Linköpings universitet
Linköping University | Department of Electrical Engineering
Master’s thesis, 30 ECTS | Vehicular Systems
2021 | ISY-EX--21/5374--SE
Battery Digital Twin
–
Modeling and Characterization of a Lithium-Ion Battery
Batteriets digitala tvilling - modellering och karaktärisering av
ett lithiumjonbatteri
Gustav Erbing
Fabian Sund
Supervisors : Arvind Balachandran, Tomas Uno Jonsson Examiner : Lars Eriksson
Upphovsrätt
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© Gustav Erbing Fabian Sund
Abstract
Electrical vehicles have become more popular during recent years due to their reduced greenhouse gas emissions. The research within Li-ion batteries is therefore moving fast. Presently, two-level converters transforming the current from DC to AC. However, an al-ternative method of power conversion is by utilizing modular multilevel converters, which can perform better harmonics than the two-level converter. This study aims to research the impact of these converters on battery cell heat generation. In doing so, developing a digital twin of the Li-ion battery cell, which in this case is a Samsung 28 Ah nickel, manganese, and cobalt prismatic battery cell, focusing on the thermal aspects such as heat generation, heat capacity, and thermal conductivity. The modular multilevel converter may also cause significant overtones, harmonics. Therefore, this study investigates the thermal impact of these frequencies. The results show that it is possible to, via experiments and simulations, determine the heat capacity and thermal conductivity of a Li-ion cell. Furthermore, the frequencies caused by the modular multilevel converter cause a temperature rise in the cell, compared to the two-level converter. Although, if the same root mean square for the modular multilevel converter current is used, the temperature rise is lower compared to DC. During the load cycles, the results show that there are slightly higher temperatures at the positive current collector side compared to the negative. It is, however, the jelly roll core that has the highest temperatures.
Keywords
Electrical vehicles, Lithium-ion batteries, Modular multilevel converter, Two-level con-verter, Thermal characteristics
Acknowledgments
First, we would like to address a big thank you to our supervisors Arvind Balachandran and Tomas Uno Jonsson for your good support and guidance throughout the whole project as well as valuable feedback on our work. Secondly, we are very thankful for all the support we have got from Scania AB, where we especially would like to mention Alexander Bessman who has brought up some very interesting points as well as valuable feedback. Last but not least, we would like to thank our examiner Lars Eriksson who brought up the thesis idea and also for his good advice along the way.
Acronyms
AC Alternating current
ARC Accelerating rate calorimeter
BOL Beginning of life
DC Direct current
DFN Doyle, Fuller and Newman
EOL End of life
EV Electrical vehicle
FB Full bridge
HB Half bridge
HDPE High-density polyethylene
HEV Hybrid electrical vehicle
IHC Isothermal heat conduction calorimeter
LCO Lithium, cobalt oxide
LFP Lithium, iron, phosphate
MMC Modular multilevel converter
NCA Nickel, cobalt, aluminum
NMC Nickel, manganese, cobalt
OCV Open circuit voltage
P2-D Pseudo two dimension
PHEV Plug-in hybrid electrical vehicle
PWM Pulse width modulation
SEI Solid-electrolyte interphase
SOC State of charge
SOH State of health
SPM Single particle model
SPMe Single particle model with electrolyte
TLC Two-level converter
TMS Thermal management system
2-D Two dimensions
"Think inside the battery, but outside the box!"
- Lars Eriksson"The more you know, the more you know that you don’t
know"
- Gustav Erbing (Aristotle)Contents
Abstract iii Acknowledgments iv Acronyms v Quotes vi Contents vii List of Figures ix List of Tables x 1 Introduction 1 1.1 Background . . . 2 1.2 Previous work . . . 2 1.3 Purpose . . . 5 1.4 Research questions . . . 5 1.5 Limitations . . . 51.6 Outline of the report . . . 5
2 Theory 6 2.1 Lithium-ion battery cell . . . 6
2.2 Battery cell principles and reaction . . . 9
2.3 Battery cell format and design . . . 11
2.4 Battery thermal characteristics . . . 13
2.5 Battery cell physics models . . . 15
2.6 Converters . . . 20
3 Method 27 3.1 Experiment with heat plate . . . 27
3.2 Experiment with DC . . . 28
3.3 Simulation of heat plate experiment . . . 31
3.4 Simulation of DC . . . 32
3.5 Simulation of MMC current . . . 32
3.6 Load cycles . . . 33
4 Results 34 4.1 Heat plate experiment . . . 34
4.2 DC experiment . . . 36
4.3 Simulation of heat plate experiment . . . 37
4.4 Simulation of DC . . . 38
4.6 Load cycles in simulation . . . 41 5 Discussion 46 5.1 Results . . . 46 5.2 Method . . . 48 6 Conclusion 49 Bibliography 51
List of Figures
2.1 A schematic view of a Li-ion battery cell. . . 7
2.2 Ragone plot for power and energy optimization in a battery cell, where "Opt" indicates the theoretical optimal solution . . . 8
2.3 A geometric view of the SPM concept . . . 15
2.4 A geometric view of the P2-D-model, where both the 2-D projection and an elec-trode particle is included . . . 16
2.5 Block diagram showing the relationships between different terms in the DFN-model 19 2.6 Operation and modulation of a TLC. . . 21
2.7 An example of a full-bridge converter . . . 21
2.8 PWM-controlled output voltage for the FB converter . . . 22
2.9 Three-phase two-level converter. . . 23
2.10 A schematic example of a basic multilevel converter . . . 24
2.11 A controlled output from a cascaded multilevel converter. . . 24
2.12 A typical topology of a Modular Multilevel Converter . . . 25
2.13 The voltage output of an MMC with five SM in each upper and lower arm . . . 25
2.14 Basic schematic of a half/full bridge . . . 26
3.1 Experimental set-up for the heat plate experiment . . . 28
3.2 Positions for the thermocouples inside the battery cell . . . 28
3.3 Experimental set-up for the DC charging/ discharging . . . 29
3.4 Experimental circuit for the DC charge experiment . . . 30
3.5 Experimental circuit for the DC discharge experiment . . . 30
3.6 3-D-thermal model in COMSOL . . . 31
3.7 3-D-thermal model in COMSOL with argon gap and current collectors . . . 32
3.8 Geometry overview of the two-dimensional battery model . . . 32
4.1 Temperature rise in the aluminum plate during heat plate experiment . . . 35
4.2 Temperature rise in the battery cell during heat plate experiment . . . 35
4.3 Temperature rise in the battery cell during DC charge experiment . . . 36
4.4 Experimental results from the discharge cycle . . . 37
4.5 Simulation of temperature change in the model compared with experimental data for the heat plate experiment . . . 38
4.6 Simulation of temperature change in the model compared with experimental data for the charge cycle . . . 38
4.7 Simulation of temperature change in the model compared with experimental data for the discharge cycle . . . 39
4.8 Cell potential for the cell during 65 A discharge . . . 39
4.9 Simulation results with same RMS for MMC during discharge cycle . . . 40
4.10 Simulation results with same DC-component for MMC during discharge cycle . . 41
4.11 Simulation results for 1 C charge cycle . . . 42
4.12 Heat dispersion in jelly roll for 1 C charge cycle in K . . . 43
4.13 Simulation results for discharge/ charge cycle . . . 44
List of Tables
2.1 Standard electrode reduction potentials at 25˝C . . . . 10
2.2 Comparison of Li-ion cell formats . . . 13
2.3 Nomenclature for the DFN-model . . . 17
2.4 Output voltages for the full-bridge converter . . . 21
1
Introduction
Most citizens around the world have over the last couple of years agreed that climate change is real [1]. Scientists have shown that there is evidence of global warming and that it is unequivocal. The current trend of global warming is a result of human activity, and espe-cially human activity from mid-20th century. As of today, the earth’s surface temperature has increased by 1.18 °C since the late 19thcentury, especially driven by the increased carbon dioxide in the atmosphere, and the years 2016 and 2020 are tied for the warmest year record since measurements started in 1880 [1]. Even though scientists have shown that climate change is real and that it is mostly affected by carbon dioxide, the car and truck industry is increasing annually. In 2018 the car industry footprint was about 9% of the annual global greenhouse emissions [2]. Therefore, some countries such as Norway are aiming for electrical vehicles (EV) and plugin hybrids (PHEV) to make up 100 % of their vehicle fleet by the year 2025. Also, the Netherlands aims to prohibit all diesel and gasoline car sales by the same year to reach the Paris agreement of 1.5 °C global warmings [2, 3]. Some critics mean that initiatives from countries such as Norway and Netherlands are not enough and therefore want to see all major automakers phase out their diesel and petrol cars, including hybrids no later than 2028 [2]. Instead, critics want the vehicle industry to build energy-efficient EVs sustainably [2].
It seems like an easy task to adapt to this electrical transformation and also agree that the sooner the change takes place the better. However, how far the vehicle electrification will go depends on how well scientists can understand and predict the batteries that the EV is built upon [3]. Comparing an EV to a combustion vehicle, all the downsides come from the battery and to mention some:
• the EV price is more expensive, • the EV’s traveling range is less,
• the charging time is longer than the re-fueling time, • the EV lifetime is shorter. [3]
1.1. Background
Considering the above factors, EVs have however, lower greenhouse gas emissions, which makes the research about batteries a necessity.
This chapter covers the motivation and introduction of the project. Furthermore, previous work will be declared as well as the aim of the project and research questions will be specified. Last, the limitations of the project will be stated together with the outline of the report.
1.1
Background
Almost all electric vehicles on the market are constructed with Li-ion batteries and the rea-son is that these batteries are rechargeable and have a high capacity [4]. A bottleneck with EVs is, though, the energy storage compared to the traditional internal combustion engine. However, over the last decade research within Li-ion batteries has increased dramatically [5] which has shown that the performance, as well as lifetime of a Li-ion cell, is dependent on the operating temperature [6]. Thus, the research about thermal management and other factors, such as high frequencies which impact the temperature, is in focus for Li-ion battery research. The heat generation of a battery cell depends on the rate of charge and discharge [5]. No thermal management results in high internal battery temperatures. Therefore, battery man-ufacturers usually dissipate the heat through a thermal management system (TMS) [4, 6]. In order to implement such a system, information about thermal characteristics are necessary to size the TMS. However, different batteries generate different heat losses for charging and discharging rates which usually depends on the geometry of the cell, packing strategies, and cell components [4]. Opening a battery cell to investigate the internal structure can be direct risky and impractical. Therefore, EV manufacturers usually deploy cooling packages based on measurements on the surface of the batteries which is not a true indicator of the cell’s in-ternal temperature [7]. The inin-ternal high temperature is however important to understand as it can result in decreased battery cell lifetime and in the worst-case scenario lead to thermal hazards. Hence, EV battery researcher needs to study thermal properties and other factors that can have an impact on the internal temperature, to ensure adequate cooling and to ex-tend the battery cells lifetime. This thesis investigates thermal characteristics such as heat capacity and thermal conductivity. Furthermore, this thesis extends the understanding of the impact of high frequencies that are generated from switches in the converter on Li-ion battery cells.
Terminology
A clarification about the battery terminology which will be used in this thesis is: A battery cell or just cell will be used to refer to a single battery cell, while a battery will be used to refer to several such cells that are connected in parallel and/ or in series.
1.2
Previous work
Relevant previous work will be presented and summarized in this section. First, literature that illustrates research about thermal characteristics such as conductivity and heat capacity. Secondly, literature that illustrates research about converters and especially modular multi-level converters (MMC) and their switching frequency’s impact on the overall cell. Finally, the thesis motivation will be established.
Thermal characteristics
Vertiz et al. [8] studied a 14Ah LiFePO4/graphite-commercial large-size pouch cell and per-formed thermal characterization using a one dimension electro-thermal model. The study
1.2. Previous work
determined two important aspects to consider for thermal characteristics, thermal proper-ties, such as heat capacity and conductivity as well as heat generation parameters, such as internal resistance and entropic factor, which were considered both analytically and exper-imentally. The study reports a good agreement between analytical calculations and experi-mentally determined heat capacity. On the other hand, the thermal conductivity calculations differ significantly, by a factor of 3. Two main reasons for the large error are the neglecting of related thermal contact resistance between solids and the fact that the experimental test does not include the in-plane thermal conductivity, due to the danger of opening up the cell. However, the research evaluated the most significant error sources such as the electrolyte, state of charge (SOC), and state of health (SOH).
• The electrolyte has a large impact as it fills air gaps between the layers and therefore increasing the global conductivity by about 50%.
• The SOC level impact on the cell was obtained to be lower when fully charged or dis-charged compared to when the SOC level was about 50%.
• The aging of the cell decreased the thermal conductivity.
However, none of the electrolytes, SOC, or SOH, had a significant impact on the heat capacity [8].
Chen et.al [9] studied large prismatic Li-ion cells and measured heat generation for a broad range of discharge rates as well as temperature range to get an understanding of the battery characteristics. An isothermal heat conduction calorimeter (IHC) method was used on an A123 prismatic LiFePO4 battery with a capacity of 20 Ah. The calorimeter was designed with the fact that prismatic cells tend to lose more heat from the front and the back faces as they have a higher surface to thickness ratio. Therefore, the battery was placed in be-tween two slabs of high-density polyethylene (HDPE) with direct contact. Accordingly, two aluminum slabs were put outside of the HDPE slabs and the whole construction into an isothermal bath of 50-50 mixture of ethylene glycol-water with the electrodes kept outside the bath at all times. Furthermore, four thermocouples were placed at strategic locations to be able to estimate the generated heat in the battery cell from temperature measurements with the assumption that the heat was transferred in one dimension between the two HDPEs through the cell. The result of the experiment failed to measure thermal parameters such as heat capacity and thermal conductivity. Although, the experiment resulted in a relationship between generated heat with known input heat from the heaters with an inaccuracy of about 20% in the temperature range 10 to 40 °C. Therefore, the generated heat can give an approxi-mation of the needed cooling system capacity but not any internal temperatures of the cell. [9] Lidbeck and Syed [5] used a customized IHC to determine thermal characteristics, such as heat generation, specific heat capacity, through-plane conductivity, and in-plane conductiv-ity for a 6 Ah NMC pouch cell and a 20 Ah LFP prismatic Li-ion cell. The study neglected SOC-level and SOH-level of the different batteries, although three customized setups were made, one for specific heat capacity, one for through-plane conductivity, and one for in-plane conductivity. Furthermore, a model in COMSOL was implemented for the pouch cell, which was used to find the internal temperatures of the cell for different load cycles. The results of the study presented a possibility to find the heat capacity for both the pouch and prismatic cell and also found the conductivity, both in-plane and through-plane, for the pouch cell. However, the conductivity was not found because of the aluminum casing of the prismatic cell. The researchers were able to ascertain the heat generation for the two different cells, which was in the range of 10 mW to 30 W [5].
1.2. Previous work
Furthermore, extensive further research within the area of thermal characterization of Li-ion batteries has been made, for example, research about thermal management for Li-ion batter-ies in a PHEV application in Lundgren, et al. [6] article. The setup in this research consisted of a 2-D electrochemical model coupled with a 3-D thermal model of a prismatic Li-ion battery cell including a heat sink on the opposite side of the terminals for thermal management. The model was validated with experimental potential and temperature data. However, the elec-trolyte is found to be the limiting part during high-current pulses, because the concentration reaches extreme values leading to uneven current distribution. Two alternative strategies of thermal management were studied, where the heat sink is placed on either the short or long side of the battery. However, these strategies are found to only impact the temperature of the cell in a minor way [6]. Sun, et al. [7] investigated a method for estimation of the cell’s internal temperature using a Kalman filter, and based the model on a simplified thermal model. Both the heat generation overpotential and the change of entropy quantitatively were analyzed, while a current pulse was used to identify the model parameters. Furthermore, data from thermocouples, placed to measure the surface and internal temperatures, were incorporated into the model. This model was found to perform with an estimation error maximal at 1 K [7].
Converters
Tolbert et al. [10] presented the purpose of the MMC concept within hybrid electrical vehi-cles (HEV) and EVs. More specifically for military combat vehivehi-cles and heavy-duty trucks that have large electric drives that will require advanced power electronics to meet the high power requirements. Stated is that MMC is suited for this type of application because of the MMCs high VA ratings. Another key benefit of the MMC in this type of application is the dV/dt rate which is much lower than the traditional two-level converter (TLC). Hence, it makes the MMC much easier to operate at high efficiency due to the possibility of switching at much lower frequencies than the traditional TLC. However, the three key takeaways from Tolbert, et al. [10] research is first, that MMC can with only fundamental switching frequency generate near-sinusoidal voltages. Secondly, the interference of the electromagnetic field is almost negligible, and third, the MMC converter makes open wiring possible for most of the EV’s power system and as a result safer and more accessible.
Bessman, et al. [11] investigates the difference of charging a Li-ion battery with either DC or different alternatives of AC. It is shown that there is no advantage of charging the cells using AC. However, there are some losses using a superimposed square AC leading to a higher mean cell temperature compared to a superimposed triangular AC, superimposed sinusoidal AC, and DC [11]. Liang, et al. [12] research MMC performance for a battery energy storage system (BESS). This highly modular application provides a good opportunity for uneven active power distribution and differences between the sub-modules. However, it is important to consider frequencies while implementing an MMC battery design. There will be a high fundamental frequency because of the injection in DC voltage which will need to be attenuated by large capacitors in the sub-modules [13].
Motivation
Even though a lot of research has been made within the area of Li-ion batteries and convert-ers there is still a lot left to explore. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] have all touched upon research in the field of either thermal modulation of Li-ion batteries or different aspects of MMCs. However, there is no previous research about modulation and simulation of a combi-nation of both heat capacity, conductivity, and high frequencies´ thermal impacts. Therefore, this thesis will investigate and fill this gap in the research.
1.3. Purpose
1.3
Purpose
To fill the research gap, the thesis primarily aims to develop a digital twin for a Lithium-Ion Cell (Li-Cell), that captures the cell characteristics concerning heating and frequency con-tents. Secondly, the study investigates how an MMC could influence the temperature in the battery cell. Thirdly, as a combination of the two previously mentioned aims, this thesis aim to collect experience and knowledge about the modeling of batteries and how this can be done efficiently by combining model knowledge, experimental tests, and methods from system identification.
1.4
Research questions
In order to achieve the requirements of the project, two research questions have been estab-lished with corresponding questions:
1. Is it possible to experimentally determine the thermal characteristics of a battery cell? a) What is the heat capacity of a battery cell?
b) What is the thermal conductivity of a battery cell?
c) How and where is the heat generated and how does it spread through a cell? d) Can internal hot spots occur that we need to monitor?
2. How do high-frequency signals from switching patterns in MMCs, i.e., first and second harmonics, thermally affect the battery cell?
a) How and where is the heat generated and how does it spread through a cell? b) Does the thermal distribution differ between an MMC and a standard TLC?
c) Does it bring any additional internal power losses?
1.5
Limitations
This study will mainly focus on the thermal aspects of the battery cell and not the electro-chemical. The study aims to produce a digital twin which will be in good agreement with thermal experiments. No experiments regarding the Li-ion density or other electrochemical aspects will be conducted.
1.6
Outline of the report
The thesis will continue with the theory in chapter 2. The theory chapter is divided into a couple of subsections whereas it starts off by deep diving into Li-ion battery cells. It then continues with battery cell principles as well as reactions and then introducing different types of cell designs. The theory thereafter moving over to covering thermal characterizations and continues by introducing different cell models. Moreover, the theory chapter finishes with theories about converters and especially about TLC, three-level converters, and MMC. In chapter 3, the different experimental setups and different simulations are generally described. In addition, the thermocouples and heat camera is explained as well as a calibration method for the heat plate experiment. Chapter 4 shows the results of the experiments as well as the simulations. Whereas the first part of the results gives proof of the built model. The second part of the results compares the MMC behavior with standard TLC. The last part of the results presents the simulated generated heat. Chapter 5 gives a brief discussion upon the results, the used method, and gives a short summarizing of the work in a wider context. Lastly, chapter 6 includes the main derived conclusions as well as ideas on future work within this domain of studies.
2
Theory
In this chapter, the theoretical framework will be presented upon which the thesis will be based. The outline of the chapter is to first introduce the lithium-ion battery cell and the two different converter types. Furthermore, there will be a presentation of previous research regarding battery thermal char-acteristics.
2.1
Lithium-ion battery cell
The main purpose of a battery cell is to transform chemical energy into electrical energy, this is done through an electrochemical oxidation-reduction (redox) reaction [16]. For the case where the battery is rechargeable, the recharge of the cell is made by reversing the process, hence the anode and cathode switches. Therefore it is common to use the terminologies posi-tive and negaposi-tive electrode to avoid confusion. During the reaction, electrons are transferred via an external circuit, from one electrode to the other, while the Li-ions are transferred in the same direction internally via the battery cell electrolyte. The basic cell schematic is shown in the figure 2.1. The battery cell includes several components:
• Negative electrode - emits electrons to the circuit during discharge and accepts electrons during charging.
• Positive electrode - accepts the electrons from the circuit during discharge and provides electrons to the circuit during charging.
• Electrolyte - provides a path for Li-ions to flow between the positive and negative elec-trodes [16].
• Separator - a mechanical separation between the positive and negative electrode, the function is to block the electrons from directly transferring between the two electrodes [17].
• Positive and negative current collectors, aluminum on the positive side and copper on the negative side, connected to the terminals and allow the electrons to go into the external circuit.
2.1. Lithium-ion battery cell
Figure 2.1: A schematic view of a Li-ion battery cell.
There are several different battery options available for EVs, but the most common and pop-ular solutions are the lithium-ion concepts. The reason is because of its high energy density and specific capacity [4]. The cell design can differ for lithium batteries, which can be either Li metal or Li-ion type [4].
Another thing to consider regarding Li-ion batteries is to not only reflect upon which ma-terial is used for the negative electrode, which often is graphite, and the electrolyte, but also to consider which material to be used for the positive electrode. The selection of which positive electrode to use will further specify the characteristics of the battery cell where some configurations are preferred in an EV. Like for the negative electrode also the positive electrode’s active material is either insertion or conversion type. The early commercialized Li-ion cell used lithium cobalt oxide (LCO) as the positive electrode, however, the use of this composition is largely reduced presently [18]. Instead, the most commonly used positive electrode material today is lithium nickel manganese cobalt oxide (NMC) and there are a few different compositions of the NMC-combination. Furthermore, nickel, cobalt, and aluminum (NCA) and lithium, iron, and phosphate (LFP) are also often mentioned in these areas [18]. Even though the positive electrode material has changed over the years, graphite is still the most common material for the negative electrode [4, 18].
During the operation there are demanding effects on the electrolyte, both thermally, electri-cally, and chemically [4]. Therefore, it is necessary to choose an electrolyte that can withstand these circumstances. Moreover, in the case of an extreme incident, for example, breakage or fire, the electrolyte and electrodes preferably should be non-toxic [4].
Battery cells are either power-optimized or energy-optimized [4]. Figure 2.2 is a Ragone plot and show the difference between power-optimized and energy-optimized cells depending on the applied C-rate [4]. The C-rate is described by the current normalized to cell capacity. 1C relates to charge/discharge in one hour and 10C relates to charge/discharge in 1/10thof an hour. In figure 2.2 it can be observed that high energy is only reachable with a low C-rate and correspondingly high power is only reachable with a high C-rate [17]. The optimal solution "Opt" is thus unachievable. Therefore, it is common practice to balance the power and energy against each other and usually manufacture the battery cell in the E/P region [4, 17], shown in green in figure 2.2.
2.1. Lithium-ion battery cell
Figure 2.2: Ragone plot for power and energy optimization in a battery cell, where "Opt" indicates the theoretical optimal solution. Image adapted from [5].
Usage and degradation of lithium-ion battery cells
For a traditional vehicle, the components are calculated to last the same amount of time as the vehicle itself, depending on the application, which is in the range of five to fifteen years. For an EV the battery is the most important, sensitive, and costly component, and therefore, it is of the utmost importance to design and adapt it to the application [4].
As soon as a battery cell is assembled it starts to degrade and the cell’s life is impacted nega-tively by several different factors, i.e. current rates, temperature, SOC ranges, and mechanical effects such as vibrations [4]. Furthermore, the use of the cell will also affect the lifespan. The lifespan of the battery cell is often compared to the beginning of life (BOL) conditions and is measured in a few different ways depending on the application, i.e. number of cycles, years, total energy throughput. When discussing the lifespan of batteries there are two main factors to consider, the capacity fade and the power fade [4, 18]. The decline caused by capacity and/ or power fade is measured using SOH and a cell reaches its end of life (EOL) when the SOH is less than 80% [8, 19]. The capacity fade relates to the fact that the cell loses cyclable lithium with the number of charges/ discharges and varies with the active materials inside. The power fade relates to the loss of active material which results in the increase in cell impedance [18]. EVs are more vulnerable to capacity fade because they are only relying on the battery for their power supply [4]. The power fade, instead, directly relates to a high power supply in short pulses, something that an HEV uses as a buffer, and when the power fade is significant this will show in the fuel consumption for the HEV.
Unfortunately, the battery performance and health will drop even though the battery is not used. When a battery is stored there will be some irreversible capacity loss, which is com-monly known as calendar aging [4, 20]. This means that the battery cannot be stored without it affecting the battery’s health and since most passenger cars are mostly parked, this has to be considered. To categorize battery life and life-span the terms calendar aging and cycle life are often used, where calendar aging relates to how many years a battery is durable, i.e the length of the life-span, and cycle life relates to how many cycles the battery can handle and still operate within the specified conditions [4]. The SOC and temperature in storage affect
2.2. Battery cell principles and reaction
the loss in capacity while the relationship between calendar aging and SOC is not linear [4]. With a cell composition including graphite as the negative electrode (the most common design), there will be a formation of solid-electrolyte interphase (SEI) layer on it [18]. The low potential of the graphite compared to lithium is the advantage of it, but this advantage brings a significant disadvantage as well: at this low potential, the electrolyte is not stable, and will therefore react at the negative electrode and decompose, where the decomposition consumes lithium which will be permanently lost for the charge and discharge reactions [18, 21, 22]. After some of the electrolytes have decomposed, the newly formed SEI layer leads to a significant reduction in further decomposition. However, during the cell reactions, the graphite will shrink and grow, leading to cracks in the SEI layer which will therefore have to be repaired by new decomposition reactions which will consume further cyclable lithium [20, 23]. Also, this effect increases with increased temperature [24]. The formation of the SEI layer and the loss of cyclable lithium yields capacity loss for the cell and the decomposition of the electrolyte also cause a rise in pressure due to gas evolution which can occasionally cause the case to break. When the graphite changes in size, there will be another effect, closely linked to SEI, called graphite exfoliation, which is when layers of graphite lose contact with each other leading to a lower capacity in the negative electrode and loss of active material. The loss of active material will negatively affect the power capacity in the cell because there will be an increase in cell impedance [18].
Another aging mechanism in Li-ion cells is lithium plating. The plating happens under spe-cial circumstances when the negative electrode’s potential is lower than the redox potential of lithium leading to metallic lithium depositing on the graphite particles [18]. At slow rates, Li-plating will only grow the SEI layers. However, with further increases in reaction rates, the plating will grow to form a dendrite. The dendrite can eventually cause a short circuit and eventually a thermal runaway [18, 21]. In commercial applications, this lithium plating is most common when the cells are charged at low temperatures. However, if the current is high, the plating will happen at higher temperatures as well which may cause problems if a fast-charge application is used [25, 26].
2.2
Battery cell principles and reaction
A battery cell (as mentioned in the section 2.1) consists of a negative and positive electrode as well as the electrolyte which conducts the ionic electricity inside the cell [16]. The cell’s ability to deliver electrical energy depends on the change in Gibbs energy∆G, which is
∆G˝ =´nFE˝, (2.1)
where n is the number of electrons participating in the redox reaction, F is the Faraday con-stant (96,487 [C/mol]), and E˝ is the standard electromotive force, i.e the electrode potential [4, 16]. Table 2.1 presents values of the electrode potential of different redox couples.
2.2. Battery cell principles and reaction
Table 2.1: Standard electrode reaction reduction potentials at 25˝C [16].
Electrode reaction E˝[V] Li++e´èLi -3.045 K++e´èK -2.925 Na++e´èNa -2.714 Al3++3e´èAl -1.67 H2O+e´è 1 2H2+OH´ 0.8277 H2O+e´è 1 2H2+OH´(seawater, pH 8.2) 0.5325 Ni(OH)2+2e´èNi+2OH´ -0.72 O2+H++e´èHO2 -0.046 2H++2e´èH2 0.000 HgO+H2O+2e´èHg+2OH´ 0.0977 CuCl+e´èCu+Cl´ 0.121 AgCl+e´èAg+Cl´ 0.2223 AgCl+e´èAg+Cl´(seawater, pH 8.2) 0.2476 Hg2Cl2+2e´è2Hg+2Cl´ 0.2682 Hg2Cl2+2e´è2Hg+2Cl´(satd KCI (SCE)) 0.2412
O2+2H2O+4e´è4OH´ 0.401
Cu2++Cl´+e´èCuCl 0.559
O2+4H++4e´è2H2O (pure water, pH 7) 0.815
O2+4H++4e´è2H2O 1.229
Cl2+2e´è2Cl´ 1.358
Ideally all of this energy would be transferred into useful electrical energy during the dis-charge of the cell. Unfortunately, this is not the case, there will be losses because of polar-ization as a load current i goes through the electrodes, complementing the electrochemical reactions. There are two cases of losses:
1. Activation polarization, this happens at the surface of the electrode and drives the elec-trochemical reaction
2. Concentration polarization, this is due to the mass transfer and arise from the difference in concentration of reactants and products, also at the surface of the electrodes.
These two different polarization effects consume some of the stored energy as generated heat. Therefore, as a result of the conservation of energy law, not all the stored energy in the cell can be transformed into electrical energy.
It is theoretically possible to calculate both the activation and concentration polarizations with the help of some electrochemical parameters and mass-transfer conditions. However, it is rather difficult to determine all the parameters and physical structures for the electrodes. Almost all battery cell electrodes are made of active material, performance-enhancing addi-tives, binder, and conductive filter and they normally have a porous structure. This means that the battery cell requires complex and extensive mathematical modeling which needs a good computer for the calculations to find the polarization components. [16]
Another factor that impacts the performance of the cell is the internal resistance [16]. The internal resistance triggers a voltage drop during the operation, which as the two different polarizations consume energy and generate heat. The voltage drop is commonly known as "ohmic polarization" or IR drop and the size of it is in proportion to the current extracted from the cell. The internal resistance can not be drawn to one of the different parts of the cell, instead, it is the sum of ionic resistance in the electrolyte, the current collectors and both of the electrodes’ electrical tabs, the active mass’ electronic resistances, and, lastly, the resistance in
2.3. Battery cell format and design
contact between the current collector and active mass. All of the above resistances are ohmic, meaning they follow Ohm’s law and they have a linear relationship between voltage drop and current [4, 16]. If the battery cell is connected to a load R, the voltage from the cell E can be described by:
E=E0´[(ηct)pos+ (ηc)pos]´[(ηct)neg+ (ηc)neg]´iRi=iR (2.2) where E0 is the electromotive force or open-circuit voltage (same as in (2.1)), (ηct)pos and
(ηct)negis the activation polarization or charge-transfer overvoltage at positive and negative electrodes respectively,(ηc)posand(ηc)negare the concentration polarization at positive and negative electrodes respectively, i is the operating current, Riis the internal resistance, and R is the external load. In (2.2) it is clear that the useful voltage from the cell is reduced with the polarization and internal resistance drop. If the current is very low, the polarizations and internal resistance drop will be small, leading to that the cell will be able to deliver a useful voltage close to the open-circuit voltage (OCV), which is close to the theoretically available energy [16].
Even though the energy available in a battery cell depends on the electrochemical reactions at both of the electrodes, there are a few additional factors to consider. These other factors also affect the cell performance such as diffusion rates, the exchange current density, and therefore the number of energy losses. The factors include the nature of the separators, electrolyte conductivity, and electrode formulation and design, etc. When designing a battery, there are some rules to consider to make sure that the highest possible performance is reached, i.e. high operating efficiency with minimal losses [16]:
1. The electrolyte should have high enough conductivity to limit the IR polarization dur-ing operation.
2. The chemical properties of the separator should be stable enough to avoid direct reac-tions between the negative and positive electrodes.
3. It is important to consider the rate of electrode reaction for both electrodes to limit the activation and charge-transfer polarization, i.e. the rate has to be fast. A commonly used method to limit this is to use a porous electrode design.
4. To limit the concentration polarization, the cell design needs to consider that there are reactants on the electrode surface which has to be diffused or transported away. There-fore, the electrolyte needs to have adequate transport to ease this mass transfer.
5. To avoid corrosion in the cell, the material of the current collector and substrate have to be chosen in regards to the materials of the electrode and electrolyte. Also, the current collector should provide low contact resistance.
6. In the case of a rechargeable battery cell, the reaction products are preferred to remain at the electrode to ease reversible reactions when the cell is charged/ discharged. It is important that the electrolyte is stable chemically with the electrodes.
2.3
Battery cell format and design
There are a few different formats suitable to use in an EV [4]. These formats may differ de-pending on which manufacturer and the application. However, there are three major group-ings that can be made for EV-battery cells, and they relate to the geometry of the battery cell: cylindrical, prismatic, and pouch. In an EV application, all of these cell formats have different performance characteristics and constraints regarding both the cell itself and the vehicle [27, 28].
2.3. Battery cell format and design
Cylindrical cells
In cylindrical battery cells, the jelly roll, i.e., the electrodes and separator, are rolled into a cylindrical shape [4]. The cell diameter relates to its capacity, a thick cell has high capacity and a thin cell has low capacity [4, 16]. The most common construction is that the terminals are placed on top and bottom of the cell and the casing is often aluminum. The cylindrical battery cell is easily produced which is one of the biggest advantages of its construction [4]. However, the temperature inside the cell format and packaging issues are disadvantages that have to be considered. Inside the cell, the temperature will increase during operation and the temperature will not disperse evenly throughout the cell. Optimal performance is reached if the temperature is uniform within the whole cell, meaning that heat needs to be removed from the core to avoid abusive situations. Because of this, there exists a trade-off between thermal management and performance, leading to thin cells are commonly used in cells that are power-optimized [16]. Furthermore, as this format are cylindrical in shape the cells cannot be packaged in an optimal way when they are placed in parallel and/ or series inside a box for the EV installation. Because of the form, there will be some voids in between the cells leading to that the final battery pack is larger than for prismatic or pouch cells. The voids are, however, good for thermal management and can act as cooling channels [16].
Prismatic cells
For the prismatic cell format, there are two ways how they are produced: wound or stacked, and the choice is depending on the performance requirements [4]. Commonly used mate-rials for the casing are aluminum or reinforced plastics. The casing of the prismatic cell is almost always connected to one of the terminals and in the case of an aluminum casing, it is connected to the positive electrode [28]. The prismatic cell format is available in a lot of different sizes depending on the manufacturer. However, there is a trend in the industry towards a thinner design (and wide enough to still have capacity) [4]. Scania is, though, moving in the opposite direction, towards thicker cells to simplify thermal management. As the cell format is a block-form the cell packaging is significantly improved when compared to cylindrical cells. However, in contrast with the cylindrical cell which can utilize the voids in the construction to manage the heat, the packaging of the prismatic cell can lead to abu-sive temperatures for the cells unless this is considered in the design [4]. In prismatic cell design, the temperature spread more evenly throughout the cell, but the close packaging of different cells may lead to cells heating each other and the heat dissipation might be an issue [28]. Therefore, when constructing the prismatic battery pack some thermal management de-vices can be incorporated between the cells, and the casing can be designed to include airflow channels that act as a cooling system. Moreover, the packaging can be optimized by placing the terminals on the same side of the prism, which is the common design trend [4].
Pouch cells
The pouch cell, commonly known as coffee-bag cells, is a special case of the prismatic cell which has a soft casing [4]. The soft casing is often made of a polymer-laminated aluminum foil and makes up a light cell design [28]. Within the cell, the electrodes are designed stacked or Z-folded to reach the desired characteristics. Like the prismatic cell, the temperature within the cell is spread uniformly but the thermal management still has to be considered between the cells in a pack. Furthermore, the position of the tabs will also impact the pack-aging of the battery pack, i.e. if the tabs are on the same side the connections will only need to be on the same side simplifying the design [4].
A summary of benefits and downsides with the different cell formats are presented in table 2.2, ’+’-sign yields an advantage, ’-’-sign a disadvantage, and ’0’-sign yields neither
advan-2.4. Battery thermal characteristics
tage nor disadvantage. The data and comparisons of the formats are very generalized and may be different depending on the manufacturer and cell chemistry.
Table 2.2: Comparison of Li-ion cell formats [4].
Cylindrical Prismatic Pouch
Packaging - + 0
Gravimetric energy and power density 0 0 +
Volumetric energy and power density 0 0 +
Cooling options + - 0
Safety + + 0
Robustness + + 0
2.4
Battery thermal characteristics
Lundgren, et al. [6] describes temperature as one of the most important characteristics of a battery. The temperature is of the utmost importance for the battery to be safe as well as to limit the aging of the cell which leads to it being more and more important as different formats of Li-ion batteries increase. The reason is that high temperature can reduce capacity or even trigger a fire that could in the worst case lead to a thermal runaway [29]. There are a few different strategies to manage the temperature in Li-ion batteries, it could be either by airflow, with a phase change material, or using a liquid, where the latter is a common strategy for large battery applications [6]. To build such heat management systems there are two essential parameters; specific heat capacity and conductivity.
Specific heat capacity
A Li-ion battery cell has a complex chemical configuration and complex reactions occur within a battery cell during charge, discharge, and also within aging, which therefore yields a change of phase structure and chemical configuration on its electrodes [29]. Research has shown that the heat capacity inside a cell depends on SOC, SOH, temperature, and electrolyte composition [29]. Furthermore, there are a couple of different methods to determine the spe-cific heat capacity [29]. Some methods to determine the heat capacity are shown as follows.
Mass-weighted average
This method is based on calculations of all materials inside a battery cell, which makes this approach in need of accurate data for heat capacity as well as mass for each component. Therefore, if there is available data it is possible to calculate the specific heat capacity of the cell according to each materials volume, density, and specific heat capacity which follows in:
CPglobal =
ř iρicivi ρřivi
, (2.3)
where CPglobal and ciare the specific heat capacity of the cell and constituent materials, corre-spondingly. ρ, ρiare densities of the cell and constituent materials, correspondingly, and viis the volume of the constituent materials [29].
Adiabatic calorimeter
The adiabatic calorimeter is the most commonly used method to find the specific heat capac-ity although the method requires complicated high vacuum equipment. The method is based on having a closed system, i.e a system where there is no heat exchange between the system and the environment outside, in which the battery cell is placed [29]. Such an isolation set-up usually includes high vacuum insulation and a radiation screen to reduce the heat transfer
2.4. Battery thermal characteristics
of gas and to eliminate the heat leakage of radiation. Using this method the specific heat capacity can then be calculated using the following expression
CPglobal = P
m¨ 1
dT/dt (2.4)
where CPglobal is the specific heat capacity of the cell, m is the mass of the battery cell, T is the battery cell temperature and t is the time when the cell gets exposed by the external heating power P.
Accelerating rate calorimeter (ARC) is one way of adapting the adiabatic calorimeter method which is based on (2.4) [8]. However, the ARC method uses a heat plate where the tem-perature is controlled so that heat exchange with the surroundings can be neglected. The plate should have a heated wire that ensures evenly spread temperature over the plate and to achieve the best result possible an adiabatic environment should be used [8, 29]. Another way of executing an adiabatic calorimeter test is to use the so-called heat flow calorimeter [29]. In a heat-flow calorimeter, the battery is placed in between two isothermal heat sources with the rest isolated, and P in (2.4) is measured by the thermocouples.
Conductivity
Heat diffusion in 3-D and Cartesian coordinates is defined as: B Bx λx BT Bx ! + B By λy BT By ! + B Bz λz BT Bz ! +˙q=ρcp BT Bt, (2.5)
where λx, λy and λz is the thermal conductivity [mKW ], depending on x, y, z-coordinates, T is the temperature at x, y, z, ρ is the mass density, cpis the specific heat capacity[kgKJ ], t is the time, and ˙q is the heat generation/ unit volume [30]. (2.5) aids in a basic understanding for the heat conduction within a body. Using (2.5), it is possible to obtain the temperature distribution T(x, y, z)from it. Moreover, in a battery cell it is common to discuss in-plane and through-plane conductivity [6]. In-plane conductivity demonstrates the plane inside the jelly roll in which the electrodes lay, while through-plane conductivity is how the heat conducts between the layers in the jelly roll.
Heat generation
In a battery cell the heat generation can be expressed in a simple way as: Qt=I(V ´ VOCV) loooooomoooooon irreversible +ITcdVOCV dTc looooomooooon reversible (2.6)
where I is the current (charge (positive)/ discharge(negative)), VOCVis the OCV, V is the cell voltage, dVOCV
dTc is the entropy coefficient, and Tcis the core temperature [7]. In (2.6), the first
term on the right side of the equal sign, I(V ´ VOCV), is the irreversible, by overpotential, generated heat, which always is positive, while the second term, ITcdVOCV
dTc , is the reversible
entropic heat. The entropic heat is either positive or negative depending on whether the current is negative or positive and also the entropy coefficient’s sign [7]. Moreover, at low C-rates, i.e. ă 1C, the reversible losses are far more dominant than the irreversible losses [31]. During operation the heat generation within the cell is higher on the negative current collector side compared to the positive [32].
2.5. Battery cell physics models
2.5
Battery cell physics models
A battery cell can be modeled in a few different ways, these include the single-particle model (SPM), the single-particle model with electrolyte (SPMe), the porous electrode model with polynomial approximation, and the pseudo-2-D (P2-D) model.
Single-particle model
Among all of the available battery cell models with reduced order, the most commonly used one is the SPM, which is a simplification of the P2-D model. Therefore, the SPM has its origin in the full-order electrochemical model, which leads to it inheriting some important properties [33]. In the model, each electrode is assumed to be one spherical particle with an evenly distributed current and the concentration of the electrolyte is assumed constant regarding both time and space [34], see figure 2.3 for an overview. Two downsides of the SPM are that it captures less dynamic behavior compared to the full-order electrochemical model and that the mechanical responses are not included. The mechanical responses are important when the electrode material characteristics include high partial molar volume and high modulus since the effects on diffusion will be significant [33].
Figure 2.3: A geometric view of the SPM concept. Image adapted from [33].
Single Particle model with electrolyte
The SPMe is similar to the SPM but here the characteristics of the electrolyte are considered. In the SPMe there are four assumptions made which make up the foundation for the model, these are:
1. in the solid phase, the Li concentration is constant for each electrode in the spatial coor-dinate x, and uniformly in time,
2. independent of the position x, the exchange current density can be approximated by the average number of it,
2.5. Battery cell physics models
4. the charge transfer coefficients for both the positive and negative terminals are the same, which is usually the case [35].
Comparing the SPM and SPMe, the SPMe model includes a few more terms, which relate to the electrolyte characteristics. For example, in the voltage equation, the SPMe includes the electrolyte concentration overpotential as well as the ohmic potential drop because of electrolyte conductivity [35].
P2-D (Doyle, Fuller, and Newman) model
In figure 2.4 the geometry of the pseudo-2-D (P2-D) model is visualized, which was intro-duced by Newman and Tiedeman [36]. In the model, electrode particles are assumed to be spherical, and the coordinate x is used for both the electrodes as well as the separator. Inside each electrode, at every point x, there is a particle with radius r. The x and r coordinates make it a P2-D model because of the macro-scale where the assumption is that at every point in the space x, a particle is present [34, 37]. The perks of using the P2-D-model are that it acknowl-edges a good prediction of electrochemical behavior during both charging and discharging of a battery cell for wide operational use. The P2-D-model still uses a reasonable amount of computation power. The reason for this is because the model includes the dominant physical phenomena. Furthermore, the model is beneficial for another reason, it is flexible and can handle other phenomena in the battery cell, for example, thermal behavior [38].
Figure 2.4: A geometric view of the P2-D-model, where both the 2-D projection and an elec-trode particle are included.
Moreover, the P2-D model has over the years been extended and is now referred to as the Doyle, Fuller, and Newman (DFN) battery model. There are some variables in the model, which are, in addition to the ones mentioned in table 2.3, the current density I(t), the specific area asand the length of each electrode L [37].
2.5. Battery cell physics models
Table 2.3: Nomenclature for the DFN-model [37, 39, 40]. Spatial variables
x Cell spatial variable m
r Particle spatial variable m
Variables
cs(x, r, t) Particle Li concentration mol m´3
csurf
s (x, t) Li surface concentration mol m´3
ce(x, t) Li electrolyte concentration mol m´3
us(x, r, t) Transformed concentration mol m´2
usurf
s (x, t) Transformed surface concentration mol m´2
φs(x, t) Potential in the solid V
φe(x, t) Potential in the electrolyte V
φdl(x, t) φs(x, t)´ φe(x, t) V
is(x, t) Current density in the solid A m´2
ie(x, t) Current density in the electrolyte A m´2
j(x, t) Reaction rate A m´3
η(x, t) Overpotential V
U(usurfs ) Open circuit potential V
i(t) Applied current density A m´2
V(t) Measured voltage V
Parameters
cmaxs Maximum Li particle concentration mol m´3
Ds Particle diffusion coefficient m2s´1
Rs Spherical radius of the particles m
Lk Length m
Lbat Total length of the battery m
ee Porosity coefficient
b Bruggeman coefficient
Deffs Effective diffusion coefficient m2s´1
as Specific interfacial area m´1
t+ Transference number
F Faraday’s constant C mol´1
αpos, αneg Positive/ negative electrode transfer coefficients
R Universal gas constant mol´1K´1
T Temperature K
k Exchange rate parameter m2.5mol´0.5s´1
σeff Effective electrode conductivity S m´1
κeff Effective electrolyte conductivity S m´1
Rctc Contact resistance Ω
Sub and super scripts
1 Denotes negative electrode parameters
2 Denotes separator parameters
3 Denotes positive electrode parameters
12 Negative electrode/ separator boundary
23 Positive electrode/ separator boundary
The inserted Li in the particles can be modeled using a spherical diffusion equation: Bcs(x, r, t) Bt = 1 r2 B Br r 2DsBcs(x, r, t) Br ! . (2.7)
2.5. Battery cell physics models
Subject to the flux boundary conditions [37, 39] Bcs(x, 0, t)
Br =0 and Ds
Bcs(x, Rs, t)
Br =´j(x, t). (2.8)
The concentration of ions on the surface of the particles serves an important role in the transmission of ions between the particles and the electrolyte, this concentration is denoted csurfs (x, t) =cs(x, Rs, t). If instead, us(x, r, t) =rcs(x, r, t), is used, (2.7) becomes
Bus(x, r, t) Bt =Ds B2us(x, r, t) Br2 (2.9) subject to us(x, 0, t) =0 and 1 Rs Bus(x, Rs, t) Br ´ us(x, Rs, t) R2 s =´j(x, t) Ds (2.10)
where the value of transformed concentration at the surface is denoted usurfs (x, t) =
us(x, Rs, t)[37, 39].
In the DFN-model, the assumption of electron-neutrality is made, which means that the con-centration of positive and negative ions, c+(x, t)and c´(x, t), is the same in the solution, (c+(x, t) =c´(x, t) =ce(x, t)). The ion transport for the electrodes is then given by:
eeBce (x, t) Bt = B Bx eeD eff e (ce(x, t)) Bce Bx ! +as(1 ´ t+)j(x, t) (2.11) where Deffe =Deebeis the effective Fickian diffusivity, the Bruggeman coefficient b accounting for porosity effects, and ce(x, t)is subject to homogeneous flux boundary conditions on both sides [37]. The volume-average concentration, eece, is the concentration which actually is studied. Within the separator, the reaction rate is zero, e.g. j(x, t) =0, which yields the ion diffusion according to [37] eeBce (x, t) Bt = B Bx eeD eff e (ce(x, t)) Bce Bx ! . (2.12)
For the Li electrolyte concentration, ce(x, t), we have the following boundary conditions: • Bce(x,t)
Bx =0, at the boundary between the current collector and electrode • ee1/3De1/3eff
Bce(x,t)
Bx =ee2Deffe2 Bce(x,t)
Bx , at the boundary of the separator and the electrodes. (Subscript 2 means the value in separator and subscript 1/3 means the value in either the positive or negative electrode, depending on which boundary) [37].
The over-potential in the circuit is the perturbed local electrical potential, which is relative to the open-circuit potential (OCP), is given by
η(x, t) =φs(x, t)´ φe(x, t)´U(usurfs (x, t)), (2.13) where the OCP is U(usurfs (x, t)), which commonly is given by an experimentally fit function of xsurfs (x, t) = c
surf s (x,t)
cmax
s , the stochiometry, after a change of variables [37].
At the particle boundary the reaction rate kinetics satisfy the Butler-Volmer equation
j(x, t) = i0(x, t) F " exp αnegF RT η(x, t) ! ´exp ´αposF RT η(x, t) !# (2.14)
2.5. Battery cell physics models
where i0(x, t), the exchange current density, is [37]
i0(x, t) =kF (csmax´csurfs (x, t))αneg(csurfs (x, t))αpos(ce(x, t))αneg. (2.15) Furthermore, Ohm’s law is assumed to hold in the solid phase
is(x, t)
σeff =´
Bφs(x, t)
Bx . (2.16)
In the liquid phase, MacInnes’ equation is assumed to hold ie(x, t) κeff(ce(x, t)) =´Bφe(x, t) Bx + 2(1 ´ t+)RT F χ cTV¯0 Bln ce(x, t) Bx , (2.17)
where κeff=κebe, the ionic conductivity [37, 40].
Another assumption that is made is that the solvent concentration is only weakly dependent on concentration. This means that χ/(cTV0¯ )«1 and this term will therefore be omitted going forth. Kirchhoff’s law yields current conservation as
is(x, t) +ie(x, t) =i(t) (2.18) where ie = 0 and ie = i(t) at the current collectors respectively the electrode separator interfaces [37].
The ionic current’s divergence is related to the reaction rate Bie(x, t)
Bx =asFj(x, t) (2.19)
where as=3es/Rsis the specific interfacial surface area [37]. Moreover, the voltage is given by
V(t) =φs(Lbat, t)´ φs(0, t)´Rctci (2.20) with Rctc, the contact resistance, in[Ωcm2], at the current collector/ electrode interfaces. In figure 2.5 all of these terms related to the DFN-model is shown, and the relationship between them.
Figure 2.5: Block diagram showing the relationships between different terms in the DFN-model. Image adapted from [37].
2.6. Converters
The porous electrode model with polynomial approximation
In the porous electrode model with polynomial approximation, the PP approach, the com-plexity of the P2-D model is retained but the mathematical calculations are more straight-forward [34]. The PP model approach is sufficient for all assumptions which are made for the P2-D model. However, the significant assumption that is made for this model is that the concentration within each of the spherical particles of each electrode approximately can be modeled as a parabolic profile [34].
2.6
Converters
In this section, the theory and principles for a few different types of converters will be pre-sented. To start, the TLC will be introduced, and after that the three-level converter. There will also be a presentation of the concept for the three-phase converter and, lastly, the MMC will be introduced and explained.
Two-level converter (Half-bridge)
The purpose of the two-level converter (TLC), sometimes named half-bridge (HB) converter, is with a given DC input produce a sinusoidal output. This is done by a switch-mode principle and therefore possible to guarantee a sinusoidal output voltage with controlled magnitude and frequency [41]. The TLC is commonly used in photovoltaic plants, AC motor drives, and AC power supply. Figure 2.6a shows the schematic and the different opera-tions of an HB-based converter. The output voltage v0, is controlled by the complementary switches S1and S1. It can be seen that when S1is ON and S1is OFF, v0becomes Vdc/2 and the opposite if S1is OFF and S1is ON, v0turn out to be ´Vdc/2. Hence, an approximately sinusoidal waveform can be produced by controlling the ON and OFF time of these switches. Furthermore, figure 2.6b shows a triangular carrier signal tri(t)and a sinusoidal reference signal V˚(t). S
1 is ON whenever V˚(t) is greater than tri(t) and OFF whenever V˚(t) is less than tri(t)and vice versa for S1. Thus, the output voltage Voutbecomes sinusoidal after filtered through the inductance. Moreover, in practice, a dead time is introduced to make sure that both S1and S1are never turned ON at the same time, which causes a short circuit [41].
2.6. Converters
(a) TLC operation
(b) TLC pulse with modulation (PWM) scheme
Figure 2.6: Operation and modulation of a TLC. Image adapted from [41].
Three-level converter (full-bridge)
The three-level converter, full-bridge (FB), is shown in figure 2.7. The TLC has two different output voltage levels, while the three-level converter has like the name suggests, three dif-ferent levels [41, 42]. This is due to the fact that the output can be either positive or negative Vdc, but also zero, see table 2.4. The diodes are used for a continuous current, with the same motivation as for the HB converter.
Table 2.4: Output voltages for the full-bridge converter. S1 S1 S2 S2 Output [V]
on off off on Vdc off on off on 0V off on on off ´Vdc
Figure 2.7: An example of a full-bridge converter.
One common modulation scheme for the three level converter is the level-shifted carrier based pulse width modulation (PWM) [41]. This approach uses a sinusoidal reference v˚(t) which is compared to two level shifted carriers triu(t) and tri
2.6. Converters
v˚(t)ątriu(t), S
1and S2are turned on, if triu(t)ąv˚(t)ątril(t), S1and S2are turned on, and if tril(t) ąv˚(t), S1and S2are turned on. This process is then repeated until the next switching cycle.
Figure 2.8: PWM-controlled output voltage for the FB converter. Image adapted from [41].
Three-phase converter
The three-phase TLC is highly popular in current EV applications [43], and these three-phases are used to reduce harmonics [42]. The three-phase TLC has three-phase output. Each phase consists of a HB or FB where each of these converters are modulated such that the output voltages of each phase are 120˝ apart. Figure 2.9a show a three-phase TLC and figure 2.9b show the output voltage for each phase. It is important that the switches T1and T2, T3and T4, or T5and T6are not closed at the same time, since this would lead to a short circuit. Since the hardware switches are not instantaneous, there needs to be some delay (dead time) in the switches to prevent short circuits.
2.6. Converters
(a)
(b)
Figure 2.9: (a) Three-phase two-level converter. (b) Output voltage for each phase after the inductance.
Modular Multilevel Converter
Modular multilevel converter (MMC) has become popular over the last years and it is be-cause of its high efficiency, modularity, and low distortion power but also bebe-cause the MMC has its attribute of controlling the flow of active and reactive power and the possibility to connect the converter to the high voltage terminal [44]. Another advantage that has been stated is that MMC does not have any electromagnetic interface which makes it safer and manageable as well as sufficient with low pulse frequency [45].
The Full bridge converter illustrated in figure 2.7 has the possibility to produce output volt-ages of Ud, 0 and -Ud. The basic full-bridge concept can be coupled together with other full-bridge converters and hence produce more voltage levels. The one shown in figure 2.10 has the possibility of producing the voltage levels 2Ud, Ud, 0, -Ud and -2Ud, where Ud il-lustrates a battery instead of a DC power supply, as above, which is used in EVs. With this type of configuration, the output will become more sinusoidal and therefore reduce harmonic content.
2.6. Converters
Figure 2.10: A schematic example of a basic multilevel converter.
Each individual FB converter operates on its own switching scheme as shown in figure 2.11c and 2.11b, and also with its own delay angle and hence producing an output which looks like the one in figure 2.11a.
(a) (b)
(c)
Figure 2.11: (a) A controlled output from a cascaded multilevel converter. (b) The controlled output from the first bridge converter. (c) The controlled output from the second full-bridge converter.
By letting N full-bridge converters be in series it is possible to get a waveform close to a sinusoidal. One way of doing so is illustrated in figure 2.12, which shows a common topology of a three-phase modular multilevel converter. With an equal amount of FBs sub-modules (SM), in the upper and lower arm, for each phase. It is possible to produce 2N+1 levels of the output voltage for each phase, as each SM can have three levels which is C0, 0 and -C0. A, B, C represents the AC terminal for each phase. iurepresents the current in the upper legs which is divided into three sub currents, one for each phase, correspondingly for ilin the lower legs.
2.6. Converters
C0represents a battery cell (it is common to illustrate a battery cell as a capacitor in the MMC case) and the diodes that were illustrated in the two- and three-level converter sections are not illustrated in this image but however included.
Figure 2.12: A typical topology of a Modular Multilevel Converter, with each sub-module as a full-bridge converter.
If for instance five SM is used in each upper and lower arm, the output in figure 2.13 is generated. If the number of SMs is increased, the closer the output looks like a sinus curve.
Figure 2.13: The voltage output of an MMC with five SM in each upper and lower arm.
