Electrochemical Model Based Observer Design for a ... - IEEE Xplore

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 2, MARCH 2013

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Electrochemical Model Based Observer Design for a Lithium-Ion Battery Reinhardt Klein, Nalin A. Chaturvedi, Jake Christensen, Jasim Ahmed, Rolf Findeisen, and Aleksandar Kojic

Abstract—Batteries are the key technology for enabling further mobile electrification and energy storage. Accurate prediction of the state of the battery is needed not only for safety reasons, but also for better utilization of the battery. In this work we present a state estimation strategy for a detailed electrochemical model of a lithium-ion battery. The benefit of using a detailed model is the additional information obtained about the battery, such as accurate estimates of the internal temperature, the state of charge within the individual electrodes, overpotential, concentration and current distribution across the electrodes, which can be utilized for safety and optimal operation. Based on physical insight, we propose an output error injection observer based on a reduced set of partial differential-algebraic equations. This reduced model has a less complex structure, while it still captures the main dynamics. The observer is extensively studied in simulations and validated in experiments for actual electric-vehicle drive cycles. Experimental results show the observer to be robust with respect to unmodeled dynamics as well as to noisy and biased voltage and current measurements. The available state estimates can be used for monitoring purposes or incorporated into a model based controller to improve the performance of the battery while guaranteeing safe operation. Index Terms—Battery management systems, electrochemical model, lithium ion batteries, PDE observer design.

I. INTRODUCTION

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HERE is an increased trend in mobile electrification in all application sectors, e.g., communication, automotive, power tools. Batteries are currently expected to be the best solution for energy storage in these devices. At the moment the most promising battery technology is based on lithium or lithium-ion (Li-ion) chemistries. The high expectations arise not only because of their high power and energy density compared to other cell chemistries, but also due to their lack of memory effect, low self discharge, and high cycle life.

Manuscript received February 17, 2011; revised September 12, 2011; accepted October 29, 2011. Manuscript received in final form November 30, 2011. Date of publication January 02, 2012; date of current version February 14, 2013. Recommended by Associate Editor Z. Wang. R. Klein is with Robert Bosch LLC, Research and Technology Center, Palo Alto, CA 94304 USA, and also with the Otto-von-Guericke University, Institute of Automation Engineering, Magdeburg 39106, Germany (e-mail: [email protected]). N. A. Chaturvedi, J. Christensen, J. Ahmed, and A. Kojic are with the Robert Bosch LLC, Research and Technology Center, Palo Alto, CA 94304 USA (e-mail: [email protected]. R. Findeisen is with the Otto-von-Guericke University, Institute of Automation Engineering, Magdeburg 39106, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2178604

The benefits of this new chemistry do not come without a cost. In particular, safety and aging of Li-ion cells are important issues which need to be addressed carefully. Li-ion battery packs usually need a complicated battery management system (BMS), which monitors the state of the battery and applies a specific charging or discharging strategy. For example, an advanced BMS heavily relies on the estimated states of the battery when deciding a specific control action [1], [2]. For this a full state estimator is needed to reconstruct the actual state of a battery based on observed voltage, current, and temperature signals. The aim of this work is to design a state observer for a battery model described by a detailed partial differential algebraic equations system (PDAE), which supplies the BMS with the needed state information to enable advanced control strategies. Using this electrochemical based model, accurate predictions of power and energy of the battery over a large range of operation are expected. Furthermore, additional knowledge of immeasurable states enables the BMS to utilize the battery more aggressively while maintaining the battery in safe and favorable conditions [1], [2]. Most BMS designs are based on simple equivalent circuit models (ECM) [3]–[5]. While this approach is successful in portable electronics, the validity of the underlying ECM is questionable for high energy and high power applications arising, for example, in the automotive industry where safety is critical for a high energy battery pack [1]. The accuracy of an ECM in predicting the input/output behavior of the battery over the entire operating regime can be improved by allowing the model parameters to vary with state of charge, temperature, and applied current. However, the complexity of the model identification problem may then be comparable with that of electrochemical models, with the difference that the identified parameters have little connection to the real physical parameters of the battery. Recently, estimation problems have been reported using empirical models [6], linearized models of the PDAE system [7], [8], and a lumped version of the PDAE system, usually referred to as the single particle model (SPM) [9], [10]. Empirical models, as used in [6] are shown to be useful for modeling and estimation purposes. However, the estimated states and parameters might not reflect the underlying physics of the battery. The same is true for reduced order linear models of the governing PDAE, as presented in [7], [8]. The approach of [9] and [10] utilizes an extended Kalman filter based on the SPM representation of the battery. However, the success of this approach is limited to an operating regime where the SPM is a good approximation of the complex behavior of the cell [1]. Especially for high power applications the accuracy of the SPM model degrades considerably.

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Fig. 1. Illustration of the lithium ion intercalation battery model. For simplicity only one type of active material is depicted. Binder, filler, conductive material, and other additives in the cell are not illustrated here since these materials are neglected in the model.

In this work we expand on the idea presented in [11], and include a temperature model for the lithium-ion battery model. As observed in [11] temperature variation affects the performance of a battery dramatically. Thus, it is important to consider the temperature as a state of the battery model, in order to improve the model’s predictive capability. The rest of this paper is structured as follows. In Section II we briefly present the full macro-homogeneous 1-D model equations of a Li-ion battery. Furthermore, a reduction of this model is introduced, such that it is suitable for estimation and control purposes. Here, we also discuss some important properties of the reduced model and compare the two models in simulations. Section III presents our approach to the design of an observer for the battery. We show that the proposed observer incorporates several structural properties of the full battery model, such as equilibrium states and conservation of mass. Furthermore, we present simulation results illustrating the performance of the proposed observer. In Section IV, we demonstrate the performance of the observer for an actual drive cycle on a commercial 18650 Li-ion battery with two active materials in the positive electrode. The developed estimation scheme is shown to be robust to noisy and biased voltage and current measurements, and to some degree also with respect to model uncertainties. Finally, in Section V we conclude the results of the work and point out future work necessary to improve the current status of estimation and control in lithium ion batteries. II. MATHEMATICAL MODEL OF THE LI-ION CELL The Li-ion battery model presented in this work is a macrohomogeneous 1D model, where variables are allowed to vary only along the -axis. However, as we shall see later, at every in the electrode, the lithium concentration in the solid phase can also vary along the radius of the active material particles [1], [2]. We restrict ourselves to explain the working of the Li-ion

battery very briefly, and refer the reader to [1], [2], [12]–[14] and the references therein for a more comprehensive introduction to electrochemical models of Li-ion batteries. Briefly, Li-ion batteries are intercalation type batteries, where intercalation and deintercalation of lithium ions in the active material of the electrodes can be modeled as a diffusion process. This model can be used to describe cells with different geometries and chemistries. Fig. 1 shows the main regions of the battery model. Note that inactive phases within the cell are not included in the illustration. The cell in Fig. 1 consists of two electrodes and a separator, where each electrode has only one type of active material that can be approximated as a continuum of spheres with a mean radius [1], [2]. The electrodes are physically separated by a porous, insulated membrane through which lithium ions ( ) can be transported, i.e., the current in the separator is only carried by ions. Lithium concentration in the active material and the electrolyte are interconnected at every distance at the solid-electrolyte interphase (SEI) by a local ionic flux of lithium ions, normal to the active particle surface. The voltage of the battery is defined as the potential difference between the two ends of the electrodes. In the following, we present the main equations relevant for estimation of a Li-ion intercalation cell with active materials in the solid phase of the electrodes. This consideration of multiple type of active materials is due to an increasing trend of cell manufacturers to produce cells with multiple active materials for improved performance, longevity, and safety. The state variables of the full battery model are lithium concentration in the electrolyte denoted by , lithium concentration in the active materials of the positive and negative electrode denoted by , potential in the electrolyte denoted by , potential in the solid phase of the positive and negative electrode denoted by , ionic current in the electrolyte denoted by , the molar ion fluxes between the active materials in the

KLEIN et al.: ELECTROCHEMICAL MODEL BASED OBSERVER DESIGN FOR A LITHIUM-ION BATTERY

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electrodes and the electrolyte denoted by , and the internal average temperature . The governing equations are given by (see also [1], [2], [12], [15])

(1) (2)

(3) (4)

Fig. 2. Open circuit potentials of the active materials in the negative and positive electrode as functions of the solid phase concentration.

of the electrolyte concentration and temperature [16]. Additionally, we consider for , , and an Arrhenius-like temperature dependency of the form

(5)

(6) In (6), the exchange current density and the overpotential for the main reaction are modeled for each material as

(7) (8) where is the th solid phase concentration evaluated at , is the open-circuit potential (OCP) of the th active material in the solid, and is the maximum possible concentration in the solid phase of the th active material which is a known constant. The internal average temperature [14] is described by

(9) where (10) is the ambient temperature, and represents the volume averaged concentration of a particle in the solid phase defined as

In the above equations, , , , , , , , , , , , and are model parameters and are constant in each region of the cell, while , and are known functions

where is a given reference temperature and is a constant parameter. The individual OCPs are measured properties of the active material in the electrodes and are assumed to be known functions. Typical shapes of the OCP function versus solid phase concentration of the materials are shown in Fig. 2. The salient feature of the OCPs presented here is that they are strict monotonic decreasing functions of the concentration. A. Model Reduction The PDE model in its current form is too complex for controls purposes. Our main goal in this section is to reduce the full PDE model to a simpler form while capturing the main dynamic features. In other words, we desire the reduced model to be simple enough for controls purposes, but detailed enough to predict the voltage response of the cell in a large operating regime. The existing approximations, e.g., ECM or SPM, provide either little information of internal states of the cell, or are valid only over a limited operating range [1]. The key assumptions to simplifying the model are a constant electrolyte concentration and an approximate solution of the diffusion equation in the active material as presented in [17]. These assumptions simplify the model dramatically, while the reduced model still compares quite well with the full model. We note that the assumptions do not affect the mass balance in the active material. It can be easily shown that the equilibrium structure of both models is the same. In our approach, (1) governing the lithium concentration in the electrolyte is reduced to , where is assumed to be a known constant. This simplifies (3) describing the potential in the electrolyte to , which now can be explicitly solved in the separator region. The solution of the PDE (2) for the mass balance in the solid material is approximated by a polynomial in the radial direction [17] in terms of the volume averaged concentration , particle surface concentration , and volume averaged concentration flux . The governing equations for the variables , , and are obtained from (2) and its initial and boundary conditions [17].

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Rewriting the reduced model for each electrode of the Li-ion battery, and using the notation in [1], we obtain for (a) Positive and negative electrode, where (11) (12)

B. Model Analysis The presented Li-ion battery model has certain properties that are important to consider when designing an observer. These properties need to be satisfied by the observer too. 1) Conservation of Mass: The total lithium, per unit area, contained in the active material of the cell is given by

(13) (22) (14) (15) (16)

and are the volume fractions of each active mawhere terial in the negative and positive electrode, respectively. We next show that for our model of the cell is a constant, i.e., . The mass conservation property can be derived by differentiating (22) with respect to time

(17) (b) Internal average temperature

(18) where , , and are given by (7), (8), and (10), written for each active material in the negative and positive electrodes, respectively. The initial conditions of the PDE system are given as

and the boundary conditions are given as (19) (20) Note that only the variables , , and have a time derivative. With as exceptions, only spatial derivatives appear in the model equations for the rest of the variables. Furthermore, the effect of the separator enters the reduced model in the boundary condition (19). The outputs of the model are the temperature and voltage, which is the potential difference in the active materials at the external boundaries of the electrodes given by (21)

where the second equality follows from (11), the third equality from (15), and the last equality from (20). 2) Equilibrium Points: We next study the equilibrium structure of our reduced cell model. At equilibrium, (11) yield the molar fluxes , where “ ” stands for equilib. From (15) and rium. Furthermore, (12) yields boundary conditions (20) we have and . Since , the input current necessarily has to be zero for equilibrium. From (18) we obtain . Also, from (17) and boundary conditions (19) we obtain and . The remaining steady state variables to be determined are the volume averaged concentrations, surface concentrations and potentials in the solid phase of each electrode. Since , (16) yield uniform potentials, i.e., . Since (16) do not have an explicit boundary condition [1], we only know that the potentials are uniform in each electrode, and thus equal to their boundary condition, but not their explicit value. To obtain their explicit


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surface. This curve is often referred to as the open circuit voltage of the cell. C. Model Comparison

Fig. 3. Equilibrium voltage of the battery as a function of the equilibrium solid phase lithium concentration of the individual electrodes. The black line represents the open circuit voltage of the cell, which is a projection of (26) on this surface.

The numerical solution of the coupled nonlinear PDAE system is a nontrivial task by itself, since we must solve an initial value problem in time for the concentrations, while satisfying the boundary value problem for the rest of the variables at all times . For simplicity, we consider in this section a battery with only one active material in each electrode. The governing PDE’s are discretized in space, resulting in a large scale index-1 DAE system, for which, in general, it is difficult to obtain consistent initial conditions. In this work we study simulations where the battery starts from a fully relaxed state and zero current. In particular, this choice allows us to compute consistent initial conditions analytically. At complete rest, the calculation of consistent initial conditions reduces to specifying the initial uniform concentration in the electrodes and the electrolyte

values, consider the Butler-Volmer kinetics in (14). At equilibrium, these equations yield

(23) For (23) to hold, we require from (8) and (16) that

to be zero. This, in turn, yields

(24) Since

From boundary conditions (19)–(20) we directly obtain the remaining initial variables as , , , and

are by assumption strict monotonic functions we have (25)

where indicates the inverse of . Finally, from (13) we can determine . Note, not only need to satisfy (25), but are furtherhowever, that more constrained by (22), which for equilibrium implies (26) For any equilibrium concentration in the solid we get the equilibrium voltage from (21) as (27) Note that for each equilibrium voltage we have a unique set of equilibrium concentrations. Thus, we have an infinite set of equilibrium points. The equilibrium voltage as a function of the solid phase concentrations is depicted in Fig. 3. For simplicity we assume here only one type active material in the negative and positive electrode. Since for the given model and satisfy (26), in (27) reduces to a single line on the surface . The black line in Fig. 3, which is the equilibrium voltage of the cell, represents a projection of (26) on this

The reference model equations are solved on a dense mesh using the Crank-Nicholson discretization scheme. The numerical code is set up such that important model properties, i.e., mass and charge conservation are maintained. The reduced PDAE model is solved with the same implicit numerical scheme with similar convergence criteria. In both cases we provide the solver with the needed Jacobians. The extra coding effort is rewarded with fast and accurate simulations of the model. The convergence criteria are usually met already in about three to four Newton iterations at each time step. The simulated voltage response to several discharge rates is presented in Fig. 4. All simulations start from an initial equilibrium voltage of 4.2 V at 25 C and stop when the cell potential reaches 2.9 V. The discharge rates presented in Fig. 4 range from C/25 up to 10C. The C-rate is defined in terms of time needed to charge or discharge a battery between specified lower and upper voltage limits. A 1C discharge rate would discharge the battery within approximately 1 hour; see [1] for an explanation of C-rate. For lower C-rates, the voltage error of the reduced model is negligible and, even at higher C-rates up to 5C, which is the highest expected C-rate in an electric vehicle application, the voltage error is still acceptable with a maximum error of less than 30 mV. At 10C the neglected electrolyte dynamics become important in predicting the performance of the battery.

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Fig. 4. Voltage response for discharge rates of C/25, 1C, 2C, 5C, and 10C between voltage limits of 4.2 and 2.9 V. Dotted lines are obtained from simulations with the reduced battery model. Note that both models predict the essentially same discharge capacity.

The instantaneous voltage drop is, however, the same for both models, since the gradients in the electrolyte concentration have not developed yet. It is worth mentioning, that both models still predict essentially the same discharge capacity. This is particularly important in electric vehicle applications. The main reason for the error in voltage are the neglected electrolyte dynamics in our reduced model, whereas errors introduced by the approximation of the solid phase model and spatial discretization are negligible. Fig. 5 shows the distribution of the electrolyte concentration during the 5C discharge. Note that in contrast to the constant electrolyte concentration assumed in the reduced model, in the full model quickly forms a gradient over the whole cell sandwich with deviations of more than 30% from the equilibrium value. In systems with poor electrolyte properties these gradients may form at lower C-rates, restricting the validity of the reduced model to lower C-rates. Thus, the assumption of constant needs to be investigated numerically for the particular electrolyte chemistry and overall cell design, including considerations of expected C-rates for that particular application. Note that the potential in the electrolyte is influenced by the gradient in the electrolyte concentration, and thus, neglecting the electrolyte dynamics increases the error in the potential calculations. This is especially important for the calculation of the reaction driving force, also known as reaction overpotential. Similar to the main reaction overpotential in (8), we define the side reaction overpotential [1] as

where is the (known) open circuit potential of the material involved in that specific side reaction. Although no side reactions are included in the current model, estimates of provide insight into whether that side reaction becomes important

Fig. 5. Electrolyte concentration distribution during a 5C discharge. At such a is far from being uniform. Thus, at very high rates high discharge rate introduces large errors. our assumption of constant

Fig. 6. Infinity norm on the spatial domain of the error in the overpotential between the full and the reduced model during a 5C discharge.

or not. Furthermore, this knowledge enables the BMS to avoid operating regimes where known side reactions start to become important [1]. We define the maximum local error of the side reaction overpotential as (28) Fig. 6 depicts the time evolution of during a 5C discharge in the negative and positive electrode. We decided to present 5C charge since the error between the full and the reduced model is expected to be largest when input currents are high. The 15 mV prediction error of the reduced model gives confidence in utilizing the estimates in a BMS algorithm. The distribution of lithium within an electrode depends on the local potentials in the solid and electrolyte phases. Since the local potentials of the full and the reduced model are slightly different, this implies that the surface concentration distribution will not be identical either. Note, however, that the overall content of lithium within each electrode is the same for the reduced and the full model, since the mass-conservation equations remained unchanged.


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The initial conditions for our observer are given as

III. OUTPUT INJECTION-BASED PDE OBSERVER In this section we first present our approach to design an observer for a Li-ion battery based on the reduced model presented in the previous section. The performance of the observer is then studied in simulations. A. Observer Design In our design of the observer we want to feedback the error between the measured and the calculated voltage and temperature, such that all estimated variables converge to the true variables [11]. Based on physical insight we propose linear corrective terms via output injection only for the volume averaged concentrations in the individual electrodes and the internal average temperature. We now present the equations of the model based observer for a cell with and active materials in the negative and positive electrode, respectively. (a) Positive and negative electrode, where

(29) (30)

(31)

(32) (33) (34) (35)

(37) (38) (39) together with boundary conditions (19)–(20). Since the observer is a PDAE system too, we require the initial conditions in (37) to be consistent and additionally to satisfy

(40) The observer output voltage is given as

(41) Note that the only differences between the model and the observer are the temperature error injection in (36) and the voltage error injection in (29). Although we do not apply a corrective term to the states, we expect the initialization error in to vanish since (30) is stable. Small errors can be reintroduced in case of model mismatch for large input currents due to mismatched calculations of . The design parameters for the observer are the gains , , and . Since the observer parameters depend on the correct temperature, should be large enough such that the estimation error in the temperature is quickly removed. The observer gains , and , as we next show, are related to each other and cannot be arbitrarily defined. 1) Conservation of Mass: In an earlier section we had obtained a conservation property for the Li-ion battery model. We will next use this property to define the gain structure for . We want to design the observer such that the mass conservation property given by (22) also holds for all time . This implies that the observer gains are to be chosen such that . Thus, as in (22)

(b) Internal average temperature

From (29) we obtain for the observer that

(36)

, , and are given by The functions (7), (8), and (10), written for each active material in the negative and positive electrodes, respectively.

(42)

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Following arguments as before, from (33) and boundary conditions (20) we obtain

The PDE given in (48) with its boundary conditions (49) is solvable if and only if (50)

(43) Substituting (43) in (42) yields

(44) Evaluating (44) we obtain that

for all time chosen as

and, in this case, has the solution . Since we know from (47) that all have the same sign, (50) implies that all . Note that since , we have . Similar arguments hold for from (47) that the positive electrode, yielding and . Substituting these results in the observer equations (29)–(36) and considering the constraint (26), it is straightforward to show that remaining equilibrium states of the observer are identical to those of the model (11)–(18). B. Observer Simulation Results

(45) . Equation (45) holds if the observer gains are

where

(46)

With this choice of we can interpret the output injection term in (29) as a fictitious flux of lithium ions from one electrode to the other, where the flux direction is dictated by the sign of the voltage error. Since the model is governed by a nonlinear PDAE system, convergence of the observer to the desired states is not easy to show and is currently under investigation by the authors. Analysis and especially stability of PDEs is an area of active research with only limited results available that are applicable to the battery problem [18], [19]. 2) Equilibrium States: Next we show that, for , the reduced model (11)–(18) and the observer (29)–(36) have the same equilibrium state. Note that this does not guarantee any asymptotic convergence, or even attractivity of the observer to the desired equilibrium. However, it does show that if the observer converges to an equilibrium, it does converge to the desired equilibrium state of the battery. Consider the negative electrode. Equating the right-hand side of (29) to zero yields (47) Suppose that are non-trivial, then (46) implies that they can only be a function of time. Then, from (33) and boundary conditions (20), we obtain (48) (49)

In this section we present simulation results that illustrate the performance of the proposed observer. For simplicity, we simulate a cell with one type of active material in each electrode. For observer studies, we consider both a high rate charge scenario and an urban dynamometer driving schedule (UDDS) [20], the latter being more common in car applications. In addition, we consider uncertain parameters, as well as noisy input and output signals, to investigate the robustness properties of the observer. In all simulations the expected noise amplitude in the voltage is assumed to be 10 mV. For the input current we consider a gain error of 2%, in addition to a C/10 bias (corresponding to 130 mA) and 25 mA white noise. Among various states of the model, one important quantity is the individual average state of charge (SOC) of the active material in each electrode defined as

The SOC of an entire electrode is then accordingly defined as

for the negative and positive electrodes, respectively. Thus, SOC is a measure of the charge contained in the active material of the electrode calculated in terms of the lithium concentration. The SOC is commonly utilized in the battery community for predicting power and energy available in the battery during operation. Thus, SOC is an important quantity for battery management systems [1]. Other quantities of interest in these simulations are the overpotential distributions across the electrodes. Accurate estimates of the SOC are needed for energy and power predictions, while knowledge of overpotentials help increase the life of a battery by avoiding operating regimes where harmful side reaction can occur [1]. In the following denotes the error between a ’measured’ quantity and the corresponding estimated quantity .


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Fig. 7. Convergence of the observer to the equilibrium point. The dashed (dotted) curves are observer trajectories with initial error of 0.2 V ( 0.2 V) in the voltage and 5 C ( 5 C) in the temperature. The black (red) curves ( ) and represent observer trajectories resulting from ( ). Note that the gains are chosen such that the temperature converges faster than the SOC.

1) Zero Current: We next show that, for a zero current, the observer converges to an equilibrium state. As shown in the previous section, the observer and the model have the same equilibrium states. The influence of the observer gains as given in (46) and on the convergence speed is illustrated in Fig. 7. The observer is initialized with 0.2 V error in the voltage and 5 C in the temperature. We present simulation results with different gains and ( ). As can be seen in Fig. 7 the convergence rate can be made arbitrary fast. While the observer gain should be chosen rather large, the observer gain should be tuned accordingly as a compromise between convergence rate and robustness w.r.t. noisy measurements and model mismatch. Similar results are obtained at different equilibrium points. 2) Non-Zero Current: In this section, we present two sets of results, where we investigate the robustness and performance of our observer with respect to noisy current and voltage measurements, current bias and model mismatch. The first set considers a 5C charge scenario. In the second set we investigate the performance of the observer for an input current profile similar to that in automotive applications. Note that the observer is based on the reduced model, while the ’measurements’ were obtained with the full model, i.e., a non-uniform electrolyte concentration. Fig. 8 presents simulation results for a 5C charging current. As can be seen, the voltage and temperature error converge rapidly to amplitudes of less than 20 mV and 0.001 C. Also, the individual SOC errors quickly converge to values less than 3% despite the large current bias. Some part of the error can be attributed to the model mismatch between the full and the reduced model. Fig. 9 presents the maximum local error in the overpotential as given in (28) between the full model and the estimated values in the negative and positive electrodes. Intuitively, the local overpotential determines the extent and direction of a reaction, and hence, a model-based control strategy can utilize this information to avoid operating regions where cell damage

Fig. 8. Fast charging with a 5C charge current. Measurement noise was added to the voltage and the current. 2% gain error and a bias of C/10 in the current is considered. “Measurements” are obtained from the full model. The voltage is tracked within less than 20 mV despite the current bias, while the errors in SOC fall quickly to values less than 3%. Note that the temperature increases about 10 C during the charge process. The initial 5 C error in the observer temperature is quickly corrected for.

Fig. 9. Infinity norm of overpotential error across the negative (upper plot) and positive (lower plot) electrode during a 5C charge. The -axis is limited to 0.05 V for illustrative purposes. The predicted overpotentials are within less than 15 and 40 mV in the negative and positive electrode, respectively.

might occur [1], [21]. Cathodic side reactions occurs whenever , while anodic side reactions occur whenever . For example during charging, lithium plating on the graphite is a damaging cathodic side reaction, and is characterized by . As can be seen in Fig. 9, the estimation error in in the negative electrode drops to values less than 15 mV and hence, it is safe to charge as long as the estimated side reaction overpotential 15 mV.

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Fig. 11. Maximum local error between estimated and “measured” overpotentials during a UDDS simulation with measurement noise and current bias. Dotted lines represent observer trajectories where 10% error in the parameter is introduced.

Fig. 10. Simulation results with a UDDS current profile. The SOC error in the negative and positive electrodes quickly falls to values less than 3%, while the voltage and temperature errors drop to less than 20 mV and 0.01 C, respectively. Dotted lines represent observer trajectories with 10% error in the . Note that while the voltage and temperature tracking remains parameter . unchanged, there is a large estimation error for

study the robustness of our observer with respect to this parameter. This parameter can be directly correlated to capacity fade. Fig. 10 presents simulation results of the UDDS input profile as before. However, this time we introduce 10% error in the total lithium of the cell . Note that the estimated SOC of the individual electrodes have a large bias, with a maximum error of almost 12% for . Although Fig. 10 might suggest that can be still correctly estimated, the estimation error depends on the operating regime. In addition to the larger SOC error, the error in the estimated overpotential in the negative electrode increases in this scenario to 30 mV, as shown in Fig. 11. IV. EXPERIMENTAL RESULTS

In the next set, we investigate the robustness properties of the observer for an input current used in automotive applications. The UDDS profile [20] is a driving profile representative of city driving conditions. The UDDS profile is a speed profile, from which one can calculate an input current. Fig. 10 presents the simulation results for a UDDS input profile. The SOC estimation error quickly drops to values less than 3% while maintaining a voltage error of less than 20 mV. The errors increase with the magnitude of the current, mainly due to model mismatch. It is worth mentioning that, although the cumulative charge transferred is quite small for a UDDS profile, high C-rates of up to 10C occur for small periods. As we mentioned in the previous section, a C-rate of 10C would lead to considerable model mismatch. However, high currents occur for very short periods, during which large electrolyte gradients cannot build up. The maximum local estimation error of the overpotential for a UDDS input profile is presented in Fig. 11. Note that the estimation error in the negative electrode drops to values less than 15 mV, and thus, a model based control algorithm could confidently use these estimates in its strategy calculations. As with any observer design, the observer presented in this paper depends on knowledge of model parameters. Uncertainties or errors in model or model parameters lead to deviations in model performance that an observer cannot correct for. Nevertheless, we next vary an important model parameter, , to

In this section we present experimental results, demonstrating the performance of the proposed observer for a real-life Li-ion cell with actual drive cycles. Experiments were carried out on a commercial 18650 Li-ion cell, where the input current to the cell is based on a ’real-world’ hybrid electric vehicle power profile measurement. The measurements were performed in an environmental chamber at 25 C ambient temperature using an Arbin BT2000 battery tester. The cell used for experiments has two different active materials in the positive electrode, which increases the complexity of the problem due to the equations describing the second material. However, multiple-material cells can be important, since there is a general trend towards multiple-material electrodes that yield higher energy and power densities, and better safety. The OCPs of the individual active materials , , and have been measured in half-cell experiments. Note that the accuracy of the OCPs directly determines the accuracy of the equilibrium state parameters of the cell. The key parameters actually fitted to the 18 650 cell are the individual volume fractions , , , and the total lithium content of the cell . These parameters define the equilibrium structure of the cell, together with the OCPs of the individual active materials. Fig. 12 presents the relaxed voltage of our 18650 Li-ion battery and the equilibrium voltage of the reduced model with two materials in the positive electrode versus discharged capacity. The remaining model parameters


Fig. 12. Comparison of the equilibrium voltage of a 18 650 cell with the reduced model equilibrium voltage.

Fig. 13. Experimental results with a current profile based on HEV-power prosince only surface temperature measurements were availfile. Note that . The SOC estimaable. The observer is initialized with 35% error in tion error remains below 3% despite the bias in the current.

have been either measured, or adopted from literature [16], [15]. The current applied to the 18 650 cell, as calculated from a measured power profile of an HEV driving aggressively in the city and on a German highway, is depicted in Fig. 13. For robustness investigations we consider again a gain error of 2%, a C/10 bias (corresponding to 140 mA), and 25 mA white noise to the input current for our observer. In addition to the noise and bias in the current, we also add white noise with variance of 10 mV

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to the output voltage. The reason for artificially adding noise is because the measurements from the Arbin unit are highly accurate compared to measurements in an electric vehicle. Thus, the magnitude of the artificially introduced noise represents noise that may be encountered in an electric vehicle setup. During the experiments we monitor the temperature of the cell by placing a thermistor on the cell can. Since the internal temperature of the cell could not be monitored in this case, we set the observer gain , and initialize the temperature for the observer to the correct value. Note that the placement of an internal temperature sensor would involve disassembly of the cell. Fig. 13 displays the measured can temperature and the observer temperature. As expected, the estimated internal temperature is higher than the can temperature, especially during the period with large input currents. As shown in Fig. 13, the observer is able to track voltage despite the bias and noise in the input current and the noise in the output voltage. Compared to the result presented in [11], the temperature variation during the experiment does not pose a problem anymore, since the observer incorporates now a temperature model. The voltage is now tracked in general within 20 mV. Larger errors of up to 50 mV occur during switching of high currents. Given accurate measurements of the current, can be monitored since it can be rewritten as a function of the applied current as

where is the calculated capacity of the negative electrode. Note that accurate current measurements are in general not available in automotive applications and any small bias would falsify the result. Additionally, this method can only be utilized to track the bulk SOC of an electrode. However, in the case of multiple active materials within an electrode, the individual SOC of these materials cannot be tracked via this method. The Arbin unit used in the experimental setup provides accurate current measurements, which enables the determination of the average SOC of an individual electrode, assuming a correct initial SOC. Since we start from a full rest, we are able to calculate the initial SOC of the electrodes from the initial equilibrium voltage using (24), (26), and (27). The estimation error in the bulk SOC of the negative electrode is presented in Fig. 13. Since we start the experiment from rest, one could initialize the observer with zero error. However, we deliberately start with an initial error of 35%, which is corrected

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fairly quick, driving the SOC estimation error to values less than 2%. Note that the observer eventually reaches an equilibrium, even with a non-zero bias in the input current. The C/10 current bias has little influence on the estimation error. Our studies (not presented here) show that even a C/2 current bias (700 mA) results in a SOC error of less than 10%. During the most aggressive part of the drive cycle, in the period of approximately 3200 to 5000 seconds, the SOC error still remains within 3%. In [11] we reported a SOC error of almost 14% during this period, caused by the large model mismatch of the isothermal model in that work. The estimation results highly depend on the accuracy of the model and the observer gain for the voltage error injection. In order to obtain similar SOC estimation results of 3% as presented in simulations, we had to decrease leading to a convergence time of approximately 80 s in the experiment compared to 20 s presented in simulations. A higher gain leads to better voltage prediction at the costs of worse SOC estimation. Among other states of the battery, the validation of the estimated overpotentials require reliable measurements of the potential distribution across the electrodes. These measurements are not trivially obtained and validating overpotentials is part of our ongoing research. V. CONCLUSION We studied a detailed model of a lithium ion intercalation battery and presented a reduced model that is simpler, but accurate enough for estimation and controls purposes. Next we design an observer for the lithium ion battery based on this reduced system of partial differential-algebraic equations. Although a detailed stability analysis of the observation error was not possible because the complexity of the problem, we presented properties of the observer, such as conservation of mass and equilibrium states. We have shown in simulations and in experiments that the observer is robust with respect to unmodeled dynamics, i.e., the observer estimates important states even though electrolyte concentration has been neglected in the model. Furthermore, the observer is robust with respect to noise in the input and output signals. Experimental results presented in this paper have confirmed that a temperature model has the potential to increase the accuracy of the estimates [11], especially for high input currents, where the internally generated heat cannot be transferred to the environment fast enough. As part of our research work, we are currently investigating the stability properties of the designed PDE observer. Since the observer accuracy depends on the accuracy of the model, a separate parameter estimation scheme is needed to account for parameter changes as the cell ages. As part of our next steps, the development of such a parameter estimation scheme is being researched. Along with parameter estimation schemes, work is also centered around model-based control strategies for optimal charging and discharging. REFERENCES [1] N. A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic, “Algorithms for advanced battery-management systems,” IEEE Control Syst. Mag., vol. 30, no. 3, pp. 49–68, Jun. 2010.

[2] N. A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic, “Modeling, estimation, and control challenges for lithium-ion batteries,” in Proc. Amer. Control Conf. (ACC), 2010, pp. 1997–2002. [3] T. Stuart, F. Fang, X. Wang, C. Ashtiani, and A. Pesaran, A modular battery management system for HEVs SAE Future Car Congr., Warrendale, PA, 2002-01-1918, 2002. [4] Y.-S. Lee and M.-W. Cheng, “Intelligent control battery equalization for series connected lithium-ion battery strings,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1297–1307, Oct. 2005. [5] M. W. Verbrugge and R. S. Conell, “Electrochemical characterization of high-power lithium ion batteries using triangular voltage and current excitation sources,” J. Power Sources, vol. 174, no. 1, pp. 2–8, 2007. [6] G. L. Plett, “Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation,” J. Power Sources, vol. 134, no. 2, pp. 277–292, Aug. 2004. [7] K. Smith, “Electrochemical control of lithium-ion batteries [applications of control],” IEEE Control Syst. Mag., vol. 30, no. 2, pp. 18–25, Apr. 2010. [8] K. Smith, C. Rahn, and C.-Y. Wang, “Model-based electrochemical estimation and constraint management for pulse operation of lithium ion batteries,” IEEE Trans. Control Syst. Technol., vol. 18, no. 3, pp. 654–663, May 2010. [9] S. Santhanagopalan and R. E. White, “Online estimation of the state of charge of a lithium ion cell,” J. Power Sources, vol. 161, no. 2, pp. 1346–1355, 2006. [10] D. D. Domenico, A. Stefanopoulou, and G. Fiengo, “Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter,” J. Dyn. Syst., Meas., Control, vol. 132, no. 6, p. 061302, 2010. [11] R. Klein, N. A. Chaturvedi, J. Christensen, J. Ahmed, R. Findeisen, and A. Kojic, “State estimation of a reduced electrochemical model of a lithium-ion battery,” in Proc. Amer. Control Conf. (ACC), 2010, pp. 6618–6623. [12] M. Doyle, T. F. Fuller, and J. Newman, “Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell,” J. Electrochem. Soc., vol. 140, no. 6, pp. 1526–1533, 1993. [13] T. F. Fuller, M. Doyle, and J. Newman, “Simulation and optimization of the dual lithium ion insertion cell,” J. Electrochem. Soc., vol. 141, pp. 1–10, 1994. [14] K. Thomas, J. Newman, and R. Darling, Mathematical Modeling of Lithium Batteries. Norwell, MA: Kluwer Academic/Plenum Publishers, 2002, pp. 345–392. [15] P. Albertus, J. Christensen, and J. Newman, “Experiments on and modeling of positive electrodes with multiple active materials for lithium-ion batteries,” J. Electrochem. Soc., vol. 156, no. 7, pp. A606–A618, 2009. -based [16] L. O. Valoen and J. N. Reimers, “Transport properties of li-ion battery electrolytes,” J. Electrochem. Soc., vol. 152, no. 5, pp. A882–A891, 2005. [17] V. R. Subramanian, V. D. Diwakar, and D. Tapriyal, “Efficient macromicro scale coupled modeling of batteries,” J. Electrochem. Soc., vol. 152, no. 10, pp. A2002–A2008, 2005. [18] J. Wu, J. Xu, and H. Zou, “On the well-posedness of a mathematical model for lithium-ion battery systems,” Methods Appl. Anal., vol. 13, no. 3, pp. 275–298, 2006. [19] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2008. [20] United States Environmental Protection Agency, Washington, DC, Jan. 2011. [Online]. Available: http://www.epa.gov/nvfel/testing/dynamometer.htm [21] P. Arora, M. Doyle, and R. E. White, “Mathematical modeling of the lithium deposition overcharge reaction in lithium-ion batteries using carbon-based negative electrodes,” J. Electrochem. Soc., vol. 146, no. 10, pp. 3543–3553, 1999. Reinhardt Klein received the Dipl.-Ing. degree in Cybernetics Engineering from the University of Stuttgart, Germany, in 2009. He is currently pursuing the Ph.D. degree at the Otto von Guericke University, Magdeburg, Germany. His research interests include modeling, estimation, and control of energy conversion systems, with a particular focus on electrochemical systems.


Nalin A. Chaturvedi received the B.Tech. and M.Tech. degrees in aerospace engineering from the Indian Institute of Technology, Bombay, India, in 2003, and the M.S. degree in mathematics and the Ph.D. degree in aerospace engineering from the University of Michigan, Ann Arbor, in 2007. He is currently with the Research and Technology Center, Robert Bosch LLC, Palo Alto, CA. His current interests include model-based and adaptive control of complex physical systems, state estimation and control with applications to systems governed by PDEs. Dr. Chaturvedi is a member of the Energy Systems subcommittee within the Mechatronics technical committee of the ASME Dynamic Systems and Controls Division and an Associate Editor of the IEEE Control Systems Society Editorial Board and Elsevier’s Journal of Aerospace Science and Technology.

Jake Christensen received the B.S. degree from the Chemical Engineering Department, California Institute of Technology, Pasadena, in 1998, and the Ph.D. degree from the Chemical Engineering Department, University of California, Berkeley, in 2005. He is currently with the Research and Technology Center of Robert Bosch LLC, Palo Alto, CA. His interests include modeling and developing new rechargeable battery systems.

Jasim Ahmed received the Ph.D. degree with a focus on spacecraft control from the University of Michigan, Ann Arbor, in 2000. He is currently with the Research and Technology Center of the Robert Bosch LLC, Palo Alto, CA. His interests include the development of model-based control for complex physical systems involving thermal-chemical-fluid interactions.

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Rolf Findeisen received the M.S. degree from the University of Wisconsin, Madison, and the Diploma and Doctorate from the University of Stuttgart, Germany. He was a Research Assistant with the Automatic Control Laboratory, ETH Zrich and a Researcher with the Institute for Systems Theory and Automatic Control, University of Stuttgart. He is currently a full Professor and head of the Systems Theory and Automatic Control Laboratory, Otto-von-Guericke University, Magdeburg, Germany. His research focuses on the method and theory development for the analysis and control of complex nonlinear systems. His main interests include optimization-based control and state estimation, set-based estimation methods with various fields of applications, spanning from robotics, batteries, to systems biology.

Aleksandar Kojic received the B.Sc. degree from the Mechanical Engineering Department, University of Kragujevac, Serbia, and the M.S.M.E. and Ph.D. degrees from the Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, in 1995, 1998, and 2001, respectively. He is currently with the Research and Technology Center, Robert Bosch LLC, Palo Alto, CA. His research interests include the area of energy storage and conversion systems.