell monitoring units of the BMS to our BPS, so the temperature related functionality of the BMS is not tested.
58
Figure 15: step response of the isolation amplifier on the rising edge (left scope image) and falling edge (right scope image) of a
block shaped pulse. The time scale of the scope images is 1 [μs/div].
Fig.15 shows a rise time of about 3 [µs] and a fall time of 2 [µs]. This is much less than the minimum
simulation time of 254 [µs] that originates from the limitations of the SPI bus. This means that the transient behaviour of the hardware hardly plays any role in the dynamic behaviour of the whole system.
59
6.3
Evaluation of SoC determination
We evaluated the SoC determination of BMS by simulating the battery pack when it is loaded by a
current profile that consists of twice the validation current profile shown in Fig.11. Figure 16 shows the
SoC, simulated by the BPS, and the SoC estimation of the BMS as a function of the time.
For our model and xPC target, it applies that the maximum sample rate decreases below 3.9 [kSample/s]
when the number of cells is more than 140.
For most battery management systems, this sample rate is sufficiently high and there is enough space to
implement more complex models that take more calculation time.
7. Conclusions and further study
Figure 16: SoC estimation of the BMS
and the SoC calculated by the model.
The difference between the SoC estimation of the BMS and the SoC calculation of the model is less than
0.5%. This corresponds to the resolution of the SoC estimation of the BMS, which is 1%.
We also tested the response of the SoC estimation by the BMS on a stepwise change of the SoC of the
BPS from 10% to 90%. We saw that the SoC of the BMS estimation did not change. Instead, we saw that
the state of health (SoH) was adapted by this change. We also entered a stepwise change of the SoC
when the BMS was switched off. Again, the BMS did not report a change in the SoC estimation after
turning it on again.
It seemed to us that the SoC estimation was only readapted to 100% after the maximum cell voltage
was reached while charging the battery.
6.4
Under and overvoltage detection
Under and overvoltage detection is tested by changing the SoC of the BPS abruptly so that the cell voltages exceeded the under and overvoltage limits. We found that the BMS detected the exceeding of the
limits correctly.
6.5
Maximum sample rate
The data transfer on the SPI bus sets the upper limit of the sample rate to 3.9 [kSamples/s]. This limit
however may be decreased by the execution time of the model. The execution time depends heavily on
the speed performance of the xPC target, the complexity of the model and the number of cells of the
battery pack. Table 3 lists the execution time of the model that is described in this paper as a function of
the number of cells. We have used a common PC with pentium 4 3.00 GHZ processor and 4 GByte RAM
as xPC target.
Table3: Execution time of the model as a function of the number of cells.
number
of cells
1
2
3
6
12
13.9
16.1
17.5
21.6
30.2
number
of cells
24
48
60
120
248
execution
time [µs]
48
84
103
207
477
execution
time [µs]
60
The BPS, proposed in this abstract, provides a flexible architecture that can be used to emulate cell
voltages and temperatures of a battery pack. We evaluated the BPS by implementing a LiFePO4 battery
pack. Test results show that relatively complex models can be simulated at high sample-rates with an
acceptable accuracy.
We connected a Lithiumate Pro BMS to our BPS. We found that the basic functionality of this BMS could
be tested well with our BPS.
A major shortcoming of the BPS now is the lacking of the capability to test the balancing feature of BMS.
Further work will focus on the implementation of this.
Acknowledgments
We wish to thank Strukton Embedded Solutions b.v. for putting an Elithion Lithiumate Pro BMS at our
disposal.
This work is supported by Raak PRO EPT.
References
[1]D. Skalny et. Al., Battery Management System (BMS) Evaluation Toolset, http://oai.dtic.mil/oai/
oai?verb=getRecord&metadataPrefix=html&identifier=ADA548816
[2]D. Bernardi. et Al., A general energy balance for battery, J. Electrochem. Soc., vol. 132 (1985), no.
1, 5–12
[3]Lijun Gao et. Al., Dynamic Lithium-Ion Battery Model for System Simulation, IEEE Transaction on
components and packaging technologies, vol. 25, no 3 (sept. 2002), 495-505
[4]Ralf Benger et. Al., Electrochemical and thermal modeling of lithium-ion cells for use in HEV or EV
application, EVS24 (2009)
[5]K. Ondar et. Al., Thermal behaviour of small lithium-ion battery during charge and discharge cycles, Journal of power sources 158 (2006), ISSN: 1873-2755, p.535-542
[6]E. Prada et. Al., Simplified electrochemical and thermal model of LiFePO4 graphite Li-Ion batteries
for fast charge applications, Journal of The Electrochemical Society 159, 9 (2012) A1508-A1519
[7]Yujie Zhu et. Al., Strain accommodation and potential hysteresis of LiFePO4 cathodes during lithium ion insertion/extraction, Journal of Power Sources 196 (2011), 1442–1448
[8]M. Roscher et. Al., Dynamic electric behavior and open-circuit-voltage modeling of LiFePO4-based
lithium ion secondary batteries, Journal of Power Sources 196 (2011), 331–336
[9]Long Lam et. Al., A Practical Circuit-based Model for Li-ion Battery Cells in Electric Vehicle Applications, Telecommunications Energy Conference (INTELEC), 2011 IEEE, ISSN: 2158-5210
[10]B. Schweighofer et. Al., Modeling of high power automotive batteries by the use of an automated
test system, IEEE Trans.lnstrum. Meas., vol. 52, no. 4, pp. 1087-1091, (2003)
[11]P.A. Veenhuizen et. Al., Experimental assessment of an energy management strategy on a fuel cell
hybrid vehicle, EVS26 (2012)
61
The modelling of the temperature at the poles and
core of large prismatic LiFePO4 cells
Stefan van Sterkenburg, Bram Veenhuizen
2. Cell assembly
In order to fully understand the thermal behavior of a cell, we opened a cell to see what is inside. We
found that the cell core exists of 112 copper current collectors, holding the Lithium carbon material and
the same amount of aluminum current collectors holding the LiFePO4. The sheets are located in between the harmonica-wise folded separator emerged in a liquid PF6 electrolyte. The battery case is 5 mm
thick polypropylene. Figure 2 shows the construction of a prismatic battery cell. Table 1 shows some of
the characteristics of the cell that are being used in our cell model.
HAN University of Applied Science, Arnhem, The Netherlands
Abstract
The safety and performance of Lithium batteries depend heavily on the cell temperature. Because of
this, battery management systems measure the temperature of the cells. When prismatic cells are used,
the temperature is often measured at the poles of the cell. This paper investigates the relation between
the temperature of the cell core and temperature at the poles of a single cell. A one dimensional model
is used to model the core of the cell. The poles are modeled as two heat capacities that are connected
to the core by thermal resistances. The model takes into account the heat dissipation because of overvoltage losses, the electric contact resistances between cables and poles and between poles and the
core of the cell and the influence of the cables that connect the battery to an external source or load.
We validated the model by comparing the measured and simulated temperatures at a load current that
gives a more or less constant power dissipation and a load current that causes a block shaped power dissipation. The validation measurements show that the model has an average error of about 5% in steady
state simulations. Both model and validation measurements show that the temperature rise at the positive pole is about 15% higher than the rise at the negative pole. The model shows that the temperature
rise of the cell core is about the same as the temperature rise of the negative pole.
We used our model to estimate the temperature of the poles and cores of a battery that consists of four
cells that and has no forced cooling. Simulations show that the difference between the temperature rise
measured at the poles and core is less than 5% in a steady state. For a current profile that has relative
many transients in it, the difference increases.
1. Introduction
The temperature is a very important quantity when it comes to the performance of a battery. When the
temperature of a cell is too high, the aging of the cell accelerates and dangerous situations may occur
when the cell temperature is out of the safety operating zone. Also, the temperature is an important factor in estimating the state of charge of a battery. Because of this, the temperatures of cells are measured
by a battery management system. When prismatic cells are used, the cell temperature is often measured
on or nearby the poles of the cell. This study investigates the difference of the temperature measured at
the poles and the average temperature inside a cell.
In the first part of this paper, we describe the modeling of a battery that consists of one cell. For this
study we used a Sinopoly lithium iron phosphate cell of 100 [Ah]. We dismounted a cell and derived a
thermal model based on the physical construction of the cell. We measured all model parameters by
relatively simple experiments that we carried out on a dismounted cell. We added a first order model of
the cable that connects the cell to an external load because we don’t want to exclude the influence of
the cable on the temperature on the poles.
In the second part of this paper we use the model to estimate the temperature at the poles and inside
the cells of a battery that consist of 4 cells in a row.
62
Figure 2: The upper figure shows in a schematic way the construction of the cell core. The pictures below show the interior of
the Sinopoly cell we have investigated.
Table 1. Characteristics of cell under study
total mass cell [kg]
3.1 [kg]
mass aluminum pole(1) [kg]
41.5 g
mass copper pole(1) [kg]
132.3g
mass and dimensions of case [kg]
500 g (l x w x h = 142mm x 61mm x 215mm)
(1)
The mass includes the weight of the nut that is used to fix the pole to the case.
63
The poles consist of a solid copper or aluminum block on which a cylindrical top is attached. The cylindrical top holds a tapped hole used for bolts to connect a current cable or connector to it. The thread on
the outside is used to fix the poles to the case. The copper and aluminum tabs on the current collector
foils are squeezed up against both sides of the poles by means of two bars.
The construction of the cell implies two electric contact resistances (ECR) at each pole. The first one is
the ECR from cable or connector to the pole; the other is the ECR from pole to current collector foils.
The contact resistance is an important phenomenon to explain the difference in temperature at the
poles and core. Literature [1] shows that ECR may cause a significant contribution of about 7% to the
power losses of a cell.
3. Thermal model
Thermal cell modeling can be done in different ways. One of the most complete and accurate models
nowadays are the three-dimensional models that couple the electrochemical and thermal behavior and
use finite element methods to obtain the three-dimensional temperature distribution inside a cell ([2],
[3], [4]).This study ,however, is basically mentioned to determine the difference in temperature measured at the poles and the average cell temperature. Therefore, we have chosen to use a relatively simple
1-D cell model. Figure 3 shows the thermal RC-network representation of the model.
The cell model consists of three heat capacities: one heat capacity represents the core of the cell (the
inner part of the cell and case); the other two heat capacities represent the two poles. The core and
poles are thermally connected to each other by heat resistances. All heat capacities are connected to
the ambient by combined heat conduction and heat convection resistances.
We added three heat sources in the model. Heat source qc represents the heat that is generated in the
core when a current is running because of the overvoltage and the reaction entropy changes. The heat
sources qap and qcp represent the ohmic heat generation in the aluminum and copper pole because of
the contact resistances between from connectors to poles and from poles to current collector foils.
4. Determination of model parameters
This section describes the determination of the model parameters of the model shown in figure 3.
4.1. Heat capacities.
The heat capacity of the aluminum and copper pole is determined from their mass and the specific heat
capacity of aluminum (903 [J kg-1 K-1], [4]) and copper (385 [J kg--1 K-1], [4]). We find:
Cap = 37.5 [J K-1] and Ccp = 50.9 [J K-1]
The heat capacity of the core is determined by measuring the heat capacity of the whole cell, including
the poles, by placing it in a calorimeter. We found a value of: Ccell = 3.05 [kJK-1] which is in fair agreement with values found in literature [5]. The heat capacity of the core is then calculated as:
Cc = Ccell – Cap – Ccp = 2.96 [J K-1]
4.2
Heat resistances to ambient.
To determine the heat resistance of the cell to ambient, we heated up a cell to about 40 [°C] and let it
cool down in our laboratory. The time constant, with which the temperature cools down, is found to
be:τ = 4.3 [ks]. This give an overall heat resistance from cell to ambient of:
Rcell→ambient = τ / Ccell = 1.42 [K W-1]
The thermal resistance from poles to cell are calculated by the inverse of the product of surface area op
the poles (13 [cm2]) with the air and the heat convection coefficient of air to a horizontal surface
(4 [W K-1 m-2], [6]). We find:
Rpa = 188 [K W-1]
The thermal resistance of core to ambient is then calculated as:
Rca = (R-1cell→ambient – 2 ∙ R-1pa)-1 = 1.44 [K W-1].
Figure 3: Thermal RC-network model of cell. The following network components are being defined:
qap = heat generated at aluminum pole
Cap = heat capacity of aluminum pole
Rpa = thermal resistance from pole to ambient
Rcap = thermal resistance from core to aluminum pole
4.3
Heat resistances from core to pole.
The heat resistance from core to pole is determined by measuring the temperature difference between
pole and core at a known heat flow injected into the pole.
Figure 4 shows the measurement setup that is used. We placed two aluminum spacers between the
poles and a steel bar that is heated by an electric resistance wire wrapped around it. To reduce the heat
convection around the spacers, we wrapped some tissue paper around them. In the spacer, we drilled
two small holes at a distance of 20 mm from each other where we measured the temperature with a
K-type thermocouple.The lower temperature hole is located 4mm above the contact area with the pole.
We also measured the temperature at the tabs on the aluminum and copper current collector foils inside the battery. To do this, we drilled two holes in the case of the cell.
qc = heat generated in the core
Cc= heat capacity of the core
Rca = thermal resistance from core to ambient
Rccp = thermal resistance from core to copper pole
qcp = heat generated at copper pole
Cap = heat capacity of aluminum pole
64
65
Figure 4: Measurement setup to determine the thermal resistance from pole to core.
Figure 5: Measurement setup that used to measure the resistance from cable to current collector tabs.
From this measurement we calculated the thermal core to pole resistance as:
Rcore→pole = (Tpole - T3) / (qspacer + qbolt)
where: qspacer = heat flow through spacer = (T1 - T2) * kaluminum * Aspacer / 20mm
qbolt = heat flow through bolt = (T1 - T2) * ksteel * Abolt / 20mm
T1 = measured temperature on upper part of spacer
T2 = measured temperature on upper part of spacer
Tpole = temperature on top of the pole
T3 = measured temperature on the tabs of the current collectors in the cell
-1 -1
kaluminum= heat conductance of aluminum [205 W m K ],
Aspacer = cross section of spacer
ksteel = heat conductance of steel bolt [17 W m-1 K-1],
Abolt = cross section of bolt
The temperature on top of the pole (Tpole) is determined via linear extrapolation:
Tpole = T2 – 4mm / 20mm * (T1 - T2)
From the measurements and the formulas above, we find the following values for the thermal core to
pole resistances:
We measured the following values:
Electric resistance cable to copper current collector tabs = 21 [μΩ]
Electric resistance cable to aluminum current collector tabs = 87[μΩ]
The measured resistance from cable to aluminum tabs is 4 times higher than that of cable to copper
tabs. This is remarkable since the specific resistance of aluminum is only 50% higher than that of copper.
The relatively high resistance of the aluminum tabs to cable might be explained because of the higher
ECR of aluminum, but we did not investigate this further in this research.
4.5
Thermal model of the cable
The cable, that connects the cell to a source or load, contributes in different ways to the thermal behavior of the cell. On the one hand, the cable is an extra heat conductor through which heat from the pole
can flow to the environment. On the other hand, the ohmic heat dissipated in the cable may heat up
the pole and cell. Also, the cable forms an extra heat capacitance that affects the dynamic temperature
behaviour at the poles.
Figure 6 shows the simple first order thermal model we have used to take into account the influence of
the cable. The derivation of this model is given in Appendix A. Table 2 shows the parameter values of the
copper cable used in our experiments. The conductor of the cable has a cross-section of 50mm2.
Rccp = 1.7 [K W-1] and Rcap = 2.2[K W-1]
4.4
Electric resistance cable to current collector tabs
We determined the electric resistance by measuring the difference between the voltage on the tabs on
the current collectors and the ring on the cable that connects the cable to the pole as a function of the
load current. In this way, the measured resistance includes the ECR between cable and pole and the ECR
from pole to tabs. The voltage on the tabs is measured by placing two test probes into two holes we
made in the case above the tabs positions (see figure 5).
66
Figure 6: First order thermal model of cable.
67
Table 2: Parameter values of cable model.
kca (1)
[WK-1m-1]
kcu(2)
[WK-1m-1]
c (3)
[J K-1m-1]
ρe(4)
[Ωm]
Rel
[Ω]
Rc
[KW-1]
Cc
[J K-1]
τc
[s]
0.52
401
170
1.8∙10-8
7.1∙10-5
9.8
19.3
188
The third measurement is the same as the second one, only we changed the pulse length to 80 [s] and
the amplitude to 150 [A]. This measurement is meant to validate the thermal behavior at a block wave
power dissipation input with a period time of 960 [s].
(3)
c is calculated from the specific heat and specific mass of copper and cross-­‐section of the cable. It applies: kca is the-­‐1thermal
resistance
to
-­‐1
-­‐3 from cable
-­‐6
2 environment.
-­‐1
-­‐1It is calculated from the rated current of the cable. For our cable, it
[kgm ] ·∙ 50·∙10 [m ] = 170 [J K m ] c = 383 [JK kg ] ·∙8900 2
applies: A=50mm , R’=0.386 [Ω/Km], Tmax=85°C, Irated=285
[A] at T=25°C → kca = R’∙ I2 / ΔT =0.52 [W K-1m-1]
(4)
-­‐8
ρ is the specific resistance of copper at 25°C (1.8·∙10
[Ωm]) (2) e
-1 -1
kcu is the heat conductance of copper (401 [W K m ]
(1)
(3)
c is calculated from the specific heat and specific mass of copper and cross-section of the cable. It applies:
Itc =is383
notable
the[kgm
electrical
-3
-6
[JK-1kg-1that
] ∙8900
] ∙ 50∙10resistance
[m2] = 170 R
[J elK-1that
m-1] determines the strength of heat source qsc in figure 6
is
copper current collector tabs and a little less than the
(4) much higher than the resistance from pole -8
ρ is the specific resistance of copper at 25°C (1.8∙10 to[Ωm])
e
resistance from pole to aluminum current collector tabs.
5. Model validation
5
MODEL VALIDATION
Model validation
validationmeasurements
measurementshave
havebeen
beencarried
carriedout
outby
byaacurrent
currentprofile
profilethat
thatexists
existsofofsymmetrical
symmetrical
Model
discharge/charge cycles.
cycles. AA symmetrical
symmetrical discharge/charge
discharge/charge cycle
cycleisisaadischarge
dischargepulse
pulsefollowed
followedby
byaacharge
pulse with
thewith
same
and pulse
The usage
symmetrical
discharge/charge
cycles macharge
pulse
theamplitude
same amplitude
andwidth.
pulse width.
The of
usage
of symmetrical
discharge/charge
kes it possible
determine
relatively relatively
easily theeasily
powerthe
dissipation
in the cellinwithout
a complex
cycles
makes ittopossible
to determine
power dissipation
the cell needing
without needing
electric
equivalent
network model
of the
cell.ofBecause
discharge/charge
cycle is symmetrical,
there
a
complex
electric equivalent
network
model
the cell.each
Because
each discharge/charge
cycle is
is no net electrochemical
conversion
of energy,
so all electrical
energy
input
of theenergy
batteryinput
is converted
symmetrical,
there is no net
electrochemical
conversion
of energy,
so all
electrical
of the
into heat
can beinto
calculated
from
current.voltage
For oneand
discharge/charge
cycle,
battery
is and
converted
heat and
canthe
be measured
calculated voltage
from theand
measured
current. For one
the following applies:
discharge/charge cycle, the following applies:
𝑊𝑊!"!#$ = !!"!#$
!
𝑈𝑈! ∙ 𝐼𝐼! ∙ 𝑑𝑑𝑑𝑑 → 𝑃𝑃!"!#$ = 𝑊𝑊!"!#$
= 𝑇𝑇!"!!"
!!"!#$
𝑈𝑈!
!
∙ 𝐼𝐼! ∙ 𝑑𝑑𝑑𝑑
𝑇𝑇!"!#$
= 𝑈𝑈!!!"#$ − 𝑈𝑈!"#$!!"#$ ∙ 𝐼𝐼!"# (1)
Where:
Where:
= the
cell current
current during
duringthe
thecycle
cycle
IIcc =
the cell
U = the measured cell voltage during the cycle
Iabs= the (absolute) value of the current during the charge/discharge cycle
ITabs= the
(absolute) value of the current during the charge/discharge cycle
cycle = time span of charge pulse and discharge pulse
T
time
of charge
and discharge
pulse
cycle = =
Pcharge
the span
average
power pulse
dissipation
during one
discharge/charge cycle
U!"#$%&
theaverage
averagepower
cell voltage
duringduring
a charge
P
dissipation
onepulse
discharge/charge cycle
charge==the
Udischarge = the average cell voltage during discharge pulse
Ucc = the measured cell voltage during the cycle
U!"#$%& = the average cell voltage during a charge pulse
U
= the
averagethree
cell voltage
during
discharge pulse
In!"#$%&'()
total we
performed
validation
measurements.
Figure 7 shows the currents and measured heat
dissipation of the validation measurements.
In total
the first
measurement,
load current
is a continuous
series
of discharge/charge
with aheat
conIn
we performed
threethe
validation
measurements.
Figure
7 shows
the currents andcycles
measured
stant
amplitude
of
125
[A]
and
pulse
width
of
20
[s].
The
average
power
dissipation
during
this
measudissipation of the validation measurements.
rement
varies
a little because
of the
gradual
rise ofofthe
cell during the measurement
In
the first
measurement,
the load
current
is atemperature
continuous series
discharge/charge
cycles with a and
the
temperature
dependency
of
the
overvoltage
([7],
[8]).
The
first
measurement
is
basically
meant to
constant amplitude of 125 [A] and pulse width of 20 [s]. The average power dissipation during this
validate the step-response of the model.
measurement varies a little because of the gradual temperature rise of the cell during the
During the second measurement, the load current is a periodic signal that consists of 3 subsequent
measurement and the temperature dependency of the overvoltage ([7], [8]). The first measurement is
charge/discharge cycles followed by a rest period where the current is 0 [A] during 120 [s]. All discharge
basically meant to validate the step-response of the model.
and charge pulses have an amplitude of 200 [A] and pulse width of 20 [s]. This measurement is meant to
During the second measurement, the load current is a periodic signal that consists of 3 subsequent
validate the thermal behavior at a block wave power dissipation input with a period time of 240 [s]
charge/discharge cycles followed by a rest period where the current is 0 [A] during 120 [s]. All
discharge and charge pulses have an amplitude of 200 [A] and pulse width of 20 [s]. This
68
measurement is meant to validate the thermal behavior at a block wave power dissipation input with a
period time of 240 [s]
Figure 7: The load current of the cell and the power dissipation calculated according to formula 1 of the three validation
measurements. The upper-left graph shows a symmetrical block shaped current that alternately is 120[A] and -120[A].
Figure 8 shows the measured and simulated temperatures on the poles and ambient of the three validation measurements. The simulated temperature of the core of the cell is also shown, as well as the
ambient and the temperature measured on the case of the cell.
From the graphs of figure 8 we conclude that the model describes the general behavior of the cell well.
Both simulations and measurements show that the average temperature rise at the copper pole is
smaller than of the aluminum pole. Also, both simulations and measurements show that the peak-peak
value of the temperature response of the aluminum pole on the block shaped power dissipation input at
measurement 2 and 3 is much higher than that of the copper pole.
Table 3 shows the measured and simulated temperatures at the poles at the end of the measurement
(in the steady state). This table shows that the steady state difference is on average about 5%. The relative difference between the measured and simulated top-top value is higher and more unsteady.
69
Table 3: Measured and simulated temperatures at the poles at the end of the measurement.
Table 3: Measured and simulated temperatures at the poles at the end of the measurement. Aluminum pole Simulated validation Simulated Measured measurement ΔTave [°C] (1)
ΔTave [°C] 1 16.0 16.0 0% -­‐ -­‐ -­‐ 2 19.1 21.2 10% 3.5 4.5 22% 3 13.0 14.0 7% 6.0 7.1 16% (1)
(3)
error ΔTpeak-­‐peak [°C]
Measured ΔTpeak-­‐peak [°C]
(2) (2) error Copper pole Measured Simulated Measured validation Simulated measurement ΔTave [°C] ΔTave [°C] 1 14.2 13.9 -­‐2% -­‐ -­‐ -­‐ 2 16.8 18.0 7% 0.83 1.4 41% 3 11.8 11.3 -­‐4% 2.9 2.6 -­‐10% (1)
(1)
(3)
error ΔTpeak-­‐peak [°C]
(2) ΔTpeak-­‐peak [°C] (2) error (1)
ΔTave is defined as the temperature difference with the ambient, averaged over a given time interval. For validation measurement 1, we averaged over a time span that lasts 6 charge and discharge pulses. For validation measurements 2 and 3, we averaged over one cycle that consists of 3 charge and discharge pulses and the rest period. (2)
ΔTpeak-­‐peak [°C] is defined in validation measurement 2 and 3 as the maximum pole temperature minus the minimum pole temperature in one cycle, which consists of 3 charge and discharge pulses and the rest period. (3) The error is defined as the relative difference between measured and simulated value. 6.
andAND
temperature
of Aa BATTERY
batteryPACK
pack
6 Core
CORE
TEMPERATUREat
ATthe
THE poles
POLES OF
We used our model to estimate the core temperature and temperature at the poles for a battery that
Figure 8: The simulated and measured temperature at the poles, ambient and case of the three validation measurements.
We used our model to estimate the core temperature and temperature at the poles for a battery that
consists of 4 cells placed in row when it is loaded by the current profile that is used in validation
consists of 4 cells placed in row when it is loaded by the current profile that is used in validation measumeasurement 1 and 3. We assumed that the cells are places in open air so the thermal resistance
rement
1 and 3. We assumed that the cells are places in open air so the thermal resistance from cell
from
cell
cores toisambient
is comparable
thefound
one we
of cell.
a single
cell. The
cells are connected
mutually by
cores to ambient
comparable
to the onetowe
of found
a single
The cells
are mutually
2
connected
by
copper
connectors
with
a
length
of
80
[mm]
and
cross-section
2 of 100 [mm ].
copper connectors with a length of 80 [mm] and cross-section of 100 [mm ].
Figure99shows
showsthe
thebattery
battery pack
packmodel.
model. The
Thepack
packmodel
modelconsists
consistsof
of44cell
cellmodels,
models,three
threeconnector
connector
Figure
models and
and two
two cable
cable models.
models. The
The cell
cell model
model and
and cable
cable model
model are
are shown
shown in
in figure
figure 3
3 and
and 66 respectively.
models
respectively.
connector
modeled
by two heat0.5∙R
resistances
0.5·R
inheat
series
and a heat
capacity.
cona
The
connectorThe
is modeled
byistwo
heat resistances
and
capacity.
We neglected
conin series
We
neglected
the
ohmic
heat
dissipation
in
the
connectors.
the ohmic heat dissipation in the connectors.
Also,we
weadded
addedthermal
thermal resistances
resistancesbetween
between the
the cores
cores of
of adjacent cells
cells and
and adapted
adapted the
the thermal
thermal reAlso,
sistance
from
core
to to
ambient
because
thethe
contact
area
with
ambient
resistance
from
core
ambient
because
contact
area
with
ambientdiffers
differsfrom
fromthat
thatof
ofaasingle
singlecell. In
our
battery
pack
model,
we
made
the
following
assumptions:
cell. In our battery pack model, we made the following assumptions:
-- The
of the
thecell
cellcore.
core. We
We use
use the
the dependenThepower
power dissipation
dissipation of
of each
each cell
cell depends
depends on
on the
the temperature
temperature of
cy
that
is
measured
from
validation
measurement
1
and
3.
dependency that is measured from validation measurement 1 and 3.
-- The
properties of
ofthe
thecase
case and
and is
is
Theheat
heatresistance
resistancefrom
from core
coreto
to core
core (R
(Rcccc))isisdetermined
determined by
by the
the thermal
thermal properties
-1
-1
calculated
as:R
=
2*d
/
(k
∙
A
)
=
6.1
[KW
]
,
where
d
=
thickness
of
the
case
(5.5
polypropylene case
case
calculated as:Rcccc = 2*dcase
case / (kpolypropylene· Acase) = 6.1 [KW ] , where dcase = thickness of the case (5.5
-1 -1
[mm]),
k
=
conductance
of
polyprolylene
(0.15
[WK
-1 m
-1 ]) and Acase = contact area of two adjapolypropylene
[mm]), kpolypropylene = conductance of polyprolylene (0.15 [WK m ]) and Acase = contact area of two
cent cases (=305 [cm2]) 2
adjacent cases (=305 [cm ])
- The heat resistance from core to ambient is inverse proportional to the contact area of the cell with
- The heat resistance from core to ambient is inverse proportional to the contact area of the cell with
the ambient. A single cell has a contact area of 873 [cm22] and a core to ambient resistance of
the ambient.
A single cell has a contact area of 873 [cm ] and a core to ambient resistance of
1.44[KW-1-1]. We then find for the corner cells: contact area = 568 [cm2]→ Rca1 = 2.2 [KW-1-1].
1.44[KW ]. We then find for the corner cells: contact area = 568 [cm2]→ Rca1 = 2.2 [KW ].
70
71
For non-corner cells, we find:contact area = 262 [cm2]→ Rca2 = 4.2 [KW-1].
- The heat resistance of a connector that connects the positive pole of a pole to the negative pole of the
adjacent cell equals: Rcon = dpoles / (kcopper∙ Aconnector) = 1.0 [KW-1] , where dpoles = distance between the
poles (61 [mm]), kcopper = conductance of copper (401 [WK-1m-1]) and Aconnector = cross section of connectors (=150 [mm2]). The heat capacitance of a connector is calculated as the mass of the connector
multiplied by the specific heat capacity and equals 31 [JK ].
Figure 11 and 12 show the simulated temperatures at the poles and of the core when the battery is
loaded with the current profile of validation measurement 1 and 3 (see figure 7).
Figure 11: Simulated temperatures of the poles and core of the 4 cells when it is loaded by the current used in validation
measurement 1.
Figure 9: Model of the battery pack. The red colored symbols are new or changed compared to the model discussed in chapter
3 and 4.
Figure 12: Simulated temperatures of the poles and core of the 4 cells when it is loaded by the current used in validation
measurement 3.
72
73
to the
ambient
Figure A1: The heat flow
Figure
in theA1:
cable
The(qheat
flow
in the
cable (qca)).and to the a
c) and
7. Conclusions
Most BMS measure the temperature at the poles. This study investigated the relation between the temperature at the poles and that of the cells. A model is made that takes among others into account the
heat dissipation at the poles, the heat resistance between poles and core and the influence of the cables
that connect a battery to an external load or source.
We found that the difference between the pole temperature and core temperature in steady state
condition is relatively low when the cell is loaded by a current profile that gives a more or less constant
power dissipation in the cell. Only at the corner cells the difference exceeds the 5%.
When the cell is loaded by a current profile that contains more transients, the difference increases
because the relative large heat capacity of the core cannot follow the transients in the same way as the
poles do. We found that the difference in temperatures at the poles and core can increase to about 20%
for battery pack of 4 cells that is loaded by a block shaped power dissipation input with a period time
of 960 [s]. Battery management systems that do not take this difference into account, may give unjust
warnings or even switch off the battery in case of heavy and short lasting loads.
A According
Calculationtomodel
of cable
According to Appendix
the law of
conservation
the
of power,
law
of itconservation
applies for each
of power,
infinitely
it appl
sm
Assume a cable with infinite length that is thermally connected to a thermal mass at position x=0. The
cable carries a current I that causes an ohmic heat dissipation. The temperature difference with the
Appendix
AA Calculation
model
cable
Appendix
Appendix
Calculation
A ΔT Calculation
model
ofmodel
cable offrom
cable
ambient
at
x=0 equals
heatof
exchange
cable to the thermal mass at x=0 equals q0. Figure
0. The
Appendix
A
Calculation model of cable
A1 shows a part c
of the cable.
c
ca
c
c
ca
c⋅dx⋅dΔT/dt = p⋅dx + q (x+dx)
c⋅dx⋅dΔT/dt
+ q (x+dx)
- q (x)=- p⋅dx
ΔT(x)⋅k
⋅dx
-q→
(x) - ΔT(x)⋅k
j⋅ω⋅c⋅ΔT
⋅dx+ kc
Assume
infinite
length
that
isisthermally
connected
totoaathermal
atatposition
The
AssumeaAssume
acable
cablewith
with
a cable
infinite
with
length
infinite
that
length
thermally
that is thermally
connected
connected
thermal
to amass
mass
thermal
position
mass atx=0.
x=0.
position
Thex=0. The
Assume
a
cable
with
infinite
length
that
is
thermally
connected
to
a
thermal
mass
at
position x=0. The
cable
aacurrent
an
ohmic
heat
dissipation.
The
difference
with
cablecarries
carries
cable
carries
currentIaIthat
current
thatcauses
causes
I that
an
causes
ohmican
heat
ohmic
dissipation.
heat dissipation.
Thetemperature
temperature
The temperature
difference
difference
withthe
the with the
cable
carries
a
current
I
that
causes
an
ohmic
heat
dissipation.
The
temperature
difference
with the
exchange
from
totothe
thermal
atatx=0
equals
qq0.0equals
ambient
ΔT
0.0.The
Theheat
heat
exchange
heat exchange
fromcable
cable
from
the
cable
thermal
to themass
mass
thermal
x=0
mass
equals
at x=0
.
q 0.
ambientatambient
atx=0
x=0equals
equals
at x=0
ΔT
equals
ΔT
0. The
ambient
at x=0
equals ΔT0. The heat exchange from cable to the thermal mass at x=0 equals q0.
Figure
shows
aapart
cable.
FigureA1
A1
Figure
shows
A1
shows
partofofthe
athe
part
cable.
of the cable.
Figure A1 shows a part of the cable.
Where: c = heat capacity
Where:
of cable
c =per
heat
length
capacity
unit of cable per length unit
ΔT = temperature difference
ΔT = between
temperature
cable
difference
and ambient
between cable a
p = (ohmic) heat dissipation
p = (ohmic)
per length
heatunit
dissipation
in the cable
per length unit in
Figure A1: flow
The heat flow
the cable (q ) and
qa = heat
qato the
toin ambient
= ambient
heat(q ).flow to ambient
c
Literature
[1]Peyman Taheri, Scott Hsieh, Majid Bahrami, “Investigating electrical contact resistance losses
in lithium-ion batteryassemblies for hybrid and electric vehicles”, Journal of Power Sources 196
(2011) 6525–6533
[2]Ahmadou Samba, Noshin Omar, Hamid Gualous, Odile Capron, Peter VandenBossche, Joeri Van
Mierlo, “Impact of tab locationon large format lithium-Ion pouch cell based on fully coupled
three-dimensional electrochemical-thermal modeling”, Electrochimica Acta147(2014)319–329
Modeling”,
[3]Meng Guo a, Gi-Heon Kim b, Ralph E. White, “A three-dimensional multi-physics model for a Li-ion
battery”, Journal of Power Sources 240 (2013) 80-94
[4]F.J.Jiang, P.Peng, Y.Q. Sun, “Thermal analyses of LiFePO4/graphite battery discharge processes”,J.
Power Sources 243 (2013) 181e194.
[5]Ahmad A. Pesaran and Matthew Keyser, “Thermal Characteristics of Selected EV and HEV Batteries”, National Renewable Energy Laboratory, 2009
[6]R. E. Simons, “Simplified Formula for Estimating Natural Convection Heat Transfer Coefficient on a
Flat Plate”,http://www.electronics-cooling.com/2001/08/simplified-formula-for-estimating-natural-convection-heat-transfer-coefficient-on-a-flat-plate/
[7]Long Lam et. Al., “A Practical Circuit-based Model for Li-ion Battery Cells in Electric Vehicle Applications”, Telecommunications Energy Conference (INTELEC), 2011 IEEE, ISSN: 2158-5210
[8]Chien-Te Hsieh, Chun-Ting Pai, Yu-Fu Chen, Po-Yuan Yu, Ruey-Shin Juang, “Electrochemical performance of lithium iron phosphate cathodes atvarious temperatures”,Electrochimica Acta 115
(2014) 96– 102
a
ambient (qa).
Figure A1: The heat flow in the cable (qc)c and to the
c
a
to
) and
ambient
to the ambient
(q ).
(q ).
Figure A1:
Figure
The heat
A1: The
flowheat
in the
flow
cable
in the
(q cable
(q the
qc = heat
q
flow
through
cable
=) inand
heat
along
along x-axis
) andthrough
to the ambient cable
(q ).
Figure
A1: The heat
the
cableflow
(qx-axis
cflow
a
c
a
According to the law of conservation of power, it applies for each infinitely small part dx of the cable:
totothe
ofofthe
conservation
ititapplies
each
dx
the
According
According
thelaw
lawto
conservation
law of conservation
power,of
power,
appliesfor
itfor
applies
eachinfinitely
for
infinitely
eachsmall
infinitely
smallpart
part
small
dxofofpart
thecable:
dx
cable:
of the cable:
the
thermal
conductance
= the
of
thermal
the
cable
conductance
of
the
cable
kca = According
kofcaofofpower,
According
to the law
conservation
of power,
it applies
for each infinitely small
part
dx of
the cable:
c⋅dx⋅dΔT/dt
==p⋅dx
- ΔT(x)⋅k
→
j⋅ω⋅c⋅ΔT
++j⋅ω⋅c⋅ΔT
kkcaca⋅ΔT
c(x)
c⋅dx⋅dΔT/dt
c⋅dx⋅dΔT/dt
p⋅dx++q=
qc(x+dx)
+-q-qcq(x+dx)
(x+dx)
- qc(x)caca
-⋅dx
ΔT(x)⋅k
⋅dx
→
j⋅ω⋅c⋅ΔT
→
⋅ΔT- -+dq
dq
kcca/dx
⋅ΔT==-pdq
p c/dx =(1)
(1)
p
(1)
cp⋅dx
c(x) - ΔT(x)⋅k
ca⋅dx
c/dx
c⋅dx⋅dΔT/dt = p⋅dx + qc(x+dx) - qc(x) - ΔT(x)⋅kca⋅dx
→
j⋅ω⋅c⋅ΔT + kca⋅ΔT - dqc/dx = p
(1)
Where:
heat
capacity
of
cable
per
length
unit
Where:cq
cWhere:
=heat
heat
ccapacity
=For
heatof
capacity
of
cableper
per
of cable
length
per
unit
lengthqunitit applies:
Where:c
==c
cable
length
unit
For the heat flow
the
heat
flow
it capacity
applies:
c
Where: c = heat capacity of cable per length
unit
ΔT
==temperature
temperature
difference
between
ambient
ΔT=
temperature
ΔT = temperature
difference
difference
betweencable
between
cableand
and
cable
ambient
and ambient
ΔT
difference
between
cable
and
ambient
ΔT
=
temperature
difference
between
cable and ambient
pp==(ohmic)
dissipation
per
unit
ininthe
cable
heat
dissipation
per
length
unit
the
(ohmic)
p =heat
heat
(ohmic)
dissipation
heat dissipation
perlength
length
per
unit
length
unit
cable
in the cable
p = (ohmic) heat dissipation per length unit in the cable
qqaaa==heat
flow
to
flow
to
ambient
heatq
flow
toambient
ambient
flow to ambient
a = heat
qa = heat
flow
to ambient
qqcc c==heat
flow
through
along
x-axis
flow
through
cable
along
x-axis
heatqflow
heat
through
flowcable
cable
through
along
cable
x-axis
along cu
x-axis
c =c
cu
q
=
heat
flow
through
cable along x-axis
c
the
thermal
conductance
of
the
cable
=
thermal
conductance
of
the
cable
kkca
caca = the kthermal
thermal conductance
of the cable
of the cable
ca = the conductance
kca = the thermal conductance of the cable
For
flow
applies:
For
the
heat
qqc
Forthe
theheat
heat
Forflow
the
flowheat
qc citititapplies:
flow
qc it applies:
applies:
qc = A ⋅ kcu⋅ dΔT /dx
⋅ dΔT k/dx= heat conductivity
where:of
kcuthe
= heat
conduct
con
q = A ⋅ kwhere:
ForCombining
theand
heat flow
q it applies:
Combining equations (1)
(2),
we
equations
find:
(1) and (2), we find:
c
==heat
ofofthe
qqc c==AA⋅ ⋅kkcuqcu⋅c⋅dΔT
=dΔT
A ⋅/dx
/dx
kcu⋅ dΔT /dx where:
where:kkcucu
where:
heatkconductivity
conductivity
= heat conductivity
theconductor
conductor
of the conductor
cu
where: kcu = heat conductivity of the conductor
qc = A ⋅ kcu⋅ dΔT /dx
Combining
equations
(1)
find:
Combining
Combining
equations
equations
(1)
and(2),
(2),
(1)we
we
and
find:
2and
2(2), we find:
2
(2)
(2)
(2)
2
Combining
(2),
(1)=
and
- A equations
⋅ kcuCombining
·(kd(1)caand
ΔT
+equations
j⋅ω⋅c)
/ we
dxfind:
⋅ΔT
p(2),-weAfind:
⋅ kcu· d→ΔT / dx = p
(kca + j⋅ω⋅c) ⋅ΔT
22
2 22
⋅ΔT
- -AA⋅ ⋅k⋅ΔT
pp / dx2 = p →
(k(kcaca++j⋅ω⋅c)
j⋅ω⋅c)
⋅ΔT
+ j⋅ω⋅c)
kcucu· ·d-dΔT
AΔT
⋅ /k/dx
· d==ΔT
→2
(kca
cudx
2
(kca + j⋅ω⋅c) ⋅ΔT - A ⋅ kcu· d ΔT / dx = p
→
→
→
- ·∙x
- ·∙x q (x) = A ⋅ k
- ·x
- ·x
-γ0·∙x
-∞γ·∙x
) )⋅e⋅e
γ γ·A·(ΔT
)0)⋅e⋅e
ΔT(x)
∞∞++(ΔT
0 0-ΔT
cucu· ·=
∞∞-· -ΔT
(ΔT
-ΔT
-ΔTand
) ⋅e
qcand
= Aq⋅ck(x)
(ΔT
⋅ kcu
γΔT
· 0(ΔT
- ΔT0) ⋅e
ΔT(x)==ΔT
ΔT
ΔT(x)
= ΔT
∞ +∞∞(ΔT
∞and
c(x)
- ·∙x
- ·x
) ⋅e
and 0
· (ΔT∞ - ΔT0) ⋅e
ΔT(x)
0
∞ = ΔT∞ + (ΔT0 -ΔT∞∞
c qc(x) =∞A ⋅ kcu · γ cu
-γ·∙x
(2)
-γ·x
-γ·x
) ⋅e = ΔT
and+ (ΔT
q (x)
-ΔT= )A⋅e⋅ k · γand
· (ΔT∞ -qΔT
ΔT(x) = ΔT∞ + (ΔT -ΔT ΔT(x)
c(x)
0)=⋅eA ⋅ kcu
γ
γ
γ
γ
γ
γ
where:
==pp/ /( (kkcaca=
++pj⋅ω⋅c
∞∞==ΔT(x→∞)
where: ΔT
ΔT
where:
ΔT∞ = ΔT(x→∞)
ΔT(x→∞)
j⋅ω⋅c
/ ( k)ca) + j⋅ω⋅c )
where: ΔT∞ = ΔT(x→∞)
= p / ( k + j⋅ω⋅c )
0.5
0.5
γ γ==( ((k(kcaca+γ+j⋅ω⋅c
⋅ ⋅kkcucu) ))/ )(A ⋅ kcu) ) 0.5 ca
=j⋅ω⋅c
( (k)ca)/ /(A
+(A
j⋅ω⋅c
0.5
γ = ( (kca + j⋅ω⋅c ) / (A ⋅ kcu) )
where: ΔT∞ =WeΔT(x→∞)
where:
= p / ( kΔT
ΔT(x→∞)
)
= p / ( kca + j⋅ω⋅c )
ca ∞+=j⋅ω⋅c
use these equations to obtain the one-port equivalent Norton network model shown in figure A2.
We use these
We use
equations
these equations
to obtainto
the
obtain
one-port
the one-port
equivalent
equivalent
Norton network
Norton model
network
shown
modelinshown
figure in
A2.figure A2.
We use to
these
equations
to0.5
obtain
the one-port
equivalent
We use these equations
obtain
the one-port
equivalent
Norton
networkNorton
modelnetwork
shown
inmodel
figureshown
A2. in figure A2.
0.5
γ = ( (kca + j⋅ω⋅c ) / (A ⋅ kγcu=) )( (kca + j⋅ω⋅c ) / (A ⋅ kcu) )
We use these equationsWe
to obtain
use these
the
equations
one-port
equivalent
to
obtain
Norton
one-port
network
equiva
mo
Figure A2:
One port equivalent
Norton
model of the the
cable.
74
Figure
One
Norton
ofofthe
FigureA2:
A2:
Figure
Oneport
A2:
portequivalent
One
equivalent
port equivalent
Nortonmodel
model
Norton
model
thecable.
cable.
of the cable.
Figure A2: One port equivalent Norton model of the cable.
75
q0 under
thetemperature
condition thatdifference
the temperature
difference ΔT0 at the
The q
Norton
heat
qsc equals
The Norton heat source
q0source
under the
condition
that the
ΔT0 at the
sc equals
The Norton heat source qsc equals q0 under the condition that the temperature difference ΔT0 at the
cable is kept toThe
0 [°C].
We heat
than
find:
cable
is
kept
to 0 [°C].
We than
q find:
under the condition that the temperature difference ΔT at the
Norton
source
q equals
sc
0 We than find:
0
cable is kept
to 0 [°C].
equals
q
under
the
condition
that
the
temperature
difference
ΔT
at
the
The Norton
heat
source
q
sc
0
0
cable is kept to 0 [°C]. We than find:
0.5
= [°C].
A⋅kcuWe
· γ · than
ΔT∞=find:
A⋅kcu· γ · p / ( kca + j⋅ω⋅c ) = qsco / (1 + j ⋅ω⋅τ)
cable is keptqto
sc 0
0.5
qsc = A⋅kcu· γ · ΔT∞= A⋅kcu· γ · p / ( kca + j⋅ω⋅c ) = qsco / (1 + j ⋅ω⋅τ)
0.5
qsc = A⋅kcu· γ · ΔT∞= A⋅kcu· γ · p / ( kca + j⋅ω⋅c ) = qsco / (1 + j ⋅ω⋅τ)
0.5
0.5
qsc0cu=· γp ·· p(A/ (⋅ kca
/ kj⋅ω⋅c
and
τ=
c / kca
· ΔT∞= A⋅k
(1
+ j ⋅ω⋅τ)
qsc = A⋅kcu· γ where:
cu +
ca) ) = qsco / 0.5
and
τ = c / kca
where: qsc0 = p · (A ⋅ kcu / kca)
0.5
and
τ = c / kca
where: qsc0 = p · (A ⋅ kcu / kca)
2
0.5
dissipaction
ρeis the specific (electical)
p · (A
⋅ kcu / kca) p per unit length
and inτthe
= 2ccable
/ k equals: p = I ·ρe / A where
where: q =The
2 (electical)
The dissipactionsc0
p per unit length
the cable equals:
p =length
I ∙ρe /cain
A where
ρe isequals:
the specific
Thein
dissipaction
p per unit
the cable
p = I ·ρ
e / A where ρeis the specific (electical)
2
resistance of
the unit
conductor.
Using
this, we
can work
the
factor qρsc0isas:
Theconductor.
dissipaction
p per
length
in
theout
cable
equals:
I out
·ρ
A where
the specific
resistance of the
Using
this, we
the
factor
qpsc0=
as:
e / work
e factor
resistance
ofcan
the work
conductor.
Using
this,
we
can
out the
qsc0 as:(electical)
2
The dissipaction
p
per
unit
length
in
the
cable
equals:
p
=
I
·ρ
/
A
where
ρ
is
the
specific
(electical)
e
e
resistance of the conductor. Using this, we can work out the factor qsc0 as:
2
0.5
=
R
·
I
where:
R
=
ρ
·(k
/
(A
·k
)
as:
resistance ofqthe
conductor.
Using
this,
we
can
work
out
the
factor
q
sc0
el
el
e
cu
ca
sc0
2
0.5
qsc0 = Rel· I
where: Rel = ρe·(kcu / (A ·kca)
2
0.5
where: Rel = ρe·(kcu / (A ·kca)
qsc0 = Rel· I
2
0.5
equals
the reatio of ΔT0 and qsc under the condition that q0 = 0 [W]:
Norton Rimpedance
(A
qsc0 = Rel· I Thewhere:
eq ·k
el = ρe·(kcu / Z
ca)
The Norton impedance Zeq equals the reatio of ΔT0 and qsc under the condition that q0 = 0 [W]:
The Norton impedance
equals theZreatio
of ΔT0 and qsc under the condition that q0 = 0 [W]:
The NortonZeq
impedance
eq equals the reatio of ΔT0 and qsc under the condition that q0 = 0 [W]:
-0.5
-0.5
Rc/ (1 the
+ j ⋅ω⋅τ))
where:
· kcaq) 0 = 0 [W]: -0.5
Zeq = ΔT∞ / qZsceq=equals
reatio of ΔT0 and
q under
theRcondition
The Norton impedance
c = (A ⋅ kcuthat
-0.5 sc
where: Rc = (A ⋅ kcu · kca)
Zeq = ΔT∞ / qsc = Rc/ (1 + j ⋅ω⋅τ))
-0.5
-0.5
Zeq = ΔT∞ / qsc = Rc/ (1 + j ⋅ω⋅τ))
where: Rc = (A ⋅ kcu · kca)
-0.5
-0.5
The
source and impedance
can
approximated
= RNorton
j ⋅ω⋅τ))
where:
Rcbe
= (A
⋅ kcu · kca) by first-order networks as:
Zeq = ΔT∞ / qsc
c/ (1 + heat
The Norton heat source and impedance can be approximated by first-order networks as:
The Norton heat source and impedance can be approximated by first-order networks as:
The Norton heat
source
and· Iimpedance
can be approximated
by first-order
networks as: where: τ = τ/ √3
2
qsc≈R
/(1
+ jimpedance
⋅ω⋅τ2 c)
Zeq≈ Rc /(1by+ first-order
j ⋅ω⋅τc)
The Norton heat
source
and
can and
be approximated
networks as:
el
c
qsc≈Rel · I /(1 + j ⋅ω⋅τc)
and
Zeq≈ Rc /(1 + j ⋅ω⋅τc)
where: τc= τ/ √3
2
qsc≈Rel · I /(1 + j ⋅ω⋅τc)
and
Zeq≈ Rc /(1 + j ⋅ω⋅τc)
where: τc= τ/ √3
The time-constant τcis chosen is such a way that the cut-off frequency of the first order model equals
2
qsc≈Rel · I /(1 + j ⋅ω⋅τc) The time-constant
and
Zeq
⋅ω⋅τ
where:
τc= frequency
τ/ √3
c /(1 + jis
c) a way that the
τc≈isRchosen
such
cut-off
of the first order model equals
the cut-off frequency
of the
original
model.
The time-constant
τ
is
chosen
is
such
a
way
that
the
cut-off
frequency
of
the
first order model equals
the ccut-off frequency of the original model.
The time-constant
τcis chosen
is such
a way
that the cut-off frequency of the first order model equals
the cut-off
frequency
of a
the
original
model.
The time-constant
τcFigure
is chosen
is
such
way
that
the
cut-off frequency of the first order model equals
the thermal
the cut-off frequencyA3
ofshows
the original
model.RC-network model of the cable.
the cut-off frequency of the original
Figuremodel.
A3 shows the thermal RC-network model of the cable.
Figure A3 shows the thermal RC-network model of the cable.
Figure A3 shows the thermal RC-network model of the cable.
Figure A3 shows the thermal RC-network model of the cable.
Figure A3: First order thermal model of cable.
Figure A3: First order thermal model of cable.
Figure A3: First order thermal model of cable.
Figure A3: First order thermal model of cable.
Figure A3: First order thermal model of cable.
76
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