Akku4Future. Report for Workpackage 2. Florian Niedermayr

Transcription

1 Akku4Future Report for Workpackage 2 Florian Niedermayr Summary This progress report explains the decision to focus on the widely used Lithium cells within this research project. Their fundamental properties are discussed in detail. The investigated electrochemical cell and its major properties are introduced. The fundamental chemical processes in a cell are described, with a special focus on the dynamic behavior. A number of different modeling approaches suitable to simulate the dynamic processes of a cell are discussed. The report concludes with a short outlook on the models used in the upcoming work- packages.

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3 Table of Contents 1 Motivation of choosing Li-ion batteries General aspects about lithium cells and there application Description of the used cell (LiNiCoMnO 2 ) Parameters describing the cell [2] [5] [9] [10] [11] Electrochemical processes in cells The cell in equilibrium Dynamic behavior of cells Models for Electrochemical Cells Simulation and Measurement with equivalent circuits Impedance based modeling approaches Impedance spectroscopy Impulse response measurements Zarc Element Warburg Impedance Used equivalent circuits Bibliography... 31

4 Abbreviations Li - ion EMF Li pol NiCD NiMH SOC SEI SOF SOH OCV Lithium Ion Electromotive Force Lithium Polymer Nickel Cadmium Nickel Metal Hydride State of Charge Solid Electrolyte Interface State of Function State of Health Open Circuit Voltage Symbols and Constants F H G S R z R C R E R AM R DC j D Li U CL C D R D CPE Faraday Constant Enthalpy of Reaction Free Enthalpy Reaction Entropy Universal Gas Constant Charge Number Ohmic Resistance of the Conductor Ohmic Resistance of the Electrolyte Ohmic Resistance of the Active Material Direct Current Resistance Diffusion Current Density Diffusion Coefficient Clamp Voltage Double Layer Capacity Charge Transfer Resistance Constant Phase Element

5 1 Motivation of choosing Li-ion batteries Lithium batteries were developed primarily in Asia. Their innovation is strongly coupled to the uprise of the consumer electronic. The yearly market volume of 10 billion dollars is solely produced in Asia. The interest in research related to batteries however, increased in the western world in recent years, due to the increased interest in electro mobility [1]. Lithium-ion batteries are regarded as key technology for future electrical vehicles and drives in general. These kind of batteries are used in areas where high power or energy densities are needed (see Figure 1). Figure 1 Ragone diagram, illustrating the relation between the specific energy (Wh/kg) and the specific power (W/kg) at cell level. Applications such as hybrid cars demand a high power density, whereas purely electrical vehicles demand a high energy density. Future generations of lithium batteries, such as Li- sulfur and Li-Air, will have an even higher specific energy. The overall opinion is that the market for lithium batteries will grow steadily, with up to three time s higher sale numbers in 2015, whereas the costs per kwh will drop by 50% [1]. The niche with the highest growth potential is the power tool market, where NiCd cells will be replaced by Li-ion cells [2]. Whit these facts in mind we decided to restrict the investigated chemistries to the Li- ion technology. The developed models will be adapted. We will include further battery types (Li-pol and NiMH) if some measurement time is left at the end of the relevant work packages. Seite 5/32

6 2 General aspects about lithium cells and there application Lithium batteries might either be single use or rechargeable. The element combines the advantages of being the lightest solid element and the material with the highest electrochemical potential. The biggest disadvantage however is its strong reaction with water, resulting in gaseous hydrogen (2Li + H 2 O 2 H2O + H 2 ) [2]. This explains, why even traces of water limit the live time of such batteries. Today lithium batteries are used in a variety of fields including: hybrid cars (since 2008), cell phones, portable computers, wireless tools (2005), video cameras, digital cameras and others [2]. The storage of electrical energy is fundamentally different to other technologies such as NiCd for example. Figure 2 Charge and discharge of lithium batteries [3]. The electrodes store the gaseous lithium atoms in a host lattice. Different host materials can be used for this purpose. Figure 2 illustrates the transport of Li ions from the cathode to the anode during the charging process. These ions react with e - transported in the outer circuit (not shown in the figure), to form Li atoms which are finally stored in the cathode layers. The inverse process happens during the discharge of the battery. The lithium atoms dissipate an e -, the so created lithium ions move to the cathode. Arrived there, they are neutralized by e - and stored in the layers of the active material [4]. Figure 2 illustrates the major components of a Li-ion battery. The positive electrode consist of a metal oxide (in this case LiCoO 2 ), a electrolyte containing Li- ions and a negative electrode made of graphite. A large number of different materials are used for the electrodes. Figure 3 illustrates some examples. The electrical potential of the cell is defined by the ability of the materials to store lithium. The used material and its properties are chosen based on the final requirements of the battery. In general two materials are chosen based on their potential which should be as apart as possible and feature a large electrical capacity (in mah/g). These properties lead to a high energy density, being a basic requirement for all electrochemical cells. Seite 6/32

7 Figure 3 Potential of different electrode materials [2]. The electrolyte consists of a conducting salt (e.g. lithium salt) plus fluent and solid additives such as organic solvents. The solvents are chosen to meet the requirements of the cell (e.g. optimization for low temperatures, high currents). The rather low conductivity leads to pronounced internal impedances and a poor high current ability of the cells. The active material of the electrodes is bonded to a slim metallic foil, necessary for the flow of the electrical charges to the pole. A separator avoids the direct contact (short circuit) of the electrodes; it has to be permeable to Li- ions and highly porous (e.g. polyethylene, polypropylene). All cavities are filled with the electrolyte [4]. This kind of battery needs a robust enclosure, featured by the so called cell and different prismatic formats Figure cell prismatic cell pouch cell Figure 4 Different formats of lithium batteries. Beside the Li- ion cells, Li- polymer (gel) cells and Li- (solid-) polymer cells are available. The structure of Li- polymer (gel) is basically identical to Li- ion cells. The same electrode materials can be used, Seite 7/32

8 leading to the same chemical reactions. An adaption of the electrolyte, meaning the combination of solvents and salts, is needed. It is possible to use the more cost efficient pouch cells with this kind of technology. Li- (solid-) polymer differs significantly from the previously described cell types. Their electrolyte contains no fluid components. This makes the ion transport slower. The cell unfolds its functionality at C, which makes them not applicable for consumer electronics [2]. A major advantage of all kinds of lithium batteries is there high cycle lifetime (up to 2000) and there excellent charge discharge efficiency in combination with extremely small self-discharge current of a few percent per month [5]. The self-discharge is reversible but increases as a function of temperature. The chemistry of the lithium cells defines the course of the electromotive force (EMF) as a function of the capacity. The curve has a characteristic course and it s strongly related to the state of charge (SOC) [2]. The EMF changes as a function of the SOC with only a small hysteresis if charged or discharged. The cell clamp voltage is a function of the drawn current. This is evident because the voltage drop over the inner impedance increases linearly to the drawn current. In general it is possible to remove the whole charge from the cell with within one hour at the nominal current (this equals 1C). For high current applications special cells are needed [4]. The optimal temperature operation area is 0-40 C. In this area more or less the whole charge can be removed from the cell. The maximal available capacity increases with the ambient temperature but decreases for higher currents [2]. Lithium cells have unlike other cells a calendric lifetime of 5 years. Moreover the cell ages with the number of performed cycles. The first charging process forms a solid electrolyte interface (SEI), which protects the negative electrode and alters the inner impedance. The SEI growths over time. By that the discharge cut-off voltage is reached faster [1]. Similar the electrical contact between active material and metallic film worsens over time, limiting the lifetime of the cell as a whole [6]. Most aging mechanisms increase the inner impedance as previously described. High currents, temperatures and cell voltages accelerate these processes. The volume of the active material changes as a function of the stored Li-ions (9.2% for graphite). The changing volume leads to a dilatation and detrition of the cell [2]. Commonly the cell is regarded to be fully consumed, if the available capacity drops below %. A misuse of the cell lowers the lifetime of a cell dramatically. This includes overcharging, continuous charging, to high charge voltages, high current densities, depth discharges, extreme ambient conditions especially in the cell package [2]. Seite 8/32

9 3 Description of the used cell (LiNiCoMnO2) A special Li- ion cell was chosen for this project. The negative electrode is composed of graphite while the positive electrode is made of Li(Ni x Co y Mn 1-x-y )O 2. The active material features a crystal structure, made of cobalt, nickel and manganese in a defined proportion (typically 1/3 each). This configuration leads to an EMF of V. The chemical composition features a number of advantages; the high capacity of LiCoO2, the good high current ability of LiNiO2 and finally the cost efficiency of LiMn 2 O 4 [2]. Moreover is this type of Li- ion cell safer as others and has the lowest selfheating rate. These properties make it the first choice for electrical vehicles [7]. Figure 5 summarizes these properties and puts them in relation to other Li- ion cells. Table 1 specifies the most important technical information of the used cell. Figure 5 Properties of different Li- ion cells. Seite 9/32

10 Li(NiCoMn)O 2 Typical Capacity 2150mAh (0.2C, 2.75V discharge) Minimum Capacity 2050mAh (0.2C, 2.75V discharge) Charging Voltage 4.2V±0.05 V Nominal Voltage 3.62V (1C discharge) Charging Method CC-CV Charging Current Standard charge: 1075mA Rapide charge : 2150mA Charging Time Standard charge : 3hours Rapid charge : 2.5hours Max. Charge Current 2150mA Max. Discharge Current 10A (Continuous discharge) Discharge Cut-off Voltage 2.75V Cell Weight 44.5g max Cell Dimension Diameter(max.) : Φ mm Height : 65mm max Operating Temperature (cell surface) Charge : -10 to 50 Discharge: -20 to 70 Table 1 Technical Information about the used cell (ICR P Samsung) [8] 4 Parameters describing the cell [2] [5] [9] [10] [11] The following definitions summarize commonly used terms to characterize electrochemical cells. Most of these parameters are interrelated and crucial for the understanding of the cell itself and potential mathematical models. Cell: Basic unit used to store electrical energy in form of chemical energy and to deliver electrical energy from the chemical energy. The cell consists, in its simplest form, of two electrodes, an electrolyte and a separator. Battery: Connection of more cells to obtain the needed voltage or capacity. Often times the term is also used for single cells. The cells can be connected in series or parallel, creating a so called battery pack. Energy density: The volumetric energy storage density of a cell (Wh/l) Power density: The volumetric power density of a battery (W/l) Rated capacity (Q N ): The capacity of a battery (Ah) obtained from the fully charged cell under the discharge conditions defined by the manufacturer (e.g. discharge current, discharge time, temperature) or relevant standards. Often time the discharge current is the current needed to discharge the cell in five or ten hours (e.g. stationary batteries, drive batteries). The charge that can be drawn from a cell is smaller as the rated capacity and related to the cell history, temperature, discharge current and the age of the battery. Peukert [12] proposed an empiric formula to describe the relation of the battery capacity and the discharge rate: (1) Seite 10/32

11 I is the discharge current, t the discharge time, n and K are parameters. K equals Q N and n is one for an ideal cell. The discharge time can be computed for a given n and known nominal conditions (nominal charging current (I N ) and nominal charging time (t N )) and arbitrary discharging currents: (2) Nominal Voltage: Is defined as the difference between the potential of the positive and negative electrode during steady state. Specific values have been defined for all kinds of cells (e.g. NiCd 1.2V, NiMH 1.2V). The combination of cells defines the overall nominal voltage. Rated energy: Is a simplified quantity describing the operation time of a cell. It is defined by the product of the nominal voltage by the rated capacity. C-rate: A C-rate of 1 C equals a current that discharges a cell in one hour. From a cell with a capacity of 2150 mah a current of 2150 ma has to be drawn. Multiples are used to express larger or smaller currents. This normalization makes it also easier to compare cells with different rated capacities. Cycle life: Number of time that a cell can be charged and discharged, before the cell fails a defined criteria (often times the criteria is 80 % of the rated capacity). Cut-off-voltage: Voltage that defines the end of the charge/ discharging process. It is important especially for safety reasons. Self-Discharge: Autonomous loss of a battery. It is expressed in percent of the rated capacity per month for a given temperature. State of Charge (SOC): The reference point for the state of charge can either be the fully charged or fully discharged cell. In general the latter definition is used. The state of discharge can be defined in various ways (see Figure 6). Important is the capacity that the SOC is related to. The measurable capacity is the charge that can be drawn from the cell, if the cell was charged according to the specifications of the manufacturer. The usable capacity is the available capacity for a given application. It is important to notice that the rated capacity keeps constant over the lifetime of a cell, while the usable capacity as well as the available capacity diminishes. The SOC is defined by the following equation as a function of the discharge current (I di ): (3) Seite 11/32

12 0% 0% rated capacity Q N SOC relative capacity Q relative SOC 100% 100% 0% available capacity practical SOC 100% Figure 6 Different definitions of the SOC according to [10] State of Health (SOH): This measure describes the age condition of the cell. Different parameters like the Capacity, the self-discharge current (I SD ) and the load change behavior can be used to derive this quantity. In the literature a variety of approaches are used. In [10] a weighted average is proposed to describe the SOH. The equation puts the initial parameter in relation to its diminution over time: (4) The equation can easily be expanded by other parameters. The parameter diminishes from 100% (begin of life) to 0% (end of life). It is obvious that the intended application of the cell strongly affects the weighing factors. A declining battery capacity is of great interest for a battery powered electronic vehicle, while for a hybrid electrical vehicle a decline of the available power is more important. State of Function (SOF): Defines the capability of a battery to deliver a defined task. An exact definition is rather tricky. A simple fail/pass criterion is a simple but reliable criterion to define the parameter [5]. But this measure does not allow a forecast with regard to critical cell states. The following equation is more suitable for this purpose [13] [11]: (5) This equation relates to a defined power profile used for the specific cell. During the discharge of the cell a minimum electrical voltage U min is measurable; U low is defined by the user as the lower voltage limit under load. U fresh is the lowest voltage of a fresh cell of the same type. With this definition the SOF becomes a quantity declining from 1 (for fresh cells) to 0 (for consumed cells). The shortcoming of this approach is that it is limited to a specific load profile. Seite 12/32

13 A set of efficiency factors can be used to describe the performance of the cell: Voltage efficiency factor ( V ): Describes the rate of the clamp voltage (during the charge and discharge process) and the open circuit voltage in that case the EMF. It can be larger or smaller as 1. (6) Charge efficiency factor ( CH ): Rate of the expended charge during the discharge process in relation to the charging processaccounts for the self-discharge and comparable processes. (7) Energy efficiency factor ( EE ): Rate of the expended energy during the discharge process in relation to the charging process. This parameter considers, unlike to the previous, the voltage losses (e.g. ohmic internal resistance). This means that EE is always smaller as CH. (8) These factors are strongly affected by the used charging procedure and the discharge current. Seite 13/32

14 5 Electrochemical processes in cells The following chapter describes the fundamental mechanisms found in all kinds of electrochemical cells, not limited to Li- ion cells. For special phenomena s (e.g memory effect) dedicated literature must be considered [2] [4]. 5.1 The cell in equilibrium The cell is in equilibrium if no current is drawn from the cell. For simplicity the self-discharge is neglected. Thermodynamic laws can be used to describe the steady state. The dissipated or absorbed energy during a chemical reaction is referred to as the standard enthalpy of reaction ( H). The free enthalpy ( G), describes the maximal electrical convertible energy and is a fundamental measure of electrochemical cells. Not the entire energy is transformed into electrical energy. The reaction entropy ( S) is derived by the difference between these measures: T S describes the dissipated or absorbed energy to / from the environment. This process is reversible and induced only by the chemical reaction and not by an overpotential inside the cell [2]. These measures allow a precise physical and chemical description of the cell. The following cell parameters can be derived from them: The theoretical equilibrium voltage, the theoretical energy density and others. The equilibrium voltage can be defined with the following relation, based on faradays law. (9) The parameter n defines the quantity of used material (mol) and F (C/mol) is the faraday constant. The free enthalpy is a function of temperature and the concentration of the reacting agents. Considering the latter case the Nernst equation can be derived: (10) [( ) ] (11) R (Jmol -1 kg -1 ) is the universal gas constant, T (K) the absolute Temperature, a i describes the activity of the reacting compounds and finally j i the number of equivalents of the reacting compounds. U 0S is the standard potential at 25 C for a given concentration of 1 mol/l. The equation shows the relation of the equilibrium potential and the concentration of the charge carrier. The open circuit voltage (OCV), measurable at clamps of a cell, is not equal to the equilibrium voltage. The difference is mainly attributed to side reactions that are not directly involved in the charging or discharging reactions. The self-discharge is a result of these processes [11] [10]. Seite 14/32

15 5.2 Dynamic behavior of cells The last chapter showed that the electrode potential can be described with the Nernst equation. The voltage at rest equals the OCV (no current is drawn from the cell). The clamp voltage however, differs from the OCV if a current is drawn from the cell. The deviation is directly linked to the energy needed to start the redox reaction and commonly referred to as overpotential ( ). Faradays law describes the deviation from the equilibrium stage. It buts the mass m (kg) involved in the electrochemical reaction (which is the reason for the deviation from the equilibrium in the first place), in relation to the produced electrical charge Q F (As): (12) The charge number (z) defines the number of electrons a given material absorbs or delivers; F is the Faraday constant and m the molar mass. The equation describes the transition from chemical to electrical energy and vice versa and is accompanied by a number of mechanisms (e.g. diffusion, charge transfer). Each of these mechanisms leads to an overpotential and is described in more detail below. Since they are triggered by an electrical current a electrical impedance can be assigned to each of them. Ohmic resistance: The ohmic resistance of a cell can be described with the Ohm s law. It is composed of the resistance of the metallic conductor (R C ), the resistance of the electrolyte (R E ) and the resistance of the active material (R AM ) [2]. The main contribution of the overall resistance comes from the electrolyte. It is strongly affected by the temperature and the concentration of ions (SOC) [11]. The resistance of the active material changes as a function of the state of charge. The total resistance is also influenced by aging mechanisms (e.g. elevation of the transfer resistance) [2]. The measurement of the resistance is rather simple. The resistance is derived from a sudden current jump and the resulting voltage jump: (13) The contribution of the involved parameters can t be separated and are summarized in the direct current resistance (R DC ). The multiplication of this resistance with the drawn current defines the ohmic overpotential ( O ). Seite 15/32

16 Charge transfer: The charge transfer on the solid electrolyte interface results in a transition from ionic to electric current and vice versa. It is a dynamic equilibrium based on reduction and oxidation reactions [2]. At rest the same amount of ions are reduced and oxidized, which means that the net current is zero as well as the charge transfer overpotential ( CT ) (see Figure 7). The shift of the potential leads to a net current in one direction and a voltage drop expressed in terms of CT. The resulting current can be described as a function of the CT with the Butler Volmer equation. [ ( ) ] (14) The i 0 is the exchange current density, symmetry factor describing the asymmetry between charging and discharging, n the number of exchanged electrons, F the Faraday constant, R the gas constant and T the absolute temperature. Figure 7 Current - Voltage relation of a lead acid battery [2] Seite 16/32

17 Double-layer capacity: The charge carrier with different polarization (lithium ions and atoms) lead to a capacitance at the electrode electrolyte interface. The Helmholtz model shown in Figure 8 illustrates this behavior. More complex models describe this interface in more detail (e.g. Gouy- Chapman model), they show however the same macroscopic behavior at the clamps of the cell [10].This process does not cause by itself an overpotential but affects the charge transfer overpotential. This leads to a capacitor parallel to the charge transfer resistance, in terms of an equivalent circuit [11]. Typical time constants are in the range of milliseconds or a few seconds [2]. In general the electrodes are quite different; leading to two unequal, serial time elements. Figure 8 Electrochemical doublelayer capacity according to Helmholtz [3]. Seite 17/32

18 Diffusion: The production and consumption of charges at the electrode electrolyte interface creates a concentration gradient. The lithium ions counteract this process which leads to a diffusion of lithium ions. The resulting Faraday current composed of lithium ions and lithium atoms is best described by Frick s laws [5]. The ions move in the electrolyte while the atoms move in the active material. Parameters like temperature, geometry and the porosity of the separator influence this process [2]. Figure 9 shows a simplified, one dimensional distribution of lithium inside the cell. The amount of available ions around the anode declines, if the drawn current increases. This leads to a drop of the terminal voltage (see the Nernst equation which is a function of the concentration). This voltage drop equals the so called diffusion overpotential. The maximal possible current is reached, if no more ions are left around the electrode. For a limited period of time much higher impulse currents can be drawn from the battery [4] [2]. Figure 9 Diffusion of lithium (upper part) and course of the lithium concentration as a function of time (lower part). The concentration difference in the electrolyte decreases with distance from the cathode. The diffusion current density j (As/m²) can be described with Frick s first law: (15) D Li is the diffusion coefficient and d cli the ion concentration along the x axis. A stationary diffusion gradient adjusts if a constant current flows in the outer circuit. Similar concentration differences can be observed within the active material. The diffusion in the solid electrodes however is much slower with much higher time constants. Seite 18/32

19 Fick s second law is used to describe the changing number (concentration) of available lithium atoms as a function of time [5]: (16) These equations describe the simplified case of a one-dimensional, stationary cell. The diffusion potential can be computed by solving these differential equations and substitute them into the Nernst equation. At the boundaries (electrolyte - electrode interface) a defined diffusion current density and a defined lithium concentration has to be assumed. This leads to the following equation [10]: (17) The equation is only valid for small concentration variations, e.g. during electrical impedance spectroscopy. The impedance caused by the diffusion is called Warburg impedance and defined by the quotient of diffusion overpotential and current [14]. W (Om²/ s) is the so called Warburg coefficient: (18) (19) The diffusion process of ions can be compared to an electrical low pass. An abrupt change of the drawn current is followed by a slow adaption of the diffusion overpotential. Equation (17) describes this process, which leads to the formation of new concentration profiles. This explains the ongoing adaption of the open circuit voltage even after a number of hours. Seite 19/32

20 The resulting voltage that can be measured at the clamps of a cell (U CL ) is derived by the summation of the described mechanism (Nernst potential and the overpotentials). A model that describes the different contributions would give a good diagnostic overview of the current status of the cell. The implementation of such a model is rather tricky since a lot of parameters (e.g. temperature, drawn current, age of the cell, state of charge) have to be considered. Figure 10 highlights the corresponding, simplified electrical circuit. D O CT U 0 U C Figure 10 Simplified electrical circuit of a electrochemical cell [2] Seite 20/32

21 6 Models for Electrochemical Cells The proper modeling of electrochemical cells is still not solved. The rise of upcoming technology such as the breaking by wire and the steering by wire, demand highly robust models that are able to predict measures like the SOC, SOF and SOH in real time and with a large accuracy. Figure 11 illustrates the state of a cell in terms of a barrel. This model was originally introduced by Jossen et. al [2]. It gives a pretty intuitive, although simplified understanding of an electrochemical cell. Figure 11 Barrel model for the state of the cell / battery The water in the barrel equals the stored energy. The barrel can only be filled at infinite speed, regulated by the diameter of the inlet and the outlet. Similar the cell can only be charged / discharged with a maximal current, defined by the internal resistance of the cell. A new barrel has a capacity of 100% and the water level defines the SOC. Small holes in the wall lead to leakage of water, similar to the self-discharge of a cell. A number of other effects (e.g stones in the barrel, constriction of the inlet/ outlet, increasing leakage do to rusting) find there counterpart in the electrochemical cell (decreasing capacity with age, increase of the internal resistance, increase of the self-discharge) and define ultimately the aging respectively the SOH of the cell. The ultimate goal of a model for an electrochemical cell is the precise prediction of the defining parameters over the whole lifetime. The fact that some of the parameters (e.g. SOH and SOF) are depending on each other make things even more complicated. The defining parameters can t be measured directly at the clamps of the cell. They can be derived by a number of measures (e.g. clamp voltage, drawn current and temperature) and an underlying mathematical model. The simplest method relies on the measurement of the clamp voltage [2]. This however requires a sufficient rest period after electrical current is drawn from the cell, plus measurements with repetitious and significant accuracy. Li ion cells are suitable for this method because the voltage differ- Seite 21/32

22 ence between a full and empty cell are pronounced, the degeneration of the over potentials is fast and the cell behaves almost the same if charged or discharged. The ambient temperature and the drawn current however influence the open circuit voltage. Figure 12 highlights the typical course of the OCV as a function of the SOC. Figure 12 Typical course of the OCV as a function of the SOC [3] To shorten the needed rest period extrapolation techniques or look up tables can be used. An advantage of this approach is that such a system provides immediate results, after the cell is connected to it. The measurement of the underlying value, namely the electromotive force (the clamp voltage after all transient process have come to rest) as a function of the SOC takes a lot of time because they must be performed for different C rates and temperatures. Interpolation and extrapolation methods can reduce the needed time significantly [15]. A very basic put powerful method is the balancing technique also known as coulomb counting [2]. The assumptions are that both the empty state (SOC = 0%) and the full state (SOC = 100%) are clearly defined. The idea is to start with such a defined state and monitor the SOC by integrating the drawn current (= electrical charge) and scale it to the maximal capacity. The self-discharge of a cell reduces the accuracy of this method. The error increases over time due to the integrating nature of the method. The self-discharge for Lithium-ion cells is very small and can be neglected. The defined states (full, empty) can be used to recalibrate the method and reduce the influence of any kind. The current is measured indirectly across a very small shunt resistor. The combination of the previous two methods ensures a time stable, precise indication of the state of charge. Therefore we plan its implementation in the course of this project. The following paragraphs should give a short overview of methods used to derive cell parameters like SOC and SOH based on the measured current, voltage and temperature. The choice of the used model may be based on the geometric configuration of the used cell. Models of plate electrodes are suitable for cells with a parallel arrangement of the active material and the electrolyte [10]. The commonly used equivalent circuit for this kind of cells was originally introduced by Randles [16]. The idea is that each mechanism described in the previous chapter is accounted for in the equivalent circuit. The model accounts for the conductivity of the materials R E [ m²], the Seite 22/32

23 charge transfer R D [ m²], the electrochemical double layer C D [Fm²] and finally the diffusion Z W [ m²] (see 7.3 for an explanation). Figure 13 illustrates the model. The model is able to describe the chemical processes described in the previous chapter macroscopically. R`D Z`W R`E C`D Figure 13 Randles equivalent circuit [16] Modifications of the model (e.g. parallel connection of resistors and capacities) are frequently added. This ensures a better agreement between the model and the measured data. Additional elements can be added to describe other physical processes and to cover all kinds off cells [3]. This approach is used in this research project. The following chapter illustrates some key elements for the improvement of such equivalent circuits. The concept of porous electrodes is also often used. Different approaches are used for this kind of modeling. Physical and chemical models try to reflect the real geometric composition and behavior. The cell and its internal processes have to be known precisely for this kind of models. Equivalent electrochemical circuits consider electrochemical as well as electrical effects. Electrochemical mechanisms alongside with fundamental material parameters (e.g. chemical capacity) are considered in these models. These models are more suitable to characterize the used materials and not so much for the characterization of the status. A disadvantage is also the needed computation power to solve the electrochemical equivalent circuits of such configurations. The use of partial differential equations is not limited to porous electrodes. This method is suitable for all kinds of problems that try to give a numerical solution for the distribution of the current, potential and other quantities [10]. The needed computation time and required detailed knowledge makes them impractical for dynamic, responsive models. Fractal models are a good way to describe geometric or material inhomogeneities. The fractals of the electrodes are described in terms of impedances [17]. The latter models are only mentioned for the sake of completeness and are not investigated within this project. The described models are directly coupled to the measured values of an electrical cell. The intrinsic problem with the state determination of an electrochemical cell however, is the unpredictable behavior of the user and the battery. Adaptive methods try to overcome this shortcoming, by combing the previously described methods and using smart signal processing algorithms. These methods condense all the Information into meaningful parameters. Kalman filters, originally used in navigation and other fields, are a highly mathematical way to predict the SOC and other parameters [18] [19] [20]. The advantage of this method is its ability to compute the cell parameters based on measurements performed during runtime. The behavior of the cell is learned by the filters online. The estimation error of the SOC is in the range of 10% [10]. Some authors argue that this method is the best for long term SOC estimations [15]. Seite 23/32

24 Neuronal networks and fuzzy logic are other well-known methods used for the characterization of electrochemical cells. For neuronal networks a training phase is needed. This phase is followed by an evaluation phase, to ensure the needed accuracy of the method. Estimation errors in the rage of up to 3% could be demonstrated for the SOC estimation [15] [21]. Much simpler as the previous method is the counting of effects that influence the aging of cells (deep discharge, overcharge). The challenging part here is the evaluation of the different factors and there relative impact. Moreover the spread among different cells of the same type complicates things even more. The method is only suitable to predict the steadily ongoing aging and is not able to detect sudden failures [2]. The described adaptive systems are not subject of this research project. 7 Simulation and Measurement with equivalent circuits The used model should be able to map the measured input parameters like current, temperature and clamp voltage to meaningful cell parameters like the EMF, the SOC, the SOH, internal temperature and others. Figure 14 highlights this relation. Figure 14 Translation of the measured data to valuable cell parameters via the used model Discrete elements like resistors and capacitors in combination with voltage sources are used to model the dynamic behavior of electrochemical cells. For certain high frequency aspects even inductivities have to be considered. The advantage of this method is that the level of detail can be adapted easily. This means that the influence of the temperature, the aging or even of other effects can be neglected or accounted for if needed. Such dependencies do increase the computation time dramatically [3]. Another advantage is the applicability to a wide range of energy storage formats like supercaps and fuel cells [10] [22]. The dynamic behavior of the cells has to be known for the parameterization of the equivalent circuits and its discrete elements. We are going to use the two impedance based methods for the determination of the values. The first is based on the impulse response approach and the second on the impedance spectroscopy. Seite 24/32

25 7.1 Impedance based modeling approaches Impedance spectroscopy This method is used in a variety of areas to describe the dynamic behavior of systems respectively its periodic processes. These processes can be of mechanical, thermal, acoustic, electromagnetic nature [10]. Figure 15 illustrates the different steps of this method with the ultimate goal to extract information. Figure 15 Application of the impedance spectroscopy method The impedance of the cell is determined for a wide frequency range, a defined SOC, current and temperature. An accelerated current is superimposed to the direct current. The voltage response defines the spectra. Both measures lead to the impedance of the cell, which can be illustrated in a Nyquist diagram. Figure 16 [23]highlights such a spectra. The internal resistance is defined for Z im =0, which equals the crossing of the real axis. The following semi-circle is defined by the SEI. The diameter of the circle defines its resistance. The corresponding capacity can be derived based on the cutoff frequency. The small semi-circle is followed by a more dominant semi-circle defined by the charge transfer. The corresponding resistant is strongly related to the current and increases dramatically for small values (see chapter 5.2). The capacity is again defined by the cutoff frequency of the circle and equals the double layer capacity of the cell. The next part of the spectra describes the diffusion branch. It is often times difficult to derive the corresponding values, because of the long measurement time. All these parameters are a function of the SOC, temperature and current. The fact that the described effects do not result in perfect semi-circles complicates things even more. To overcome these problems improved elements are used to derive the parameters of the equivalent circuits (see chapter 0 and 7.3). Seite 25/32

26 Figure 16Typical spectra of an electrochemical cell Impulse response measurements The impulse response measurement is a more simplistic approach. A defined current is drawn from the cell for a short, defined period of time. The idea is to derive the values for the equivalent circuits from the voltage response of the cell. Commonly used fitting algorithms (e.g. optimization toolbox by Matlab ) can be used to determine these parameters. Measurements at different SOCs, temperatures and currents allow a complete description of the dynamic behavior of the cell. Seite 26/32

27 7.2 Zarc Element The semi-circle in the impedance spectra, caused by the SEI and/or the charge transfer, are often times squeezed. Ordinary RC- elements are however not able to reproduce this behavior. The passivation film impacts the impedance spectra in the high frequency area and is much more pronounced on the anode [3]. A second semi- circle is attributed to the charge transfer reaction. This effect is seen at lower frequencies compared to the previous. In principle both electrodes should be modeled in separate, it is however also possible to combine them in one model [3]. The porosity of the electrodes leads to a distributed relaxation time (around a mean value), resulting finally in a depression of the corresponding semi-circle in the impedance spectra [24]. Such behavior can be modeled with Zarc elements, defined by the following equation: ( ) ( ) (20) This equals a parallel circuit configuration of a constant phase element (CPE) and a resistor. The dimensionless defines the gradient of the impedance curve (1= capacity, -1= inductivity), A instead defines the generalized capacity (Fs -1 ), R ( ) defines the diameter of the circle. Figure 17 illustrates the complex- plane representation of the Zarc impedance for different. Figure 17 Complex-plane representation of a Zarc element [3] The Zarc element can be modeled with a series of RC elements, it was shown that five RC chain links (see Figure 18) are a good compromise between accuracy and computation time [3]. Figure 18 Equivalent circuit of a Zarc element in terms of an RC-chain The following set of equations is used to compute all needed parameters [3]. All chain elements (index A-E) can be computed based on the cut- off frequency 0_Zarc. The simplest approximation is defined by a single RC element. Seite 27/32

28 ( ) (21) ( ) (22) (23) ( ) (24) ( ) (25) ( ) (26) (27) ( ) (28) ( ) (29) The optimization factors f 1 ( ), f 2 ( ) and f 3 ( ) are computed offline as a function of the depression factor. Table 2 highlights the used optimization factors f1(ξ) f2(ξ) f3(ξ) Table 2 Optimization factors f1( ), f2( ) and f3( ) Seite 28/32

29 The optimization factors have to be fitted with a polynomial function if the parameters of a Zarc element have to be computed in time domain. Therefore following equations have been derived, based on an a polynomial fitting algorithm. ( ) (30) ( ) (31) ( ) (32) 7.3 Warburg Impedance The diffusion branch of the impedance spectra is not modeled properly with a sole RC element; instead so called Warburg impedances are used. In the literature three different kinds are used (see Figure 19). Figure 19 Complex plane plots of a) Z W, b) Z WI and c) Z WL Z W equals an infinite diffusion path. Z WI and Z WL correspond to finite diffusion paths with different boundary conditions. Z WI assumes an ideal pool, while Z WL defines a limiting non permeable wall. The latter models are used for the simulation of batteries [2]. The Warburg elements can also be modeled by a chain of RC elements, similar to the Zarc elements (see Figure 20). Seite 29/32

30 Figure 20 Approximation of the Warburg element (Z WI ) by a chain of RC elements It could be shown that seven RC chain elements are already a good compromise between accuracy and computation time [10]. 7.4 Used equivalent circuits The equivalent circuits investigated within this research project will be based on the Randles model. To best describe the dynamic behavior of the cell it will however be improved with Zarc and Warburg elements. Different configurations will be tested in the next work package in order to identify the optimal equivalent circuits in terms of computation time, accuracy and complexity. 7.5 Measurements To derive the parameters of the equivalent circuits a number of measurements are needed. A detailed measurement plan has been developed with the Carinthian University of Applied Sciencesin Villach. So far the planed measurements do only include impulse responses. The CUAS- Villach is working on a measurement setup that allows the measurement of the impedance spectra in the near future. The elaborated plan lacks of minor details and will be included in the next status report. Seite 30/32

31 8 Bibliography [1] A. Thielmann, R. Isenmann and M. Wietschel, "Produkt-Roadmap-Lithium-Ionen-Batterien 2030," Franuhofer-Institut für System- und Innovationsforschung ISI, Karlsruhe, [2] A. Jossen and W. Weydanz, Moderne Akkumulatoren richtig einsetzen, München: Inge Reichardt Verlag, [3] S. Buller, Impedance-Based Simulation Models for Energy Storage Devices in Advanced Automotive Power Systems, Aachen: Univ.-Prof. Dr. Rik W. De Doncker, [4] C. Daniel and J. O. Besenhard, Handbook of Battery Materials, Weinheim: Wiley- VCH Verlag & Co. KGaA, [5] M. Roscher, Zustandserkennung von LiFePO4-Batterien für Hybrid- und Elektrofahrzeuge, Aachen: Univ.-Prof. Dr. Rik W. De Doncker, [6] J. Vetter, "Ageing mechanisms in lithium-ion batteries," Journal of Power Sources, pp , [7] "Battery University," [Online]. Available: [Accessed ]. [8] "TME," [Online]. Available: HV.pdf. [Accessed ]. [9] H. J. Bergveld, Battery Management Systems Design by Modeling, Dordrecht: Springer, [10] U. Tröltzsch, Modellbasierte Zustandsdiagnose von Gerätebatterien, Düsseldorf : VDI-Verl., [11] O. Bohlen, Impedance-based battery monitoring, Herzogenrath: Shaker Verlag GmbH, [12] D. Doerffel, "A critical review of using the Peukert equation for determining the remaining capacity of lead-acid and lithium-ion batteries," Journal of Power Sources, pp. 155: , [13] E. Meissner and G. Richter, "Vehicle electric power systems are under change!: Implications for design, monitoring and management of automotive batteries.," J. Power Sources, pp. 95:13-23, Mar [14] E. Warburg, "Ueber das Verhalten sogenannter unpolarisirbarer Elektroden gegen Wechselstrom," Annalen der Physik und Chemie, pp. 3: , Seite 31/32

32 [15] V. Pop and H. J. Bergveld, Battery management Systems: Accurate State-of-charge Indication for Battery-Powered Applications, Springer Science + Business Media B.V., [16] J. E. B. Randles, "Kinetics of rapid electrode reactions," Discuss. Faraday Soc., pp , [17] B. Sapoval and J.-N. Chazalviel, "Electrical response of fractal and porous interfaces," Physical Review, p. 38: , [18] G. Plett, "Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 1. Background," Journal of Power Sources, p , [19] G. Plett, "Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 2. Modeling and identification," Journal of Power Sources, p , [20] G. Plett, "Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 3. State and parameter estimation," Journal of Power Sources, pp , [21] M. Charkhgard, "State-of-Charge Estimation for Lithium-Ion Batteries Using Neural Networks and EKF," IEEE Transactions on Industrial Electronics, pp , [22] J. Müller and P. Urban, "Impedance studies on direct methanol fuel cell anodes," Journal of Power Sources, [23] J. R. Macdonald and E. Barsoukov, Impedance Spectroscopy : Theory, Experiment, and Applications, London: John Wiley & Sons, [24] E. Barsoukov, Impedance Spectroscopy: Theory, Experiment, and Applications, John Wiley & Sons, Seite 32/32