SECONDARY BATTERIES – LITHIUM RECHARGEABLE SYSTEMS – LITHIUM-ION | Lifetime Prediction

Lifetime Prediction S Santhanagopalan, Celgard LLC, Charlotte, NC, USA J Stockel, Quandary Solutions LLC, Derwood, MD, USA RE White, University of South Carolina, Columbia, SC, USA & 2009 Elsevier B.V. All rights reserved.

Introduction Design of lithium-ion cells is complicated by several factors such as inherent intolerance to abuse conditions such as overcharge and safety issues when the cells are exposed to excessive heat. Hence, a more effective approach toward designing a safe system is to incorporate a battery management system that can predict the performance of the cell. The prime function of such a device is to ascertain that the battery can support the stipulated application for the required period of time. With an increase in the volume and sensitivity of applications that the lithium battery system is used for, there is an increasing demand for accurate estimation of the life of the battery. The number of techniques used for predicting the performance of a battery has grown exponentially from Peukert’s equation to modern genetic algorithm– based techniques. Each technique claims its own advantages and suitability for a specific application. Design of an efficient power source for the increasingly complicated and ever-increasing power demand of today’s applications relies to a great extent on identifying the best-suited methodology that provides a reliable estimate of the life of the battery. Availability of a tool to predict the life or performance of a battery is highly advantageous and desirable to improve the design of cells for a particular application and for making specific maintenance plans during the course of the life battery’s life. The first step toward understanding how to design a life estimator for a lithium-ion battery is to define life. Several definitions prevail in the literature from the various regulations for life expectation of batteries. For example, the Freedom Cooperative Automotive Research (FreedomCAR) program describes the life expectation of a battery as follows: ‘The cycle life goals depend on the power-assist ratings – 240 000 cycles at 60% of rated power, plus 45 000 cycles at 80% of rated power, plus 15 000 cycles at 95% of rated power. A cycle consists of a power profile that includes the vehicle-operations of engine-off, launch, cruise, and regenerative braking.’ This set of three operating conditions corresponds to the 90th percentile of automotive customer requirements. Also cited as a primary objective is the requirement to verify or predict the performance of the pack within 90% confidence intervals. The Partnership for a New Generation of Vehicles (PNGV) postulates the end of life of a battery as that point when the battery reaches 23% of power fade

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in the application of interest, whereas the United States Advanced Battery Consortium (USABC) goals cite the requirements for the batteries as 1000 cycles at 80% depth of discharge (DoD) with a capacity loss no greater than 30 Wh kg1 within 10 years. The Medical Design magazine states that this requirement for implantable batteries is a mere 300 cycles when the end-of-charge voltage (EoCV) is 4.2 V and 400 cycles when the EoCV is 4.1 V. It is not surprising to find such vivid descriptions of the end of life of a cell because these definitions are application specific and some applications are more lenient in their tolerances compared with the others. Also, the cell designs used for each application are very different from the other. Inherently, some cell designs are safer than others and hence may have a more stringent life requirement. A few definitions of the battery end of life are illustrated in Figure 1. Although there exists no universal definition cell life, there are a few measures that are commonly used to characterize performance. These include the state-ofcharge (SoC), the state-of-health (SoH), and the state-offunction (SoF). The SoC corresponds to the stored charge available to do work relative to that which is available after the battery is fully charged. This parameter is popularly visualized as a thermodynamic quantity, enabling one to assess the potential energy of the system. State-of-health is a term that is becoming increasingly popular within the battery community, as a measure of a battery’s life, but as described above, the definition of the life of a cell is context-specific, and hence, SoH is not clearly defined in many cases. In general, SoH is a measure of how well the battery system is functioning relative to its nominal (rated) and end states (where the term ‘end state’ is defined as that point when the battery has failed one or more criteria formulated to describe its expected performance). Knowing the change in the SoH with time can be perceived as an assessment of the increase in irreversible losses owing to aging of the cell. The SoF is a more sophisticated term usually used to assess the performance of a battery, with specific constraints on the performance. The SoF is usually defined as a function in terms of the SoC and the SoH.

Choice of Models During the past two decades, investigators have proposed hundreds of models to simulate the performance of a lithium-ion cell. Some of these are comprehensive models

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction

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Figure 1 Different definitions for the end of life of a lithium-ion cell: (a) based on a specific number of charge/discharge cycles, (b) based on a specific period of time until which the cell can provide the required voltage, (c) based on the decay in the cell capacity, and (d) based on the rate capability of the cell after cycling.

that address a wide range of experimental observations – thermal effects, volume changes, and capacity fade to name a few. With an increase in the degree of complexity, the time and effort required to obtain results that are of practical significance becomes excessive. Several simplified models were also proposed. However, their scope is usually limited to specific data sets or operating conditions, and hence, these models cannot be used to develop a generalized protocol for making life predictions. In general, there are a few criteria that govern the suitability of a model for a particular application: i) The model must be a reasonable representation of the physical process. In other words, one or more of the experimentally observed phenomena must be explained based on a hypothesis. ii) The time required for solving the model must be practical. For example, if an estimation algorithm is expected to update the predictions every few seconds, the solution time for the model must be on the order of milliseconds. iii) Some models/techniques require specific types of data from a battery. However, it is not always possible to conduct experiments on a battery used in a practical application. For example, if the estimation technique relies on impedance measurements, the frequency-response data must be measurable from

the system with a reasonable addition of extra hardware. Hence, it is strongly advocated that the techniques used to make predictions be restricted, as much as possible, to data readily available from the system – e.g., the cell voltage or the skin temperature as a function of time. iv) The memory and computational requirements must be as small as possible. This criterion allows for building an onboard device to make predictions. The cost of circuitry involved in implementing such a device increases exponentially with the complexity of the model. v) The model must provide a tangible correlation between a parameter that can be measured and the life expectancy of the battery. In other words, a change in the cell performance (e.g., a decrease in the capacity) must be readily captured by the model in terms of a change in an observable parameter (e.g., the cell voltage or temperature).

Observability of a Model The term observability is frequently used in the literature as a primary concern while choosing a model for making predictions of the SoC or the SoH. In the broadest sense, observability refers to the set of criteria that determine whether or not the parameters can be

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obtained, from the experimental measurements available, from a system. For example, if the only set of data available from a battery for making SoC predictions is the voltage as a function of time, then the model must provide a relationship between the voltage of the system and its SoC. Also, if the experimental voltage remains fairly constant throughout a considerable portion of the discharge curve, the SoC predictions based only on this voltage will not be reliable. Hence, additional measurements are required for this case. Unobservability of a system can arise from one of the following three factors: i)

The perturbations caused by a change in the load may be so small that the system noise interferes with the response of the battery. Usually, this issue is addressed by the use of a suitable filter and/or by adjusting the frequency of data measurements. ii) The model used to characterize the response of the battery to a perturbation in the load may not be sensitive to the kind of load imposed by the system. For example, if the cell voltage is maintained constant for a major part of the charge or discharge process during the actual application and the model relies upon voltage measurements for updating the SoC of the system, then the model cannot effectively update the SoC of the cell. In order to overcome such difficulties, the parameter or function used to update the SoC of the cell must be sensitive to the forcing function offered by the load. This can be achieved by using an alternative combination of parameters or by using a different model.

iii) The perturbation that the load produces in the measured parameters (e.g., the cell voltage) may not be large enough to be tracked by the model. In this case, the model used to describe the system may be fairly accurate and have a sound physical basis; however, still the estimates for the SoC or SoH may not be accurate. A usual reason for this type of unobservability is that the battery/cell has not been perturbed sufficiently to evoke a detectable response. This issue is usually addressed by carrying out periodic capacity check or rate capability measurements, wherein the cell is forced to discharge and/or charge to the maximum (or minimum) SoC or voltage possible of the chemistry and the response is used to update or match the predictions made using the data collected during the normal cycling of the battery. The unobservability of a system can be detected mathematically by reformulating the model into the statespace form and calculating the rank of the observability matrix. Different types of unobservabilities commonly reported in the literature are illustrated in Figure 2. In Figure 2(a), according to the model used for life prediction, the cell voltage depends on two parameters: p1 and p2. A measurable change in the cell voltage is caused by a considerable change in both the parameter values p1 and p2. Hence, the model can be used to estimate either p1 or p2 or both from measurement of the cell voltage; Figure 2(b) shows the case where p1 changes appreciably during a measurable change in the cell voltage, but a

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Figure 2 Observability illustrated: (a) a well-observed model; (b) model exhibiting lack of observability for parameter p2; (c) poor choice of model; and (d) poor design of experiments.

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change in p2 is very small and hence p2 cannot be estimated; Figure 2(c) illustrates the scenario when the model assumes that a change in the cell voltage is caused by a change in the parameter values p1 and p2, whereas the experimental cell voltage is insensitive to the parameter values: this is an example of a poor choice of the model; and finally, Figure 2(d) shows the case where the parameter values do not change much in the range of data collected, nor does the change in the cell voltage cover a broad domain: this case illustrates a poor design of experiments.

Tools for Life Estimation Some common estimation techniques used for predicting the life of a lithium-ion cell are illustrated in the following sections with examples. The list of techniques described here is more representative than exhaustive. However, the set of tools outlined in this section provide a sufficient background to assimilate the essence of any protocol from the literature. Classification of Methods The problem of life prediction has thus become a vital ingredient in the recipe for developing an efficient design, either for an individual cell or for a pack. Over the last decade, the number of methods used for this critical task has grown significantly. Depending on the requirements (e.g., how often should the model predict the life of the cell), the assumptions behind developing the models, the expected outcome of the predictions, and the degree of complexity of the algorithm, these methods can be classified in many different ways. Every technique used in the life prediction of a battery has some mathematical representation of the battery. This model of the battery is a key factor that determines the accuracy of the predictions. One major classification of the techniques used for life prediction is based on the type of model a technique uses. The models used can be arbitrary mathematical expressions that best represent some experimental data, in which case, the technique is classified as an empirical fit. Another way to develop a battery model is to identify the individual processes that one encounters during the functioning of the cell, such as transport, reaction, and thermodynamics; describe each of these phenomena in terms of mathematical equations; and then combine these individual components to predict the overall performance of the cell. Because the development of such equations usually follows a well-established set of descriptions of each physical process taking place inside the cell, these second set of models are said to be physicsbased or first principles-based models. An intermediate collection of models that are a hybrid between the above

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two classes of models is described in a later section. These are referred to as semiempirical models. A second classification of models is based on the algorithm used for making the predictions. Some utilities use a battery pack for a routine or preprogrammed load. Examples include the battery packs used onboard the satellites during the lack of solar energy or cells used in watches. For these applications, because the load is known a priori, it is possible to develop a model knowing the operating conditions and the design parameters of the cell that can be used readily to predict the cell performance (e.g., maximum allowable number of cycles before reaching the end-of-life criterion, optimum load per cell to maximize the number of cycles). Because the operating conditions are already known, the model can accommodate an inbuilt mechanism to calculate the extent of degradation. Such a model does not require frequent updates for the parameters, unless there is a significant change in the operating conditions. In some other applications, the cell (or the pack) is subject to a dynamic load which changes as frequently as every few milliseconds. In these cases, the degradation mechanism and hence SoC or the SoH of the power system depend on the load conditions imposed in the immediate past, and it is necessary to monitor the cell on a regular basis. There are some differences between the algorithms used to make life estimates for the case with the known operating parameters and those used in the dynamic load case. For instance, the latter situation is more demanding in terms of the calculation time. The set of algorithms that update the parameters in the model periodically based on the experimental data as and when they become available are classified as online estimation techniques. A third classification of the methods used for making life predictions is based on the outcome of the algorithm. Most techniques provide with just an absolute numerical value for the estimates and do not take into account factors such as the scatter in the experimental data used to build the model or the error introduced by the approximations made during the model development process. As opposed to these ‘absolute’ techniques, some models provide a range of trustworthiness for the predicted parameters or a window of operation – showing the best and worst possible scenarios. Such techniques may incorporate uncertainties in the parameter values or provide an estimate of the robustness of the model, for example. These techniques are classified as probabilistic techniques. It is pertinent to mention here that this classification is not universal. Quite often, a particular technique may be accommodated in more than one classification. For example, a first principles-based model can be used either with an online or with an offline estimation technique. It can be used to provide an absolute or a probabilistic estimate of the SoC or SoH. A summary of the classification is provided in Figure 3.

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Figure 3 Classification of the different types of models and tools used in making life predictions of lithium-ion cells.

Empirical Fits Empirical fitting of data to make model predictions is particularly useful in cases where extensive testing is carried out prior to the intended application. The availability of simple and ready-to-use curve-fitting tools in commercial packages such as Microsoft Excels has enhanced the popularity of these methods. In the present work, any model that does not incorporate a mechanism for the functioning of the cell is considered as empirical. Typical examples include curvefitting, circuit-analog models, and pole–zero models. The latter two are grouped into this category because the actual elements of the circuit or the distinct time constants do not exist in the physical system. Alternatively, these models do not describe in detail the physical phenomena such as adsorption, diffusion, and migration that take place in the system. A few commonly used empirical techniques for life prediction are described in the next few pages. Curve fitting

Fitting experimental data to an arbitrary equation to extract coefficients for the equation is the essence of this technique. The simplest case, namely linear extrapolation, typically involves plotting the capacity of the cell versus the cycle number and regressing the slope and the intercept of a straight line. When a cell is subjected to repetitive cycling under operating conditions that do not cause severe wear out of the cell until the end-of-life criterion is reached, linear extrapolation has been found to provide a good degree of confidence in predicting the end of life of the cell. The primary advantage of using this technique is the ease of extracting the coefficients. The models do not require extensive use of mathematical tools. Depending on the range of operating conditions, more than one set of coefficients may be required for successful prediction of the cell performance. For example, if two cells are subject to different depths of discharge, naturally one would expect the

Figure 4 Cell capacity vs cycle number data fit to empirical expressions: the linear equation fits the data at milder conditions (cycling at 5 1C) much better than the data at a more rigorous condition (cycling at 25 1C). For the data collected at 25 1C, a more complicated expression (in this case, a fifth degree polynomial) is required to represent the data more accurately.

degradation rate of one cell to be faster than the other, and consequently, the coefficients in the empirical fits for each case are different. The accuracy of the method relies on the functional terms in the expression used. A complicated polynomial expression may provide a better prediction compared with a linear equation; however, estimating the coefficients for the former case requires a significant amount of computation compared with a linear fit. Tools for nonlinear regression are also readily available in the form of commercial packages. Whereas the minimum number of coefficients required for a good fit is not usually known beforehand, the statistical significance of each coefficient can be obtained by performing a confidence interval analysis. If the confidence interval for a parameter is larger than the value of the parameter in itself, it is implied that the parameter does not contribute significantly to the fit and hence can be removed. The success of the technique in making life predictions depends entirely on the prior knowledge of the system at hand. In other words, the curvefitting technique is used more often to interpolate to an unknown operating scenario rather than to make predictions beyond the limiting cases for which experimental data are available. This shortcoming is typical of any empirical prediction technique and does not prevent curve fitting from being the most popular choice in the industry. Some predictions made using empirical models are shown in Figure 4. The technique of estimating the coefficients is discussed in a subsequent section. Circuit-based models

The circuit-based models use electric circuit elements such as resistors and capacitors to represent the various physical phenomena that take place during the degrada-

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tion of a cell. For instance, the ohmic drop across a film of side-reaction products deposited on the surface of the anode can be represented by a resistance component in the circuit model. A typical equivalent circuit used widely in the literature to represent a variety of chemistries is represented in Figure 5. The resistor R1 represents the ohmic drop across the cell and R2 the faradaic resistance. Frequently, the capacitor C is replaced by a Warburg element to simulate any mass transport limitations. The resistance and capacitance values are obtained from alternating current impedance measurements. The apparent opencircuit voltage (V0) is experimentally measured at various rates of discharge (to estimate the SoC) or at various cycles (to obtain correlations for the SoH). The relationship between the current I and the cell voltage V for the circuit shown in Figure 5 is given by V ðt Þ ¼

  Q ð0Þ t =R2 C e þ V0  IR1  IR2 1  et =R2 C C

½1

where Q(0) is the nominal capacity of the cell. The above equation can readily be used in conjunction with voltage versus time data and cell impedance data to fit the resistance values R1 and R2 as functions of the SoC of the cell. Empirical correlations (e.g., polynomial functions) are

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obtained for R1 and R2 in terms of the SoC. The change in the coefficient values of the polynomial functions is then used to predict the SoH of the cell. It must be mentioned that not every element in the circuit presented in Figure 5 readily identifies with a physical phenomenon occurring inside a cell – although usually a change in the value of a particular set of parameters is associated with a specific change in the cell or the operating conditions. For example, the change in the rate of decay with temperature can be captured by adjusting a specific coefficient in the empirical expression relating the resistances to the SoC. Pole-placement type of models

A dynamic model for a cell is often represented by a set of differential and algebraic equations. An alternative representation of the system is in the Laplace domain using the convention of poles and zeros. The location and values of the poles (which are the roots of the polynomial in the denominator of the transfer function used to represent the response of the system) are used to determine the stability of the battery. This approach gains its popularity mainly from the prior art in the process control community, where the frequency response of a system is typically analyzed by plotting the lead or lag in the response of the system (i.e., the complex part of the response) versus the C

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Figure 5 Circuit-based model for life prediction: (a) a popularly used equivalent circuit used to model capacity fade in a variety of chemistries; (b) change in parameter Ri obtained by fitting experimental impedance data to empirical expressions; and (c) predictions of cell voltage vs state-of-charge (SoC), made using the circuit-based model.

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Figure 6 Time required for convergence when using a pole-zero model for making predictions of the state-of-charge (SoC) of the cell: current pulses from 150 to 1 A followed by periods of rest were used to perturb the system. The initial SoC was incorrectly initialized to 80% whereas the experimental data was collected from a cell initially at 100% SoC. (a) Shows the current pulse used and (b) the reduction in the time required for convergence to within 2% error. Note that the time required for convergence is reduced exponentially with the increase in the number of poles and that the use of more than 3 poles is redundant because the convergence time is not improved any further.

response of the system at that instant (referred to as the ‘real’ part of the response). Once the model for the battery is cast into the pole–zero form, the problem of life prediction is reduced to determining the position of the poles of the transfer function. The stability of the battery then depends on the eigenvalues of the transfer function used to describe the battery performance. Thus the task of life prediction for a battery is reduced to a classic poleplacement problem in the process control literature. Like the higher order polynomial providing a better approximation, a model with a greater number of poles reduces the time required for convergence of the predicted SoC to the experimentally measured values. Figure 6(b) shows the time required for convergence as a function of the number of poles used in the model for the load profile shown in Figure 6(a). The actual convergence of the SoC values is also shown in Figure 6(a). The identification of the poles of the system is similar to estimating the coefficients of an empirical expression. The accuracy of the model is improved with the increase in the number of poles; however, large systems are difficult to represent in the pole–zero form, and the computations become quickly formidable for a complicated model. Other limitations for the pole–zero type of models arise from the lack of basis to establish theoretical formulations for nonlinear systems. As a result, accuracy of the predictions cannot be guaranteed to a prespecified range, for highly nonlinear responses from the battery. Advantages and disadvantages of using empirical models

There are several advantages in using empirical models: i) These models are relatively easier to implement on hardware circuitry, compared with the elaborate

mechanistic models that often require solution of differential equations. Hence, empirical models make an ideal choice for embedded estimators. ii) As mentioned earlier, the time required to solve an empirical model is typically far less compared with the solution of a rigorous model. Hence, online estimators that are limited in the solution time permissible often use an empirical model. iii) Another significant edge that empirical models have over physics-based models is their wide popularity in the industry. They are easy to use and provide quick back-of-the-envelope type estimates. iv) The development of an empirical model does not require an in-depth study or understanding of the chemistry of the system. There are also some limitations associated with the use of these models: i) Before an empirical model that can make predictions over a reasonable range of operating conditions is constructed, extensive testing is essential over the entire operating range to obtain the data points for the fit. ii) Empirical models often do a good job interpolating within the test window than predicting the performance under a set of conditions not evaluated at the time of developing the model. iii) The actual operating conditions of the cell are often different from the test conditions used to develop the empirical correlations. Care must be exerted in using such test results because the response of the system varies, especially when the actual operation involves an extreme operating condition not tested a priori.

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iv) Because an empirical model has incorporated no component of the chemistry of the system, the results cannot provide a feedback on the design of the cell (e.g., the choice of materials, the porosity of the electrodes). This limitation prevents the extension of the results predicted by an empirical model to other cases, e.g., an alternative cell design.

Physics-Based Models The lithium-ion cell functions based on a unique intercalation mechanism, wherein lithium diffuses out of the cathode matrix during the charge process, is subjected to a charge transfer reaction at the electrode–electrolyte interface, followed by diffusion in the solution phase to the anode, wherein another set of charge transfer and diffusion processes occur. Each of these stages offers some resistance to the charge/discharge process. As the cell ages, additional reactions such as growth of a film at the electrode–electrolyte interface, buildup of the electrode impedance, and mechanical disruption of the intercalation matrix complicate further the charge/discharge mechanism of the cell. Most of these physical phenomena have been widely studied in the past, using a variety of materials under similar circumstances. Consequently, such well-understood processes have been represented by physical laws. For example, diffusion of a particle inside a solid is represented by Fick’s laws and charge transfer at an interface is modeled using the Butler–Volmer equation. Physics-based models use such an established mathematical treatment to simulate each physical phenomenon happening within the lithium-ion cell. The performance of the cell, e.g., the rate capability after several cycles, is calculated as a net outcome of these individual phenomena. The current in a physics-based model is carried by electrons in the solid phase, and hence, Ohm’s law is used to model the solid phase potential. In the solution phase, the charge is carried by the ions. Unlike the electrodes, the conductivity of the electrolyte is a function of the concentration of the ions. Hence, transport limitations of the ion within the electrolyte constitute a considerable portion of the potential drop in the electrolyte. The solution phase potential is governed by a modified form of the Ohm’s law that includes conductivity as a function of the Li þ concentration in the electrolyte and the concentration gradients in the solution phase. Butler–Volmer kinetics is used to model charge transfer at the electrode–electrolyte interface. To illustrate the predictive capability of a physics-based model, a degradation mechanism based on solvent reduction on the anode is also included. Finally, the cell voltage is calculated as the difference between the solid phase potentials at each electrode–current

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collector interface. Use of the porous electrode framework requires volume averaging the equations over each domain. The major difference between empirical models and the equations in a physics-based model is that the parameters that appear in the physics-based model represent actual phenomena and can be measured independently. Figure 7 shows the predictions from a model that incorporates transport, kinetic, and thermodynamic limitations within the electrodes as well as the electrolyte, for various rates. As observed, once the physical parameters such as the rate constants and diffusion coefficients are fixed, the model can be used to predict the end of discharge for a wide range of scenarios. Understanding the mechanism of degradation is a critical step in developing a physics-based model for a system. For example, in the nickel-based cathodes, surface oxidation of the cathode particles results in an additional impedance created at the cathode, whereas such an increase in the impedance of the cathode for a cobalt-based system is attributed to phase changes at higher voltages. Similarly, volume expansion is considered important for the manganese- and iron-based chemistries of the cathode, whereas the mechanical strain has little effect on the cobalt oxide cathode. Another major challenge in using physics-based models for life prediction arises from determining the values of the parameters used in the model. The number of parameters used in a physics-based model is considerably higher than an empirical fit. Although most of these parameters are available from the operating conditions and the design of the cell components, it is rather time consuming to measure some parameters such as the rate constants and transport coefficients. Also, many of the parameters change with the operating conditions. Developing a mechanism to model changes in the parameter values is difficult and including such elaborate mechanisms for modeling the change in each parameter increases the computational time significantly. However, a physicsbased model does not require such a mechanism. A typical example is incorporating the solution phase conductivity as an empirical function of the concentration of lithium ions in the electrolyte. Semiempirical Models Mechanistic models incorporate the chemistry of the system, and hence, the same set of fundamental equations can be used to model a different system once the set of parameters and the key degradation mechanism for the new system are deduced. The failure of a cell is due to a combination of several phenomena. It is a tedious task to formulate the physical equations for several degradation mechanisms, identify values for the parameters associated with each, and solve the resultant set of equations. Semiempirical models present a trade-off between an

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Figure 7 Predictions from the physics-based model described in the section entitled ‘Physics-Based Models’ compared with experimental data: note that the same model equations are used to fit all the experimental curves, by just altering the current supplied to the cell, akin to experimental measurements. Reproduced from Santhanagopalan S, Guo Q, and WHite RE (2007) Parameter estimation and model discrimination for lithium-ion cell. Journal of the Electrochemical Society 154(3): A198–A206. Copyright with permission from the ElectroChemical Society.

entirely empirical set of equations and the puristic first principles approach. Although the active material and the electrolyte degrade owing to several factors, the physical phenomena that take place during the insertion/deinsertion of the lithium ion do not change even as the cell ages. A semiempirical model capitalizes on this phenomenon. It does not include any mechanism to explain the capacity fade in a cell. A physics-based model is used to model every charge/discharge cycle. However, the key parameters are periodically extracted from the test data using an empirical equation such that the cell performance as a function of time (or cycle number) matches the experimental value. For example, a physical model may be used to describe the charge/discharge performance of an individual cycle and one or more parameters in the physics-based model can be periodically updated using an empirical trend obtained from limited test data. A typical example of the empirically varied parameter is the thickness of the film formed on the anode as a result of the deposition of salt formed by reducing the electrolyte during the charging of a lithium-ion cell. Using the semiempirical model thus developed, the discharge curve after N cycles can readily be simulated by calculating the value of the film thickness at the end of N cycles using the empirical expression and using it directly with the physics-based model to simulate the

discharge curve for the Nth cycle. This avoids the tedium of solving the model from cycle 1 to cycle N and reduces the number of equations to be solved. Figure 8 shows the predictions made using such an approach for 1.8 Ah Sony 18650 cells up to cycle 800. Semiempirical models combine several aspects of empirical as well as first principles-based models. For example, in order to predict the performance of the cell after 400 cycles using a semiempirical model, one does not have to calculate the charge/discharge profiles of the cell from cycle 1 to cycle 400. Using the empirical expression for the respective parameters evaluated at cycle 400, the potential/ capacity curve for that cycle can be readily computed using the model, even though it does not include an explicit degradation mechanism. A potential drawback in using a semiempirical model is that it cannot handle circumstances when the trend in the parameter values changes with cycling. Also because the cell components decay to varying degrees depending on the operating conditions, monitoring the change in a parameter, e.g., the diffusion coefficient during a particular operating condition, cannot be used readily to predict the performance at a different operating condition. The advantage that the semiempirical models exert over empirical models is that the change in the property can be measured physically and verified using independent experiments. Using a semiempirical model

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Simulation 4 Experiment

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Figure 8 Predictions made from a semiempirical model: the physics-based model is used to make predictions of a given cycle number (N) as shown in (a). The adjustable parameters in the model are the state-of-charge (SoC) of the negative electrode at the beginning of N discharge (yN n ) and the resistance of the film formed on the anode (Rf ). The parameter values were obtained by adjusting the experimental curves at various cycle numbers as shown in (b). Copyright with permission from the ElectroChemical Society.

avoids the complications of reinitializing the physics-based model between cycles and circumvents issues related to numerical convergence. The major limitation in using the first principles–based model arises from the huge computational cost associated with solving the elaborate set of equations. Much of the complication associated with solving a first principlesbased model is retained in the semiempirical models. In order to overcome this difficulty, several simplifications are used in the model equations. For example, the

concentration profile within the solid particles is approximated to be parabolic and a simplified set of equations is used instead of solving rigorously the Fick’s laws of diffusion at each point within the electrode. Another approximate model involves the assumption that each electrode can be represented by a spherical particle of equivalent area. The use of simplified models has proven to be very effective particularly in models used for making life predictions because the model needs to be run several times repeatedly both for estimating parameters and for

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making predictions. In addition to providing very small solution times, simplified models often provide analytical solutions that can be readily used in calculations. These closed-form solutions are of great value in repetitive calculations such as predicting cycle life and regression of parameters using the model from experimental data. When using the simplified models, one must consider the limitations associated with the approximations involved in developing the models. For example, the model may not provide good predictions at very high charge or discharge rates or it may not consider the thermal effects associated with the discharge of a cell.

Parameter Estimation and Life Prediction The predictions made from a model based on first principles rely to a great extent on the list of parameters used for the physical properties in the model. Hence, these parameters need to be estimated from experimental data accurately to ensure confidence in the model predictions. The use of empirical and semiempirical models in life prediction relies on obtaining a trend in the change of one or more parameters estimated from experimental data. Thus, parameter estimation plays an integral part in life prediction. Linear Estimation Linear-least squares is the simplest and the most popular of techniques used to estimate parameters. As described in a previous section, a simple example is the fit of cell capacity data versus the number of cycles the cell is cycled for. Figure 4 shows experimental data from a lithium-ion cell cycled at 5 and 25 1C. The spikes in the data are due to the capacity check performed after every 100 cycles. The experimental cell capacity (CN ) is shown on the ordinate. This set of data is fit to a linear equation of the form CN ¼ mN þ C0

½2

where CN represents the cell capacity at the end of N cycles as predicted by the model and C0 represents the initial capacity of the cell before the first cycle. The parameter m represents the rate of decay in the capacity, which the model assumes to be constant throughout the life of the cell. For simplicity, let us assume that the initial capacity of the cell C0 is accurately known. The objective of the model is to find the value of m such that the predicted capacity CN matches the experimental cell capacity CN . The total error between the predicted and the actual cell capacities after N cycles is given by e¼

N X ðCk  Ck Þ2 k¼1

½3

The square on the right-hand side of the equation ensures that the overshoots in the predictions at some cycles are not counterbalanced by the underestimates. Substituting eqn [2] into [3] results in e¼

N X ðmk þ C0  Ck Þ2

½4

k¼1

If one were to try different values for the unknown parameter m, the error e varies depending on the choice of m. In order to minimize the error e so that the model predictions accurately match the actual set of data, the first derivative of the error with respect to the unknown parameter m is calculated and is set equal to zero: 2

N X ðmk þ C0  Ck Þ ¼ 0

½5

k¼1

Equation [5] can now be solved for the unknown parameter m: 2 m¼ N ðN þ 1Þ

N X

! Ck

 NC0

½6

k¼1

The value for the parameter m that fits the data at 5 1C is  0.001 736 and that for the data collected at 25 1C is  0.004 042. The model predictions based on these values for the parameter are shown in Figure 4. Also shown in the figure is a fit to a fifth degree equation of the form CN ¼ a0 þ a1 N þ a2 N 2 þ a3 N 3 þ a4 N 4 þ a5 N 5

½7

where ai (i ¼ 1,y, 5) are the unknown parameters calculated based on a similar procedure as that described above. Figure 4 illustrates the ease of using linear fits as well as the limitations in using a simple linear model. At 5 1C, the experimentally measured capacity varies in a linear fashion with the cycle number. Hence, a linear model is a good choice; however, at 25 1C, the higher order equation provides a better fit. The visual observation is further substantiated by the numerical values of the error e for the linear and the higher order fits for the data at 25 1C: 0.1965 and 0.02344, respectively. Nonlinear Estimation Model equations such as eqns [2] and [7] exhibit a linear dependence on the parameters. In more complicated situations, the relationship between the measured variable and the parameter of interest is nonlinear (e.g., the transfer coefficient appears in the exponential term of the Butler–Volmer equation). Under such circumstances, the evaluation of the parameters involves nonlinear parameter estimation. Software packages such as MATLABs and Octaves present extensive algorithms and modules that can be used readily for nonlinear estimation. A

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction

detailed account of the several techniques is out of the scope of this article. The branch-and-bound algorithm and the Levenberg–Marquardt algorithm are two widely used approaches. The steps involved in estimating parameters using each of these algorithms are outlined below.

429

for the optimal solution throughout the parameter space. This is especially tedious because the dimensions of the parameter space increase significantly with the addition of more and more parameters. The Levenberg–Marquardt algorithm

The branch-and-bound algorithm

Step 1: Identify the feasible region for the parameter set; i.e., find the minimum and maximum values for the parameter set that are allowable by physical constraints. Step 2: Evaluate the objective function (i.e., the difference between the model equation and the experimental data squared; see eqn [4], for an example) at the lower and upper bounds. Step 3: If the values of the objective function evaluated at the lower and upper bounds match, the value for the parameter set is already optimal. In other words, the parameter set being investigated does not influence the measured variable within physically reasonable limits. Step 4: If the values of the objective function evaluated at the bounds do not match, the feasible region is divided into two or more subregions and steps 1–3 are repeated for each subregion until the optimal values for the parameters are reached. The steps are illustrated in Figure 9. The branch-andbound algorithm has one potential disadvantage: The rate of convergence toward the optimal solution is relatively slow, because the method involves searching extensively Objective function

This algorithm calculates the steepest slope from any given starting point and moves the parameter vector toward that direction in the parametric space. There is a scaling factor (l) used to adjust the rate of convergence according to the change in the vector of parameters. The steps involved are summarized below: Step 1: Obtain the value of the measured variable (usually the cell potential or the cell capacity) with an initial set of guess values (h0) for the unknown parameters. Step 2: If the difference between the model predictions and the experimental values is less than the required tolerance, then the initial guess values present an optimal solution for the set of unknowns; if not, the Jacobian matrix (J) is evaluated at the initial guess values for the parameters. The components of the Jacobian matrix are calculated as follows: Ji ðhÞ ¼

JD1;n ðD1;p ; D1;n Þ ¼

Initial guess

Converged value

Figure 9 Illustration of the branch-and-bound algorithm: A linear single minimum objective function is shown in this illustration. The objective function is estimated at an arbitrary set of initial guess values, to form a triangle. The initial guess points are then moved along the corresponding median to the opposite bases and the objective function is reevaluated at the new set of points. The minimum value of the three residuals is chosen and the corresponding vertex is shifted across the median. This process is continued until the convergence criteria are met.

½8

The Jacobian is a measure of the extent of influence each of the unknown parameters yi has on the measured cell voltage or capacity (Y). For example, if the cell voltage is measured experimentally and these values are used to back-calculate the diffusion coefficients of lithium at the anode and the cathode, the Jacobian of the cell voltage with respect to the solid phase diffusion coefficients of lithium JD1;p ðD1;p ; D1;n Þ ¼

Iterations

@Y @yi

@VCell @D1;p @VCell @D1;n

½9

represents the relative significance of each of the parameters D1,p and D1,n. Step 3: The correction vector for the list of unknowns (h) is calculated as follows:  1 Dh ¼ JT J þ lI JT ðY   Y Þ

½10

where l is the step-size correction factor, which is assigned a large value of 100 initially in the regression and a very small value on convergence, e.g., 106, I is an identity matrix, and the superscripts T and  1 represent the transpose and the

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Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction

inverse of a matrix, respectively. Note that the correction vector is the product of two terms: There is a term showing the error between the model prediction (Y) and the experimentally observed value (Y *), and there is another term involving the Jacobian matrix (J), which weighs the error (Y  Y *) according to the relative significance of each parameter. Step 4: The parameter vector is updated using the following equation: hmþ1 ¼ hm þ Dh; m ¼ 0; 1; 2; y

½11

where m is the iteration counter. Steps 1–4 are repeated until the required tolerance criteria are met. The major disadvantage associated with the Levenberg– Marquardt method is the tedium associated with calculating the Jacobian matrices. However, the convergence rates are significantly higher than the branch-and-bound algorithm. A common concern in using any nonlinear estimation technique is that the set of parameters obtained may not provide the global optimum. The physical analogy is that one set of parameters may provide a lower value for the error compared with the values in the neighborhood; however, this set may still not give the lowest value for the error. This is an open topic where active research is still in pursuit. Recent algorithms such as simulated annealing and genetic algorithms propose more effective search techniques that are claimed to ensure global optima. Figures 7 and 10 show some interesting results on predicting life from a physics-based model utilizing the Levenberg–Marquardt technique outlined above. The cells were cycled at 15 1C. The cell voltage was used as the measured variable (Y *) and the list of parameters (h) included the SoC at the beginning of the cycle inside each electrode and the weight of active material inside each electrode. These parameters were estimated at the end of every 100 cycles. These estimates not only provide information about how much capacity is lost during every 100 cycles from the beginning of life, but also give an insight into the possible failure mechanism(s) is. In this particular example, the capacity fade is dominated by the loss of cyclable lithium for the first few hundred cycles. The cell capacity at later cycles is controlled by the amount of active material available at the anode. By using the trend in the values for these parameters, a semiempirical model can be formulated. Online Monitoring of a Lithium-Ion Cell Some applications require periodic monitoring of the SoC and SoH of the cell. Such requirements typically arise owing to changes in the operating conditions of the

cell, arbitrary load conditions, or simply owing to a need for more accurate predictions. For example, Figure 4 shows that a linear model does not yield a good fit for the data at 25 1C. One alternative is to use a more complicated model to represent the data. However, there are limitations associated with implementing complex models in practical systems to monitor the status of the cell. Under such conditions, it is required to monitor accuracy of the predictions made by the linear fit and to periodically update the parameter set when the predictions do not match the tolerance requirements. How often the parameters need to be updated depends on the load conditions, the memory constraints imposed by the onboard devices, and the complexity of the algorithm implemented. Very frequent updates require highly sophisticated circuitry with very fast clock times and robust memory handling capabilities: These are often associated with huge financial constraints. There are a few applications where updating the SoC of the cell at intervals of milliseconds is critical for the safety and performance of the system. A first principles–based model is invaluable in designing a cell for such applications; however, the numerical solution of the model and the associated optimization problem for extracting the parameters is a formidable task for an onboard device. Empirical equations are typically employed in these situations. This section elaborates on a few algorithms used to update parameters periodically using experimental data as and when it become available. Such techniques are often referred to as online estimation or monitoring algorithms in the literature. Moving horizon estimation

The moving horizon algorithm estimates the parameters used in the model using an initial set of data points (e.g., from the start of the experiment to some time t). These values for the parameters (yt) are then used to predict the cell performance for the next few data points (e.g., between times t and t þ Dt). The error between the model predictions and the actual data points collected between t and t þ Dt is then used to calculate the updated set of parameters yt þ Dt. This process is repeated at periodic intervals of time or load. More elaborate moving horizon estimates include the influence of several sets of parameters from the past on the current estimates. One example is the use of exponential forgetting functions. In this example, the effect of the parameter values yt , yt þ Dt , yt þ 2Dt , etc. on the current estimate yt þ kDt is assumed to decay exponentially. The steps are summarized below: Step 1: Choose a subset of data points N0 at the end of which the parameters need to be updated. Calculate the initial value for the SoC. Step 2: Calculate the value of the exponential forgetting function at the end of N0.

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction

431

0.35

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Capacity fade (Ah)

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Figure 10 Comparison of capacity fade in a lithium-ion cell calculated using a physics-based model and the Levenberg–Marquardt algorithm: The line indicates the total experimental capacity fade as a function of cycle number at 25 1C. The bars indicate capacity fade from various factors after every 100 cycle: indicates capacity fade owing to loss of cyclable lithium at the anode, indicates capacity fade owing to loss of cyclable lithium at the cathode, indicates capacity fade owing to loss of active material at the anode, and indicates capacity fade owing to loss of active material at the cathode. The identification of the origin of capacity fade is a unique feature of the physics-based models. Reproduced from Santhanagopalan S, Zhang Q, Kumaresan K, and White RE (2008) Parameter estimation and life modeling of lithium-ion Cells. Journal of the Electrochemical Society 155(4): A345–A353. Copyright with permission from the ElectroChemical Society.

Step 3: Use the next set of data points N0 þ 1 to N1 to calculate the updated values for the parameters in the model equations. Step 4: Update the SoC for the next set of data points using the parameter values from the previous step. Step 5: Update the exponential forgetting function based on the data points N0 þ 1 to N1, new values for the parameters, and the current value of the SoC. Step 6: Repeat steps 1–5 until the end of the data set. This results in a set of values for the SoC updated whenever the error between the model and the experimental data is significant. The exponential forgetting function algorithm is highly effective in providing rapid estimates of the parameters because a simple model equation can still be used effectively to make accurate predictions. Figure 11 provides an example of online estimates made using the circuit model shown in Figure 5 using the moving horizon algorithm. The technique is not limited in terms of the linearity of the model equation. If a nonlinear

equation is used, iterations are involved in extracting the parameter values. Convergence of these techniques still remains an open question. No rigorous criteria have been formulated to ensure convergence of the parameters. Moreover, because these techniques are often used for online update of the parameters, in conjunction with real-time data, convergence of the parameters within a limited time is critical. Filter-based techniques

When a battery is used as a part of a vehicle or satellite, there are limitations on the degree of sophistication employed in measuring the voltage or current. Often, there are other electronic components whose response interferes with the quality of the measurements. The term ‘filter’ refers to a weighted least-squares algorithm that can be used to minimize the effects of measurement noise. The filter-based techniques specialize in isolating the response of the cell from the background noise. The working principle of a filter is illustrated further in Figure 12. The load fed to the actual experimental setup (the battery) is determined by the application for which the battery is used. The same load

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Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction 425

260

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Figure 11 Moving average estimation: The circuit shown in Figure 5 is used to make predictions. The parameters V0 and R are periodically updated to make predictions of the state-of-charge (SoC) and the pack voltage (V). The current profile is also shown. Reproduced from Verbrugge M, Frisch D, and Koch B (2005) Adaptive energy management of electric and hybrid electric vehicles. Journal of the Electrochemical Society 152(2): A333–A342. Copyright with permission from the ElectroChemical Society.

Background noise Q

Load

Update parameters (SoC/SoH)

P

Model

Estimator

Experiment Error

Figure 12 Illustration of a filter: experimental cell voltage is compared with the model predictions and an estimator is used to update the parameter values such as the state-of-charge (SoC) and the state-of-health (SoH). A weighting function P is used to correct the updated values for any inadequacies in the model before feeding the updated values into the model to make predictions for the next time step. Any background noise is eliminated by a suitable weight function Q. The functions P and Q are largely determined from statistical covariances between the state variables and the noise data, respectively.

function is supplied to a mathematical model that can calculate the SoC and/or SoH as well as the voltage of the battery as a function of the SoC. The predicted voltage is compared with the experimentally measured value of the voltage. A suitable error metric is implemented to estimate the parameters using the techniques outlined earlier. The experimental noise and the model predictions are weighed relatively, based on a priori knowledge of the degree of sophistication involved in the model development process

as well as the data acquisition step, using weight functions (represented as P and Q in Figure 12). Finally, the updated values of the SoC are used to calculate the cell voltage at the next time step in the model. This cycle is repeated at periodic time intervals. The results from this approach are shown in Figure 13. Figure 13(a) shows synthetic voltage versus time data generated by adding a zero mean Gaussian noise to the current fed to the cell. The noise added to the flux simulates the case in which there is an issue with the cell leading to fluctuations in the cell voltage. Figure 13(b) shows the case in which the synthetic voltage is generated by adding the noise directly to the voltage to simulate any background noise. As observed, for the first case, the model predicts that the SoC of the cell fluctuates with the cell voltage. This is true because in generating the voltage versus time data, the noise was added to the flux. In the second case, although the voltage versus time data is noisy, the filter eliminates the background noise, and hence the SoC versus time curve is smooth. Figure 14 shows the predictions made for a typical UDDS cycle using this approach. One widely criticized aspect of these filter-based models is the arbitrariness involved in estimating the weight factors (P and Q). Although there exist algorithms that claim to estimate the weight factors from training data, this area remains open for improvement. Another critical issue associated with the filter-based techniques is the time required for convergence of the predictions to the actual data set. For example, Figure 14

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction

433

4 Data Vcell (V)

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Figure 13 Predictions made using an extended Kalman filter (EKF): case (a) is an example of the algorithm tracking the voltage closely to include any oscillations in the system and case (b) shows the capability of the technique to reject any background noise.

shows that for the error in the SoC (yn) to reduce to 0.01, the time taken for convergence is about 2000 s. In the cases where it is critical to achieve the required precision within very short times, such a filter does not fit the needs. One major reason widely attributed to such issues is an inherent assumption in the Kalman filter, which involves linearizing the model equations to formulate the state-space representation. Hence, to overcome this problem, a recent approach involving an advanced transformation of data points from the state variables space to the observed variable space has been proposed. An outline of the algorithm is shown in Figure 15. The filter design uses the relationship between the concentrations (c1,j) and the cell voltage (VCell) at several points chosen beforehand (known as the Sigma points) in order to develop a correlation without using the linearization required when using an extended Kalman filter (EKF). Hence, the predictions are more accurate and provide rapid convergence. Probabilistic Methods Most mathematical models provide life estimates as a certain number of operating cycles or a specified period of time. However, what is practically significant is a range for the estimates, and the level of confidence associated with the predictions made. In other words, a model must provide an interval encompassing the best possible performance achievable by the cell and the worst possible scenario. An overview of some statistical tools used in making such predictions is provided in this section.

Confidence interval estimation

The section entitled ‘Nonlinear Estimation’ introduced several techniques to estimate parameter values from a given model. In obtaining these estimates, the experimental scatter in the data points was not considered. Quite often, the parameters are extracted using data from more than one cell and inevitably there are measurement errors that result in error between the measured voltages from two or more cells operated under identical conditions. Hence, a measure of the degree of confidence in these estimates is required. The confidence interval estimates provide the boundaries on the parameter values, taking into account the statistical variance of the experimental data. The confidence interval for a parameter yi estimated from a set of data is calculated as follows: pffiffiffiffiffi pffiffiffiffiffi yi  tð1a=2Þ SE aii oyi oyi þ tð1a=2Þ SE aii

½12

where (1  a/2) is the degree of confidence to which the parameter is known. For example, if one wishes to estimate the parameter with 95% confidence, a ¼ 0.05 (so that (1  a/2) ¼ 0.95). The term t(1  a/2) is the statistical value of Student’s t distribution with (N  ny) degrees of freedom, N being the number of data points and ny the number of parameters estimated, and aii is the ith element of the principal diagonal of (JTJ)1 (see eqn [8]). The term SE refers to the average error between the predicted and the experimental values of the cell voltage given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffi XN h 1 Expt 2 Model SE ¼ ðVCell Þj ðVCell Þj j ¼1 N  ny

½13

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Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction Prediction of SoC during a typical UDDS cycle

Current (A)

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Figure 14 An extended Kalman filter (EKF) is used to update the state-of-charge (SoC) of the cell as a function of time. Note that the current pulses are steep, and hence, the system is perturbed considerably. For such cases, the parameter values must be updated rapidly. The error and the associated 3-sigma outliers are also shown in the figure.

Parameter domain Sigma point

Covariance Mean Experimental data

Linearized EKF

Unscented filter

Predicted variance Predicted mean Measured variable domain Measured variance (a)

(b) Measured mean

Figure 15 The unscented Kalman filter (UKF): (a) shows how the experimental cell voltage is mapped from the parameter domain (comprising parameters such as SoC and film resistance). (b) Shows the predictions made using an extended Kalman filter (EKF). The linearization of the model introduces a significant difference in the mean and covariance of the predicted voltage from the corresponding experimental values. The UKF does not involve any linearization of the model. The use of carefully chosen sigma points minimizes the error between the model predictions and the experimental values.

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction 4.2 4.1 4

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3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 0

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1 Cell capacity (Ah)

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435

A typical example is the spread in the particle size within an electrode. Classically uncertainties associated with a system have been addressed by analyzing the limiting case scenario. In other words, if one were to study the effect of the particle size distribution within the electrode or the pore size distribution within the separator membrane on the cell capacity or the cell voltage, in the past this problem had been addressed by considering a mixture of particles or pores of a specified number of size. The physical model then computes the aging effects for various cases: all particles of equal size, the largest difference in particle sizes, etc. However, the realistic scenario involving a continuous distribution of the parameter could not be addressed. This limitation is addressed with the use of the polynomial chaos theory (PCT), which assumes that the uncertainty associated with the parameter of interest is structured. For example, a perfectly Gaussian particle size distribution can be represented by a function of the form

Cell capacity (Ah)

1.5

Rj ¼ r0 c0 ðxÞ þ r1 c1 ðxÞ 1

0.5

0

0 (b)

100

200 300 Cycle number

400

500

Figure 16 Illustration of confidence intervals: In (a), the solid line represents the predicted cell voltage at cycle number 1 and the dotted lines represent the upper and lower confidence limits in the predictions. Part (b) shows the confidence intervals for the cell capacity predictions with cycling. As observed, after about 300 cycles, there is a change in the pattern of decay in the capacity, which the model apparently fails to capture very well. As a result, the confidence intervals are widened, which signifies that the confidence in the predictions made from the model decreases beyond 300 cycles.

pffiffiffiffiffi The term tð1a=2Þ SE aii provides the interval within which the parameter value (e.g., the SoC or the SoH) can lie, as predicted by the scatter in the data. Figure 16 illustrates the confidence interval predictions made for the SoC within a single discharge as well as for the SoH during cycling. Polynomial chaos-based approach

The models described thus far have all assumed that the parameter set associated with the model equations is precisely known. Often, the parameter values measured from an actual experiment do not hold a unique value. There is always a certain degree of uncertainty or a distribution associated with the value of the parameters.

½14

where ck ðxÞ are multivariate polynomial basis functions and r0 and r1 are coefficients that can be determined using the mean and variance of the Gaussian distribution. Then the PCT representation of the battery model maps this distribution in the parameter into the output variable, that is the cell voltage. Figure 17 shows a typical distribution for the particles present at the anode and its effect on the capacity of the cell during a constant current discharge. Fuzzy logic-based approach

There are several models that employ fuzzy logic-based algorithms in order to estimate the SoC and SoH of the cell. This methodology involves basically three steps: an initial training step that involves use of exhaustive amount of data collected as to cover the wide range of the operating space. In other words, the extreme conditions to which the battery may be subjected in its course of use are provided in advance as training data, to the algorithm. The second step is the building of the model. In this step, the inputs as well as the response of the system are grouped into several packets. For example, if one were to use the cell voltage and the impedance of the cell measured at various SoC values as the input to the algorithm, the model-building stage categorizes these inputs into several groups such as low impedance and high voltage, low impedance and low voltage, high impedance and low voltage, and high impedance and high voltage. The third step is to establish a mapping function that relates the changes in the operating conditions and the SoC of the cell, for the training data with one or more of these packets or groups. By the end of this step, the algorithm learns the response of the system under a variety

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Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction 0.9 Gaussian fit Particle size distribution

0.8 Fraction of particles

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3

4

5 6 Particle size (µm)

7

8

4.5

Cell voltage (V)

4

All particles are of 4 µm size

3.5 PCT predictions for the particle size distribution shown above All particles are of 6 µm size

3 2.5

Mixture of 4 and 6 µm size particles

2 1.5 0

1000

2000

3000

4000

5000

6000

7000

Time (s)

Figure 17 Effect of particle size distribution on the discharge curve of a lithium-ion cell: the particle size distribution is approximated using a Gaussian function. The use of a smooth distribution function predicts that the actual decrease in the cell capacity is much smaller than that predicted using a discrete mixture of particle sizes. PCT, polynomial chaos theory. Copyright with permission from the ElectroChemical Society.

exist today to address the problem – from linear fit of the capacity versus time data to complicated regression algorithms. With the increasing availability of packages to perform advanced calculations readily, the rigor of the models and algorithms has greatly been enhanced, and the predictions have drastically improved over the recent years. Judicious choice of the models implemented and the techniques used are critical for developing a successful battery management system. The key challenges in the near future include improving the robustness of the predictions and implementation of these algorithms in a real-time device. The former challenge needs work in the areas of developing tools that can withstand the rigors of the practical environment – e.g., noisy data, skewness of the Jacobian matrix in the estimates and determination of a suitable scheme to deweight erroneous data. With respect to the challenge of implementation, the models and the algorithms need to be simplified as to reduce the memory requirements to as low as 1% of the existing features. Only then a reliable implementation of these mechanisms on an actual hardware circuit is feasible. Current improvements in the field of electronics also help address these issues of scaling these techniques from an ideal simulation to an actual gadget. The increasing popularity of terms such as battery management systems and smart batteries in the literature stands testimony to the hope that accurate design of battery systems for safe operation and adequate performance will materialize.

Nomenclature Symbols and Units aii ak C c1,j

of conditions. Hence, the next step is to measure the impedance and cell voltage of the system. Then, using the trend in the training data and the operating conditions together with the measured responses of the system, the model can make predictions of the SoC for the measured cell impedance and voltage. Commercial packages such as MATLABs offer inbuilt toolboxes for developing this type of models and hence these are gaining increasing popularity.

Conclusions Life estimation for lithium-ion battery systems is critical with the increase in demand and development of increasingly higher cell capacity. A wide variety of options

Ck C*k CN D1,j e I I J JD1;n

diagonal elements of the covariance matrix used in the Box method coefficients in empirical equations (k ¼ 0, 1, 2, etc.) capacitance (F) concentration of lithium within the particles inside electrode j (mol m  3) predicted capacity of the cell at the end of k data points (Ah) (k ¼ 0, 1, 2, y, N) experimental cell capacity at the end of k data points (Ah) (k ¼ 0, 1, 2, y, N) cell capacity diffusion coefficient in the solid phase of the electrode j least squares error current (A) identity matrix Jacobian matrix sensitivity of the cell voltage to the diffusion coefficient of lithium in the anode (V sm  2)

Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion | Lifetime Prediction JD1;p

Ji JT m nh N Nj P Q Q Q(0) rj R R1 R2 Rf Rj SE t t V V0 VCell Y Y* a Dx h h h k n wk (n)

sensitivity of the cell voltage to the diffusion coefficient of lithium in the cathode (V sm  2) i th component of the Jacobian matrix transpose of Jacobian matrix parameter employed for data fitting number of parameters estimated maximum number of data points number of data points collected in the interval between tj and tj + Dt covariance matrix of the noise originating from the system covariance matrix of the noise originating from the measurement weighting factor used to filter background noise initial capacity of the cell (Ah) coefficients of the particle size distribution function resistance (O) resistor – ohmic drop across the cell faradaic resistance resistance of the film radius of the particles inside electrode j (m) average value of the sum squared error statistical parameter time (s) voltage (V) open-circuit voltage (V) cell voltage model prediction of the dependent variable experimentally measured value of the dependent variable confidence limit change in the parameter value x state-of-charge parameter extracted from the experimental data to fit a model vector of parameters step-size correction factors; scaling factor arbitrary variable space multivariate distribution function

Abbreviations and Acronyms EKF EoCV FreedomCAR PCT

extended Kalman filter end-of-charge voltage Freedom Cooperative Automotive Research polynomial chaos theory

PNGV SoC SoF SoH UDDS UKF USABC

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partnership for a new generation of vehicles state-of-charge state-of-function state-of-health urban dynamometer driving schedule unscented Kalman filter United States Advanced Battery Consortium

See also: Batteries and Fuel Cells: Lifetime; Batteries: Charge–Discharge Curves; Modeling; Secondary Batteries – Lead–Acid Systems: Lifetime Determining Processes; Secondary Batteries – Lithium Rechargeable Systems – Lithium-Ion: Aging Mechanisms; Overview; Secondary Batteries: Overview.

Further Reading Bergeveld HJ, Kruijt WS, and Notten PHL (1999) Battery Management Systems: Design by Modelling, Philips Research Book Series. Norwell, MA: Kluwer Academic Publications. Constantinidis A and Mostoufi N (1999) Numerical Methods for Chemical Engineers with MATLAB Applications. Upper Saddle River, NJ: Prentice-Hall. Crassidis JL and Junkins JL (2004) Optimal Estimation of Dynamic Systems. New York: Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Hansen T and Wang C-J (2005) Support vector based battery state-ofcharge estimator. Journal of Power Sources 141: 351--358. Huet F (1999) A review of impedance measurements for determination of the state-of-charge or state-of-health of secondary batteries. Journal of Power Sources 70: 59--69. Liaw BY, Jungst RG, Nagasubramanian G, Case HL, and Doughty DH (2005) Modeling capacity fade in lithium-ion cells. Journal of Power Sources 140: 157--161. Plett GL (2004) Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs. Journal of Power Sources 134: 252--292. Plett GL (2006) Sigma-point Kalman filtering for battery management systems of LiPB-based HEV battery packs. Journal of Power Sources 161(2): 1356--1384. Piller S, Perrin M, and Jossen A (2001) Methods for state-of-charge determination and their applications. Journal of Power Sources 96: 113--120. Santhanagopalan S and White RE (2006) Online estimation of the stateof-charge of a lithium ion cell. Journal Power Sources 161(2): 1346--1355. Singh P, Fennie C Jr, and Reisner D (2004) Fuzzy logic modelling of state-of-charge and available capacity of nickel/metal hydride batteries. Journal of Power Sources 136(2): 322--333. Verbrugge M, Frisch D, and Koch B (2005) Adaptive energy management of electric and hybrid electric vehicles. Journal of the Electrochemical Society 152(2): A333--A342. Verbrugge M and Koch B (2006) Generalized recursive algorithm for adaptive multiparameter regression. Journal of the Electrochemical Society 153: A187--A201.