Method and apparatus for evaluating battery cells containing materials exhibiting voltage hysteresis

Abstract

A method of charging a battery cell of a direct current device, wherein the battery cell includes an electrode, the method comprising: determining a rate-invariant charge / discharge ratio between an open circuit voltage (OCV) and a state of charge (SOC) for the battery cell, including: performing a voltage sampling at a first finite rate associated with the first state of dynamic equilibrium in which the sampled voltage follows a reduction branch of the relationship between the OCV and the SOC, and performing a voltage sampling at a second finite rate associated with the second state of dynamic equilibrium in which a sampled voltage follows an oxidation leg of a relationship between the OCV and the SOC; determining a rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a period in which the sampled voltage moves between the reduction branch and the oxidation branch; the dynamic determination of an electrical potential for the battery cell; dynamically determining, via a controller, a current SOC state for the battery cell based on the electrical potential for the battery cell, the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, and the rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a reversal of the voltage sampling that occurs when the sampled voltage moves between the reduction and oxidation branches; and controlling the charging of the battery cell based on the current SOC state for the battery cell.

Description

INITIATIONLithium-based battery cells may be used to provide electricity to vehicles and electronic consumer devices.US 2013 / 0 271 089 A1 describes systems and methods for precisely characterizing thermodynamic and material-related properties of electrodes and electrochemical storage and conversion systems. The increased sensitivity of the described methods and systems, as well as measurement conditions reflecting thermodynamically stable electrode states, enables highly accurate measurements of thermodynamic parameters, including state functions such as Gibb's free energy, enthalpy and entropy of electrochemical reactions. This allows prediction of important performance characteristics of electrode materials and electrochemical systems, such as energy and power density, current rate, life time and state of health of an electrochemical cell. In addition, systems and methods for charging electrochemical cells are provided, for example depending on the health of the cell.SUMMARYA method and apparatus for determining a state of charge (SOC) and for charging a rechargeable battery cell including an electrode is described. The method includes determining a rate-invariant charge / discharge ratio between an open circuit voltage (OCV) and a state of charge (SOC) for the battery cell, including performing voltage sensing at a first finite rate associated with a dynamic equilibrium state within which a sensed voltage follows a reduction branch of a ratio between the OCV and the SOC, and performing voltage sensing at a second finite rate associated with a dynamic equilibrium state within which the sensed voltage follows an oxidation branch of a ratio between the OCV and the SOC. A rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell is determined during a period in which the sensed voltage moves between the reduction and oxidation branches. An electrical potential for the battery cell is determined in a dynamic manner. A present SOC state for the battery cell is determined based on the electrical potential for the battery cell, the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, and the rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a reversal of the sensing of the voltage that occurs as the sensed voltage moves between the reduction and oxidation branches. The charging of the battery cell is controlled based on the present SOC state for the battery cell.One aspect of the disclosure includes the electrical potential for the battery cell (U) relative to the SOC for the battery cell (x) indicated according to the following ratio:The rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during the reversal of the sampling of the voltage is determined according to the following ratio:The rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell during the reversal of voltage sensing is determined according to the following ratio:The terms include α representing a time-unit fitting factor and associated with a load function χ representing the SOC; and t representing time.Another aspect of the disclosure includes a change in the load function ξ with time indicated according to the following ratio: andThe terms include α, which represents an adjustment factor relative to the stress function K, which is a constant, χ, which represents a boundary between rate invariant and rate dependent behavior, χ, which represents the stress, x, which represents the SOC, and t, which represents time.Another aspect of the disclosure includes the rate-invariant charge / discharge ratio including a hysteresis ratio between the OCV and the SOC.Another aspect of the disclosure includes determining the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, including determining a mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, and generating executable algorithm code that mathematically represents the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell.Another aspect of the disclosure includes the mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell that is a common differential equation, wherein generating the executable algorithmic code representing the mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell includes executing, via the controller, a numerical method to determine a numerical solution for the common differential equation.Another aspect of the disclosure includes voltage sampling at a first finite rate, which is rate-invariant sampling.Another aspect of the disclosure includes voltage sensing at a first finite rate, the rate being -0.01 mV / s.Another aspect of the disclosure includes voltage sampling at a second finite rate, which is rate-invariant sampling.Another aspect of the disclosure includes voltage sensing at a second finite rate, the rate being 0.01 mV / s.Another aspect of the disclosure includes voltage sensing at a second finite rate associated with the dynamic equilibrium state in which the sensed voltage follows the oxidation branch of the ratio between the OCV and the SOC associated with lithiating the electrode of the battery cell.Another aspect of the disclosure includes voltage sensing at a first finite rate associated with the dynamic equilibrium state in which the sensed voltage follows the reduction branch of the ratio between the OCV and the SOC associated with delithiating the electrode of the battery cell.The foregoing features and advantages, as well as other features and advantages of the present teachings, will become apparent from the following detailed description of some of the best modes and other ways of carrying out the present teachings with reference to the accompanying drawings.BRIEF DESCRIPTION OF THE DRAWINGSOne or more embodiments will now be described by way of example with reference to the accompanying drawings, in which: FIG. 1 schematically illustrates information regarding a charge / discharge ratio between the OCV and the SOC for a rechargeable battery cell, including a beginning and subsequent breakthrough of rate-invariant behavior as a sweep rate or current, in accordance with the disclosure; FIG. 2 graphically shows voltammetry data associated with a rechargeable battery cell, the data being acquired at 0.01 mV / s, starting at 1.5 V, and cycling until 0.16 V, at which point the current is interrupted for a period of 24 hours; after which the cycling rate is inversely but held at the same sampling rate until it again reaches 1.5 V, in accordance with the disclosure; FIG. 3( a) graphically illustrates voltammetry data associated with a rechargeable battery cell, the data including maintaining the voltage and a 12-hour period under open circuit voltage at 0.01 mV / s, in accordance with this disclosure; FIG. 3( b) graphically shows data associated with a rechargeable battery cell, the data including a portion of the open circuit voltage shown with reference to FIG. 3( a) that has been increased to show the increase in potential beginning at about 63 hours, in accordance with the disclosure; FIG. 4( a) graphically shows data associated with a rechargeable battery cell, the data including voltage vs. capacitance plots corresponding to the data shown in FIGS. 2, 3( a), and 3( b), in accordance with the disclosure; FIG. 4( b) graphically shows data associated with a rechargeable battery cell, the data including both a portion of the open circuit periods shown in FIG. 4 that have been increased and the multiple hold of the voltage, in accordance with the disclosure; FIG. 5( a) graphically illustrates data associated with a rechargeable battery cell, the data including plots of voltage and current versus time for three different cycles in accordance with the disclosure; FIG. 5( b) graphically shows data associated with a rechargeable battery cell, the data including the corresponding data plotted as voltage vs. SOC in accordance with the disclosure; FIG. 6 graphically illustrates a comparison of the sign(y) and tanh(αy) functions in accordance with the disclosure; FIG. 7 shows a schematic diagram of the rate constant k as a function of rate, in accordance with the disclosure; FIG. 8 shows data associated with a rechargeable battery cell, the data including a model comparison with the data shown in FIG. 2 using the values α=30 hr. and K=25, in accordance with the disclosure; and FIG. 9 shows data associated with a rechargeable battery cell, the data including model simulations of a current cycle at C / 40 and linear voltammetry at 0.01 mV / s using the same values α=30 hr. and K=25 as in FIG. 8, in accordance with the disclosure.DETAILED DESCRIPTIONThe components of the disclosed embodiments described and illustrated herein may be arranged and constructed in a variety of different configurations. Therefore, the following detailed description of the embodiments is not intended to limit the scope of the disclosure as claimed, but is merely representative of possible embodiments thereof. Moreover, although numerous specific details are disclosed in the following description in order to provide a thorough understanding of the embodiments disclosed herein, some embodiments may be practiced without some or all of these details. Moreover, for clarity, certain technical material known in the art has not been described in detail to avoid unnecessarily obscuring the disclosure. Moreover, as illustrated and described herein, the disclosure may be practiced in the absence of an element not expressly disclosed herein.The drawings relate to a direct current device 15 in communication with a controller 12. The direct current device 15 can be used to supply a vehicle with direct electric current. This includes use to supply current to an electric motor that provides mechanical power to the traction power. The vehicle may include, but is not limited to, a mobile platform in the form of a commercial vehicle, industrial vehicle, agricultural vehicle, passenger car, aircraft, watercraft, train, off-road vehicle, personal movement device, robot, and the like, to accomplish the purposes of the present disclosure. Alternatively, the DC device 15 may be deployed in any suitable device that uses electrical power to accomplish a task, including, but not limited to, consumer electronic devices such as cellular phones, portable computing devices, etc. The terms "DC device", "battery", "battery cell", and "cell" may be used interchangeably throughout the specification. In an embodiment, the DC device is configured with silicone electrodes that cause an electric charge by migration of lithium. The term "curve" is used herein to describe a relationship between identified parameters, the curve also indicating a graphical representation of the relationship between the identified parameters.Hysteresis behavior can be observed in lithiated silicon electrodes by the development and implementation of a model that describes the major characteristics of the observed voltage hysteresis and is capable of being reduced to an executable algorithm. In particular, slow scan voltammetry at 0.01 mV / s can be used to study hysteresis in lithiated silicon thin film electrodes. At higher sampling rates, the curves U(x) where U is the voltage and x is the state of charge depend on the sampling rate, but no difference of the curves can be noted if sampling is at sufficiently slow rates. For example, no difference is found at 0.01 mV / s or at 0.005 mV / s, although the same hysteresis behavior is observed for both sampling rates; i.e., the lithiating curve deviates substantially from the delithiating curve. The rate of 0.005 mV / s is the slowest rate that could be measured in prior art devices. However, if a sample of lithiating at 0.01 mV / s is suddenly set to open-circuit conditions, the voltage slowly moves upward, and the voltage slowly moves downward when delithiating is discontinued. It can be seen from this that the rate invariance observed at 0.005 and 0.01 mV / s would have to be broken up at a lower sampling rate if this could be measured. A semi-empirical model is used to describe many aspects of this behavior based on the assumption of a range of rate invariance that collapses at higher sampling rates due to transport and kinetic losses, but that collapses even at slower sampling rates due to other transients that can be observed.Silicon may be used as an anode electrode material in a lithium-based battery. Examples of lithium-based batteries include, by way of non-limiting examples, Li-Si, LiPO 4, Li-SiO x and others. When silicon is used as an anode electrode material in a lithium-based battery, hysteresis may occur in open-circuit voltage (OCV) when it is recorded with respect to a state of charge (SOC) of a lithiated silicon electrode depending on whether in the charging mode or the discharging mode.When an electrode material does not have hysteresis, the OCV of a partially lithiated host material with respect to a lithium reference electrode depends on the thermodynamic variables of the host material, including the SOC or molar fraction of the available sites occupied by lithium x, stress, temperature, and possibly other intense variables. In contrast, the OCV of the material exhibiting hysteresis depends on whether the material is lithiated, i.e., charged or delithiated, i.e., discharged. Since OCV should be considered a measurement taken at thermodynamic equilibrium, the existence of hysteresis poses difficult questions regarding the interpretation of such measurements. Hysteresis arises in many different contexts, e.g., in proportion describing magnetization versus applied field strength for ferromagnetic materials and force-displacement curves for different structural and semi-structural materials, and much work has been invested to understand the causes of hysteresis and its ratio to thermodynamic equilibrium. Regardless of the type of physical system contemplated, hysteresis phenomena are an indication that the system is in conditions far from thermodynamic equilibrium for long periods of time. For this reason, the observed behavior should in principle be interpreted as being within the scope of a thermodynamic imbalance. As a result, simplified approximate models based on reasonable phenomenonological assumptions may be used to interpret and employ the hysteresis phenomena.Data associated with slow scan voltammetry data taken from thin silicon films can be described as follows. Hysteresis can be attributed to the coexistence of regions of different states to generate time-independent hysteresis control loops. A Kranach model can be used to describe the hysteresis phenomenon associated with the open circuit voltage of a nickel hydroxide electrode and associated with the open circuit voltage of a lithium iron phosphate electrode. These represent true rate invariant approaches; in particular, slower rates (lower currents or slower potential sampling rates) do not remove or alter hysteresis from the model results. This is a fact that may appear anomalous in the context of thermodynamics. A rate-invariant term may be established for nickel metal hydride battery systems to which self-discharge has been added. The same model may be developed for characterizing a plurality of lithium ion cells, and a close variant of the same model may be applied to the lithium-silicon electrode operating in a constant current mode. While the comparison with the Li-Si experiment data failed well, the model may not be able to sufficiently describe the sample data. As described herein, hysteresis may occur in the charging and discharging electrodes manufactured by using Li-Si (including lithium silicates).The causes of the OCV hysteresis may have a phenomenological character in the case of Li-Si. The concept of rate invariance is somewhat more complicated than the description found above and can be generalized to describe the types of data observed. This can be best illustrated within the context of the specific data as follows.The slow sample voltammetry at rates of 0.01 mV / s and 0.005 mV / s was compared for silicon films in the voltage range of 1.5 V to 0.16 V and it was found that the same path of voltage U vs. state of charge x was followed at each sample rate. Further, the same voltage vs. lithium concentration diagram was observed at voltage sweep rates of 0.01 mV / s and a constant current sweep of C / 40, where the term C / 40 is a C rate, which is a term used to describe the cell state of charge, where the battery is charged from 0% SOC to 100% SOC over a period of time measured in hours, e.g., 40 hours in this example. This condition may be referred to as a "dynamic equilibrium" and a formula relating current density to the sweep rate may be derived under the same conditions. The 0.005 mV / s sweep rate represents the lower limit of what can be accurately measured using prior art instruments. At higher flow rates, the plot of voltage vs lithium concentration begins to deviate from that at dynamic equilibrium, presumably by voltage losses from kinetics, solid phase diffusion and other rate-dependent irreversible processes. If the thin films were fully balanced at these flow rates, it would be assumed that the voltage vs lithium concentration plot would remain unchanged at flow rates less than 0.005 mV / s if the data could be measured at such slow rates, but the data to be presented here indicate that this is not the case. Rather, it seems to be a range of rate invariance where the voltage vs. SOC ratio is independent of the voltage sweep rate or current, although other transient effects at slower rates may be observed, and therefore the voltage vs. SOC ratio changes. The most explicit indication of this is the decay of the voltage that occurs when the voltage sweep is suddenly interrupted and maintained under open circuit voltage conditions, as illustrated with reference to FIG. 2. Figure 2 graphically shows voltammetry data acquired at 0.01 mV / s, starting at 1.5 V, and running through to 0.16 V, at which point the current is interrupted for a period of 24 hours. The sweep rate is then reversed, but held at the same sampling rate until 1.5 V is reached again.At currents anywhere between what is seen in the voltammogram at 0.005 mV / s and idle, the rate invariance is broken. If this is not the case, then the open loop voltage would remain constant since the SOC is no longer subject to any change. A schematic representation of this situation is shown in Figure 1, which includes a schematic diagram illustrating the onset and subsequent breakthrough of rate invariant behavior as the sweep rate or current varies. The truly reversible behaviour referred to above can be occluded by secondary reactions. FIG. 1 is used to map the effects of progressively increasing charging rates associated with the DC device 15. When a rechargeable battery is exposed to open circuit conditions for a longer period of time, e.g., for more than 24 hours (10), reversible behavior occurs without hysteresis. When the rechargeable battery is subjected to charging conditions at a low rate, e.g., less than 0.005 mv / s (20), the rechargeable battery exhibits rate dependent behavior with detection of hysteresis. When the rechargeable battery is subject to moderate charging conditions, e.g., between 0.005 and 0.01 mV / s (30), the rechargeable battery exhibits rate-invariant behavior with detection of hysteresis. When the rechargeable battery is subject to high charging conditions, e.g., greater than 0.05 mV / s (40), the rechargeable battery exhibits rate-dependent behavior associated with kinetics, diffusion, or other irreversible processes.The voltammetry data presented here correspond to a sampling rate of 0.01 mV / s in the region of the rate invariance. The results are then plotted in Figures 4 and 5 as voltage vs SOC and the results are classified as follows. FIGS. 4 and 5 graphically show plot data as voltage versus SOC, where the voltammetry data corresponds to a sampling rate of 0.01 mV / s in the region of the rate invariance, with the sampling rate being immediately reversed at the end of each sweep. During a voltage scan at -0.01 mV / s, a dynamic equilibrium state is finally reached by the voltage following the lithiating branch of the OCV vs SOC curves. The behavior along this path is rate invariant. When the sampling is reversed so that the voltage is increased at 0.01 mV / s, a dynamic equilibrium state is finally reached in which the voltage follows the delithiation branch of the OCV versus SOC curves. The behavior along this path is again rate invariant. Just after reversing the sampling of the voltage, a transition period takes place in which the voltage moves between the above-described lithiating and delithiating branches.The plots of voltage vs vs SOC in the transition period have a different shape depending on whether linear sweep voltammetry or constant current sweep is used, as can be seen in FIG. 9. During the constant current sweep, the voltage direction switches at the same time that the current direction switches, and during linear voltammetry after reversing the sampling of the voltage, a time delay occurs before the current changes sign (see, for example, FIG. 9 below). It is understood that at least one of the above situations, if not both, must break through the rate invariance.An empirical approach is applied to the model to develop a modeling strategy that can be used to detect not only rate-invariant behavior, but also transition regions. The result is a model that describes the rate-invariant behavior and also predicts the manner in which this behavior develops into rate-dependent behavior at very low currents or very slow voltage sampling rates. Very small currents or very slow voltage sampling rates refer to time scales of practical interest. For example, in applications involving electrified vehicles and consumer electronics, the battery charge and discharge times last longer than 40 hours, i.e., C / 40 are not of interest.Experimental data were collected using thin silicon films which were applied to copper current collectors in a high-frequency magnetron sputtering system. The deposition was carried out without heating the substrate to obtain amorphous Si. The film thickness was monitored by a quartz crystal microbalance and joined with the thin silicon film as the working electrode and a lithium film as the counter reference electrode using a microporous membrane as a separator, and the electrolyte was a mixture of ethylene carbonate and diethylene carbonate (1:1 volume ratio) with 1 mLiPF 6.FIG. 2 graphically shows data plots of cell voltage 210, capacitance 220, and current 230 versus time 240 during clocking with scales of cell voltage 201, capacitance 202, and current 203 indicated on the vertical axes. The cell begins at time t 0 242 at a voltage of 1.5 volts, goes down to a voltage of 160 mV at a sampling rate of -0.01 mV / s, as indicated at time t 1 244. At this time, the cell was switched to the open circuit state and allowed to sleep for 24 hours while the voltage change was detected. After 24 hours, as indicated at time t2 246, the sampling of the voltage was reversed and clocked back to 1.5 V at a rate of 0.01 mV / s at time t3 248, at which the voltage 210 was held constant. During the 24 hour rest time between times t 1 242 and t 2 244, cell voltage 210 is observed to decrease, although a equilibrium point may not have been reached at the end of the time period.FIGS. 3( a) and 3( b) graphically show data of a similar type to that of the data shown with reference to FIG. 2, including data plots of cell voltage 310, capacitance 320, and current 330 versus time 340 during a clock with scales of cell voltage 301, capacitance 302, and current 303 indicated on the vertical axes. FIG. 3(a) graphically shows voltammetry data at 0.01 mV / s including voltage hold and a 12-hour open circuit duration. FIG. 3(b) graphically shows a portion of the open circuit period that is increased to show the rise in potential, presumably due to secondary reactions, beginning at about 63 hours. Beginning at time 342, the cell is clocked down from 1.5V to 0.1V at a sampling rate of -0.01 mV / s, and at this time the voltage is held at 0.1V for a period of 10 hours. The voltage then leads the scan back to 371 mV at 0.01 mV / s, at which time, as indicated at time 344, a 24 hour period starts for open circuit conditions. Finally, at time 346, the voltage returns to 0.1 V at -0.01 mV / s and is maintained at 0.1 V. Figure 3(b) shows data associated with the open circuit area that has been increased; the voltage first decays and then begins to slowly increase at about 63 hours. The voltage rise after 63 hours appears to be the result of secondary reactions occurring in the electrolyte. These secondary responses limit how precise a voltage decay can be measured and it is therefore difficult to evaluate where the true equilibrium point of voltage decay would occur without such responses even if the cell is open-circuited for arbitrarily long periods of time. For purposes of modeling, it is assumed in the next section that this equilibrium point would occur midway between the rate-invariant lithiating and delithiating OCV curve: where x is the state of charge, U represents the voltage, U 1( x) is the rate-invariant delithiating curve, and U 0( x) is the rate-invariant lithiating curve. This assumption seems to agree with the available data, but cannot be checked for higher accuracy due to the intrusion of the secondary reactions.FIGS. 4(a) and 4(b) graphically show data plots of cell voltage 420 versus capacitance 410 during clocking. FIG. 4( a) graphically shows voltage diagrams (V) 420 vs. capacitance (C) 410 corresponding to the data shown in FIGS. 2, 3( a), and 3( b). After initially maintaining the voltage at 0.1 V, when the sampling is resumed at 0.01 mV, a transition region is created according to the introduction before the voltage reaches the rate-invariant curve, U 1( x) associated with delithiation. When the voltage reaches 371 mV, the current is set at 0 and the voltage returns to 320 mV. After the sampling is resumed at -0.01 mV, a transition phase again occurs, but this time on the rate-invariant lithiating curve U 0( x). FIG. 4(b) graphically shows a portion of the open circuit periods as well as the holding of the voltage that is increased between voltage levels 421 and 422.FIG. 5(a) graphically shows voltages 512, 514, and 516 on V scale 502 and corresponding currents 522, 524, and 526 on I scale 504, plot vs. time on T scale 510, for the first, second, and third cycles, respectively. Each cycle begins at 1.5 V vs. Li and lithiates down at 0.01 mV / s to a minimum voltage of 0.267 V for the first cycle, a minimum voltage of 0.16 V for the second cycle, and a minimum voltage of 0.066 V for the third cycle. When the minimum voltage is reached, in each of the three cycles, sampling is immediately reversed and continued in the direction of delithiation until the original voltage of 1.5 V is reached again. Holding the voltage for 4 hours separated each cycle. FIG. 5(b) depicts the corresponding data plotted as voltage on the V scale 502 relative to SOC on the SOC scale 508. A detailed test of Fig. 5(a) shows that there is no discontinuity in the measured currents even immediately after the reversal of the sampling. Instead, there is a short period of time in which the current steeply rises from its original negative values, before reversing the sampling, to the positive values seen on the rate-invariant delithiation curve U 1( x). As a result, the SOC continues to increase for a short period of time after the sampling is reversed until the current changes sign between a positive value and a negative value. A second consequence of this is that during this period the current passes through the zero value where it has already been observed that the rate invariance is broken. If the rate invariance were to be held for the entire period of time that the current is negative, then the curves of Figure 5(b) would remain on the low voltage curve U 0( x) that they followed before the reversal of the sampling, although this is not what actually occurs. Instead, "overshoot" occurring are shown in which the SOC continues to increase, although the voltage also increases. This is again an indication of the collapse of the rate invariance after the reversal of the sampling. The type of overshoot observed after the reversal of the sampling in linear voltammetry does not occur when the clocking is performed at constant current. In the latter case, the current is discontinuous while changing sign and the voltage and current simultaneously reverse direction.A method associated with rate-invariant modeling will now be described. A timing method can be defined as rate invariant if the path taken in the x-U plane, where U is the voltage, depends only on the direction followed along the path, but not on the speed at which it is traversed. A large class of rate-invariant processes can be described as the function U(x), which function can change depending on the sign, but not depending on the magnitude of it. The specification of the value range of the values of within the framework of which the holding of rate invariances can be determined.The primary examples of rate-invariant functions are lithiating and delithiating OCVs that are measured at very slow sampling rates, either by voltammetry or clocking at a slow or constant C-rate, while the cell is clocked from zero lithium content to full lithium content and back. The measurement of the OCV may begin with a cell that is initially equalized at constant voltage before the start of the sampling in any direction, and it should be noted that the rate invariance of the sampling may occur at an unspecified time after the start of the sampling, which time may be needed to achieve a dynamic equilibrium state. Preferably, rate-invariant OCV has been measured in lithiating and delithiating, and each OCV can be described as a rate-invariant function U(x).Since the currents passed through linear voltammetry vary from those at which the clocking occurs at a constant C-rate, the fact that the two different methods track the same OCV curves is an example of rate invariance. The function U 0( x) can be adapted to lithiating OCV and U 1( x) to delithiating OCV. The resulting adjusted parameter values are given in Table 1 as follows: Table 1 Reduction (Lithiating) Table 1 Reduction (lithiating)10, 058910, 357341,1 15220, 222440, 237581,1 69130, 229730, 273702,9 95840, 413030, 131376,8 045Oxidation (delithiation)Oxidation (delithiation)10, 301920, 330911,4 09020, 481070, 277661,7 04930, 653020, 121812,7 76440, 961320, 119167,6 463Parameters are used to represent the functions U 0( x) and U 1( x). These formulae given above describe the reversal functions x(U 0) and x(U 1); must then be reversed numerically in order to obtain the functions U i( x), i=0.1.The modeling describes the transition regions between these two curves that occur when the sampling rate of the current or voltage changes direction.The definition of U avg( x) given in this last paragraph can be improved by the additionally defined variables U max( x) and ξ:The term ξ is a dimensionless stress term saturated at ±1 when the potential U is on either the rate-invariant branch of delithiation U 1( x) or lithiating U 0( x).If ξ = -1, the path U 0( x) is traversed and if ξ = 1, the path U 1( x) is traversed; for the intermediate values of ξ, the voltage U lies somewhere between the upper and lower voltage curves U 1( x) and U 0( x).To continue, the function may be defined as follows.Note that this function may be defined such that sign(0)=0, but the above definition satisfies the following equation:The starting point is the rate-invariant model as follows:The constant K is positive, so that when (lithiating, reducing), the solutions to equations (2) and (5) tend to be ξ = -1 and U = U 0( x), and when (delithiating, oxidizing), they tend to be ξ = 1 and U = U 1( x).Equation (5) can be written as differential equations as follows:The rate-invariant model includes equations (2) and (6), or equivalently equations (2) and (5).It turns out that the derivatives always have opposite signs during the rate-invariant clocking. To see this, equations (2) and (5) reveal the following:The above inequality remains because the function always has negative values and both U max and K are positive. From equation (7) it can be seen that the following ratio is created during the rate-invariant clocking:The rate invariance breakthrough can be described as follows. The data shows that the rate invariance must break at a low value of. This is apparent from linear voltammetry plots that exhibit "overshoot" during sample reversal. The overshoot means that for a short time after the scan reversal, it remains positive as it becomes positive, and this violates equation (8). When moving along the lower voltage curve U 0( x), it is evident from the rate invariance that the stay on the lower voltage curve must take place as long as although the voltage leaves the lower voltage curve during the crossing.The breakthrough of rate invariance into equation (6) can be incorporated in two different ways. First, the function is replaced with the function, and the value of α needs to be selected to match the measurement data. The model now has two independent parameters, K (which is dimensionless) and α (with time dimension). FIG. 6 graphically shows a comparison of the functions sign(y) 602 indicated on axis 601 and tanh(αy) indicated on axis 603 and including tanh(y) 604, tanh(y) 606, and tanh(y) 608; it is determined that α becomes large, the two functions approach each other according to the above equation (4). The shape shown as the argument of the tanh function recognizes the fact that the OCVs have a logarithmic singularity such as x→0. Consequently, linear samples of the voltage at very small values of x have a much smaller value of x than the same sample of the voltage at larger values of x, and the tanh argument approximates to maintaining the same extent at both large and small values of x due to the factor 1 / x. In this variation, the following relationships are used:The tanh function value never reaches an absolute value; it only approximates one for sufficiently large values of Es is therefore convenient to select a more or less arbitrary threshold as a limit for rate invariance. As such, the equation (9) can be changed to the following form:The value of the adjustment factor α may be selected to determine the threshold value of the rate invariance, where large values of α result in rate invariance at small values of α, and to the extent that the adjustment factor α decreases, the breakthrough of the rate invariance increases to higher values of α.The second change to equation (6) is to set a lower limit to the factor appearing on the right side. The modified embodiment of equation (6) is now as follows:In summary, equations (11) and (12) approach equation (6) strongly if because and in this range the rate invariance remains, but equation (6) becomes rate dependent, in accordance with equations (10) and (12). FIG. 7 shows a schematic diagram of the rate constant k 701 indicated on the rate constant axis 702 depending on indicated on the axis 704. The boundary of is indicated at point 703.The changes in equation (11) to break through the rate invariance are not absolutely arbitrary. The rate-invariant form of equation (11) cannot be reverted to arbitrarily small values of Q, as this would predict that ξ is, therefore, not degraded when the cell is in open circuit. In contrast, FIG. 2 shows a voltage drop in an open circuit. It follows that the coefficient which is part of equation (12) is to be replaced with a value other than zero at zero current. In the rate-dependent form of equation (12), the rate constant k=K χ is constant over the entire range of rate dependence.The function represents the boundary between rate invariant and rate dependent behavior, and this boundary is independent of its sign. However, during linear voltammetry when specified, the boundary between rate invariant and rate dependent behavior becomes directional at once. For example, assuming that the cell is lithiated in a rate-invariant manner and follows the lower voltage curve, where ξ = -1. Then differentiating equation (2) results in the following binding of the rate invariance:However, when the reversal of scanning takes place, the binding changes in terms of that from χ to -χ andAnd this is the boundary in the delithiation direction. It follows that reversing the sampling of the voltage under these circumstances can lead to a rate-dependent behavior even if it was rate-invariant until reversing the sampling. This fact is of substantial significance to understanding the overshoot observed during voltammetry in reversing the scan. Numerical solutions for equations (2) and (11) also illustrate the overshoot in the reversal of the scan during voltammetry.Equations (2) and (11) must be solved for either the voltage or the timing of the current. The timing of the current is known and it is straightforward to introduce this value into equation (11) in order to determine. The time derivative of equation (2) then yields the following:Equation (15) then determines the derivative of the voltage. Conversely, the derivative in voltammetry is known and the derivative must be determined. It can be shown that the solutions exist and are unique provided that it holds based on equations (2) and (11). They can be solved numerically by solving equation (11) for using the first of equations (12). If the response is of magnitude greater than or equal to χ, then this is the desired solution. Otherwise, the second of equations (12) must be used to solve. This response will then be of the order of magnitude less than χ and is the desired response.Model simulations were performed that involve fitting the model parameters to data, which includes values not only for and α, but also for lithiated and delithiated OCVs, U 0( x) and U 1( x). These characteristics vary somewhat from cell construction to cell construction and it is important to perform the matching based on data of the same cell. As mentioned previously, the methods are used to adjust U 0( x) and U 1( x), and the adjusted values for the data presented in this section are given in Table 1. Figure 8 shows a model comparison with the same ones in Figure 2, with values using the values α = 30 hr and K = 25.The data plots of cell voltage 210, capacitance 220, and current 230 are shown versus time 240 and corresponding modeled cell voltage 810, modeled capacitance 820, and modeled current 230. The assumption for the model simulation is that at the beginning of the sampling, 0( = -1 and U x) = 1.5. After the start of the scan, χ follows the lithiating OCV course until the start of the 24 hour idle period. After the open idle period ends, ξ decreases to a value very close to zero, therefore it takes some time to reach its maximum value, at which time the course follows the lithiated OCV. The actual reduction in open circuit is only approximately followed by the model, as the simplest linear locking law was used in the rate dependent form of equation (11). FIG. 9 shows model simulations of a current cycle at C / 40 910 and a linear voltammetry 920 at 0.01 mV / s plotted as potential U on the vertical axis 902 relative to a fraction of the maximum SOC on the horizontal axis 904 using the sameValues α = 30 hr. and K = 25 as indicated in Fig. 8. Each cycle begins at 1.5V and decreases to a state of charge of approximately x=0.79 and is then reversed. Most of the two scans are identical and follow the rate invariant paths U 0( x) and U 1( x); the only differences are in the transition region after the reversal of the scan where voltammetry overshoot was observed but no overshoot was detected in the C / 40 scan.The values for K and α are most important in transition range model simulations. These values are not likely to remain the same for reversals of sampling at different SOCs. Figure 5(b) compares model simulations with the data for the displayed three different samples. The reversals of the sampling take place at the voltages 0.066, 0.16 and 0.267 V. In order to allow the values of K and α to differ from each other for each of these cases, polynomial expressions for both α and K depending on x and ij of the following form are used: and the matrix of the values α ij and K agrees with the data in Fig. 5(b). Table 2 gives the coefficients that match these polynomial expressions in one embodiment.Thin silicon films were clocked at 0.01 mV / s and the behavior of the ratio between the voltage U and the state of charge (SOC) x was examined in the context of the clocking, during the holding of the voltage or open circuit conditions immediately after the reversal of the sampling. At these very low sampling rates, depending on the SOC, the voltage follows different paths in the x-U plane depending on whether the cell lithifies or delithifies, although there is a range of rate invariance, so that the clocking at C / 40 of a constant current, for example, follows the same path in the x-U plane that it follows during linear voltammetry at 0.01 mV / s, although the currents at C / 40 are different from those indicated during linear voltammetry. The simulations shown in FIG. 9 illustrate that the rate invariance breaks down in the transition region just after the reversal of the sampling if the path followed during voltammetry is different from the path followed when a constant current was clocked. Moreover, as shown in FIGS. 2-4, it is noted that the open circuit voltage drops and the SOC exhibits a transient decrease during the holding of the voltage in an apparent steady state, although it may prove difficult to accurately determine what these steady state conditions are due to the apparent secondary reactions occurring in the electrolyte. It can be concluded that the rate invariance during the agitation only occurs over a fixed range of flows; with larger flows and faster agitation rates irreversible processes attributable to diffusion or kinetic effects become noticeable, with transient effects occurring with smaller currents. The junctions associated with very low currents can be related to slow stress relaxation and creep of the active material. The latter effects are difficult to analyze because they occur at clock rates that are below values that can be achieved with the existing measurement equipment. FIG. 1 summarizes the various types of behavior that may occur at different clock rates in thin silicon films.As described herein, an empirical approach starting with a simple rate-invariant behavior model may be applied to the data model. At a starting point in the x-U, the model equations (2) and (5) predict the path U(x) followed only depending on the direction of the current but not its magnitude. When the cell lithiates, all such paths tend to a single rate-invariant path of lithiating U 0( x), and the path followed during delithiating always tends to a single rate-invariant delithiating path U 1( x).For smaller currents outside the range of rate invariance, the equations are changed as indicated in equations (2) and (11). Since direct measurements are not readily attainable at clock rates below the rate invariance, Equation (5) through Equation (11) can be generalized. In particular, equation (11) is able to reproduce the open circuit voltage drop in a qualitative manner that is not predicted by the rate-invariant analog system described in equation (5).FIG. 5 shows that the current reverses some time after the voltage is reversed when the clock is at 0.01 mV / s, thereby producing transients visible on the x-U plane. This behavior violates the rate invariance and equation (11) predicts such behavior when the model parameters meet certain inequality. The simulations shown in FIG. 5 illustrate this behavior. For both the model associated voltage 514 and the experiment associated voltage 512, these overshoot is not detected when the clocking is at a constant current.Equation (11) predicts that crossings should decrease after reversing sampling because the sampling rate increases and the crossings should disappear completely at sampling rates that are high enough. Such results can be masked by effects of interfacial kinetics (charge transfer) and solid phase diffusion resistance before they are detectable, although it is hoped that smaller crossings at higher sampling rates are still detectable in the range of rate invariance before the above effects become noticeable.Hysteresis behavior between the potential U or OCV that can be dynamically measured and a battery state such as the SOC can be characterized as shown with reference to FIGS. 4( a) and 4( b). This involves sampling a voltage of a cell that is slow but performed at a finite rate (e.g., -0.01 mV / s) to determine a state of dynamic equilibrium that is reached by the voltage following the reduction branch of the OCV vs. SOC plot associated with lithiating or charging. An example of such a result is shown in line portion 435 with reference to FIG. 4(b). The behavior along this path is rate invariant. When the sampling is reversed, a dynamic equilibrium state is ultimately achieved in which the voltage follows the oxidation branch of the OCV versus SOC plot associated with lithiating or charging. An example of such a result is shown in a line segment 425 with reference to FIG. 4( b). The behavior along this path is again rate invariant. Just after reversing the sampling of the voltage, a rate-dependent transition period takes place in which the voltage moves between the reduction and oxidation branches described above. An example of such a result is shown in line section 430 with reference to Figure 4(a).The characterization of the hysteresis ratio between the potential U or OCV and the battery state of the SOC may be expressed as a ratio in the form of a detachable mathematical expression, e.g., equation (2), above, and may be repeated herein:If ξ = -1, the path U 0( x) is traversed and if ξ = 1, the path U 1( x) is traversed; for the intermediate values of ξ, the voltage U lies somewhere between the upper and lower voltage curves U 1( x) and U 0( x).The voltage U reflecting the desired OCV value may determine the SOC. The expression of the equation (2) takes the form of an ordinary differential equation which can be reduced to an algorithm code and dynamically solved to obtain the value of using a numerical method such as an Euler method, a Runge-Kutta method, a linear multistage method or another suitable method. When the value ξ is determined, the value for U avg( x) and other terms may be determined, such as routing to determine the voltage U that may be used to determine the SOC. The concepts described herein may be reduced to be practiced as algorithm code and calibrations stored in non-volatile memory and executed in a controller to determine various parameters that may be associated with operating conditions of the battery using lithium.The concepts described herein may be applied to any hysteresis model applied to any material that conforms to the classification scheme described as follows. During a slow sampling of the voltage, but which takes place at a finite rate (e.g. -0.01 mV / s), a state of dynamic equilibrium is reached in which the voltage follows the reduction branch of the OCV vs. SOC curves. The behavior along this path is rate invariant. When the sampling is reversed, a dynamic equilibrium state is ultimately achieved in which the voltage follows the oxidation branch of the OCV versus SOC plot. The behavior along this path is again rate invariant. Just after reversing the sampling of the voltage, a rate dependent transition period occurs in which the voltage moves between the reduction and oxidation branches as described above.The term "controller" and other related terms, such as control module, module, controller, controller, processor, and the like, refer to one or more combinations of application specific integrated circuit(s) (ASIC), electronic circuit(s), central processing unit(s) such as microprocessor(s) and non-transitory memory component(s) associated therewith in the form of memory and storage devices (read only memory, programmable read only memory, random access, hard drive, etc.). The non-transitory memory component is capable of storing machine readable instructions in the form of one or more software or firmware programs or routines, combinatorial logic circuit(s), input / output circuit(s) and devices, signal conditioning and buffer circuits, and other components that can be accessed by one or more processors to provide described functionality. Input / output circuit(s) and devices include analog / digital converter-related devices that monitor sensor inputs at a predetermined polling frequency or in response to a triggering event. Software, firmware, programs, instructions, control routines, code, algorithms, and similar terms refer to controller executable instruction sets such as calibrations and look-up tables. Each controller executes control routine(s) to provide the desired functions. The routines may be executed at regular intervals, such as during ongoing operation every 100 microseconds. Alternatively, routines may be executed in response to a trigger event. Communication between the controllers and communication between controllers and actuators and / or sensors may be via a direct wire connection, a networked communication bus, a wireless connection, or another suitable communication connection. Communication involves the exchange of data signals in a suitable manner, including, for example, electrical signals via a conductive medium, electromagnetic signals through the air, optical signals via optical fibers, and the like. Data signals may include discrete, analog, or digitized analog signals representing inputs from sensors and actuator commands, as well as communication signals between controllers. The term "signal" refers to a physically perceptible display that conveys information and may include any suitable waveform (e.g., electrical, optical, magnetic, mechanical, or electromagnetic), such as DC, AC, sine waves, triangular wave, square wave, vibration, and the like, that may pass through a medium.The term "model" refers to processor-based or processor-executable code and associated calibration that simulates the physical existence of a device or process. As used herein, the term "dynamic" describes steps or processes that are performed in real-time and are characterized by monitoring or otherwise determining parameter states and periodically or periodically updating parameter states when executing a routine or between iterations when executing the routine. The terms "calibration," "calibration," and related terms refer to a result or method that compares an actual or standard measurement associated with a device to a perceived or observed measurement or a commanded position. Calibration described herein may be reduced to a storable parametric table, multiple executable equations, or any other suitable form. A parameter is defined as a measurable quantity that represents a physical property of a device or other element recognizable by one or more sensors and / or a physical model. A parameter may have a discrete value, e.g., "1" or "0", or may be continuously set.The equations may be reduced to executable algorithm code implemented by specific purpose hardware-based systems that perform specified functions or acts, or combinations of specific purpose hardware and computer instructions. These computer program instructions may also be stored in a computer readable medium that can control a computer or other programmable data processing apparatus to function in a particular manner such that the instructions stored in the computer readable medium produce an article of manufacture including instruction means that implement the function / act set forth in the flowchart and / or block diagram block or blocks. In addition, the present disclosure may take the form of a computer program product that may be deployed in a tangible medium of expression having program code usable for computers.However, while the detailed description and drawings support and describe the present teachings, the scope of the present teachings is defined solely by the claims. While some of the best modes and other ways of carrying out the present teachings have been described in detail, there are various alternative constructions and embodiments for practicing the present teachings defined in the appended claims.

Claims

A method of charging a battery cell of a DC device, wherein the battery cell includes an electrode, the method comprising: determining a rate-invariant charge / discharge ratio between an open circuit voltage (OCV) and a state of charge (SOC) for the battery cell, comprising: performing voltage sensing at a first finite rate associated with the first dynamic equilibrium state in which the sensed voltage follows a reduction branch of the ratio between the OCV and the SOC, and performing voltage sensing at a second finite rate associated with the second dynamic equilibrium state in which a sensed voltage follows an oxidation branch of a ratio between the OCV and the SOC; determining a rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a period in which the sensed voltage moves between the reduction branch and the oxidation branch; dynamically determining an electrical potential for the battery cell; dynamically determining, via control of a present SOC state for the battery cell, based on the electrical potential for the battery cell, the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, and the rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a reversal of sensing of the voltage that occurs when the sensed voltage moves between the reduction and the oxidation branches; and controlling, via the control, the charging of the battery cell based on the present SOC state for the battery cell.The method of claim 1, further comprising determining the electrical potential for the battery cell (U) relative to the SOC for the battery cell (x) according to the following ratio: U = U a v g ( x ) + U max ( x ) ς, - 1 ≤ ς ≤ 1 U a v g ( x ) = U 1 ( x ) + U 0 ( x ) 2, U m a x ( x ) = U 1 ( x ) - U 0 ( x ); 2 wherein the rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during the reversal of the sampling of the voltage is determined according to the following ratio: | tanh (α x d x d t) | < 0.97 and wherein the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell during the reversal of the sampling of the voltage is determined according to the following ratio: | tanh (α x d x d t) | ≥ 0.97 wherein: α represents an adaptation factor in relation to a load function x represents the SOC, and t represents time.The method of claim 2, further comprising indicating a change in the stress function ξ with time according to the following ratio: d χ d t = - k ( tanh ( α x d x d t ) + χ ) and k = { K | d x d t | for | d x d t | ≥ χ K χ for | d x d t | < χ where tanh ( α x χ ) = 0.97, and χ = x α 2.092... wherein: K is a constant, χ represents a boundary between rate invariant and rate dependent behavior, and χ represents a stress.The method of claim 1, wherein the rate-invariant charge / discharge ratio includes a hysteresis ratio between the OCV and the SOC.The method of claim 1, wherein the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell further comprises: determining a mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell; and generating executable algorithmic code representing the mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell.The method of claim 5, wherein the mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell comprises a common differential equation, wherein generating the executable algorithmic code representing the mathematical expression for the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell comprises executing a numerical method, via the controller, to determine a numerical solution for the common differential equation.The method of claim 1, wherein the voltage sampling at a first finite rate includes rate invariant sampling.The method of claim 7, wherein the voltage sampling at a first finite rate comprises a rate of -0.01 mV / s.The method of claim 1, wherein the voltage sampling at a second finite rate includes rate invariant sampling.An apparatus for determining a state of charge (SOC) of a rechargeable battery cell including a lithiated electrode having a hysteresis ratio between SOC and an open circuit voltage (OCV), comprising: a controller in communication with the battery cell, the controller including an instruction set, the instruction set executable to: determine a rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell including: performing voltage sensing at a first finite rate associated with the first dynamic equilibrium state in which the sensed voltage follows a reduction branch of the ratio between the OCV and the SOC, and performing voltage sensing at a second finite rate associated with the second dynamic equilibrium state, wherein a sensed voltage follows an oxidation branch of a ratio between the OCV and the SOC; determining a rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell during a period in which the sensed voltage moves between the reduction and the oxidation branches; dynamically determining an electrical potential for the battery cell; and dynamically determining, via a controller, a present SOC state for the battery cell based on the electrical potential for the battery cell, the rate-invariant charge / discharge ratio between the OCV and the SOC for the battery cell, and the rate-dependent charge / discharge ratio between the OCV and the SOC for the battery cell.

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