Abstract:
A method of managing a battery system, the battery system including at least one battery cell, at least one sensor configured to measure at least one characteristic of the battery cell, and a battery management system including a microprocessor and a memory, the method comprising receiving by the battery management system, from the at least one sensor at least one measured characteristic of the battery cell at a first time and at least one measured characteristic of the battery cell at a second time. The battery management system estimating, at least one state of the battery cell by applying a physics-based battery model, the physics based battery model being based on differential algebraic equations; and regulating by the battery management system, at least one of charging or discharging of the battery cell based on the at least one estimated state.

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
       [0001]    This invention was made with government support under ARPA-E Award Number DE-AR0000278 awarded by the U. S. Department of Energy. The U. S. government has certain rights in the invention. 
     
    
     FIELD 
       [0002]    The invention generally relates to batteries, and more particularly to the management of a secondary battery. 
       BACKGROUND 
       [0003]    Rechargeable lithium batteries are attractive energy storage devices for portable electric and electronic devices and electric and hybrid-electric vehicles because of their high specific energy compared to other electrochemical energy storage devices. A typical lithium cell contains a negative electrode, a positive electrode, and a separator located between the negative and positive electrodes. Both electrodes contain active materials that react with lithium reversibly. In some cases, the negative electrode may include lithium metal, which can be electrochemically dissolved and deposited reversibly. The separator contains an electrolyte with a lithium cation, and serves as a physical barrier between the electrodes such that none of the electrodes are electrically connected within the cell. 
         [0004]    Typically, during charging, there is generation of electrons at the positive electrode and consumption of an equal amount of electrons at the negative electrode. During discharging, opposite reactions occur. 
         [0005]    During repeated charge/discharge cycles of the battery undesirable side reactions occur. These undesirable side reactions result in the reduction of the capacity of the battery to provide and store power. 
       SUMMARY 
       [0006]    Traditional approaches to managing the undesirable side reactions in a battery include limiting (or otherwise controlling/regulating) the rate of charge/discharge of the battery in an attempt to minimize the undesired effects. These efforts result can results in extended charge times and peak power reduction. Thus, there is a need for a system and method for the determination of the states and parameters within a secondary battery allowing the battery management system to efficiently regulate the operation of the battery. 
         [0007]    A summary of certain embodiments disclosed herein is set forth below. It should be understood that these aspects are presented merely to provide the reader with a brief summary of these certain embodiments and that these aspects are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of aspects that may not be set forth below. 
         [0008]    Embodiments of the disclosure are related to systems and methods for implementing a battery management system that estimates and predicts various states of the battery by applying, for example, an extended Kalman filter. 
         [0009]    In some embodiments, the invention provides a method of managing a battery system. The battery system includes a battery cell and a sensor configured to measure battery cell characteristics such as, for example, voltage, current, and temperature. A state of health (SOH) and state of charge (SOC) of the battery cell are estimated at a first time by applying a physics-based battery model that applies differential algebraic equations to account for physical parameters relating to the chemical composition of the battery cell. The estimated state of health and state of charge of the battery are updated based on the battery cell characteristics measured by the sensor. The operation of the battery (e.g., the charging and discharging) are then regulated by the battery management system based on the updated state of health and state of charge. 
         [0010]    In another embodiment, the invention provides a method of managing a battery system, the battery system including at least one battery cell, at least one sensor configured to measure at least one characteristic of the battery cell (such as voltage, current and temperature), and a battery management system including a microprocessor and a memory. At least one state of the at least one battery cell is estimated at a first time by applying a physics-based battery model that applies differential algebraic equations to account for physical parameters of a chemical composition of the at least one battery cell. At least one measured characteristic of the battery at a first time is received by the battery management system from the sensor. The at least one state of the at least one battery cell is then updated based on the at least one measured characteristic of the battery at the first time. The operation of the battery (e.g., the charging or discharging) is then regulated by the battery management system based on the at least one estimated state. 
         [0011]    The details of one or more features, aspects, implementations, and advantages of this disclosure are set forth in the accompanying drawings, the detailed description, and the claims below. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1A  is a block diagram of a battery system including a battery cell and a battery management system with sensing circuitry incorporated into the battery cell, in accordance with some embodiments. 
           [0013]      FIG. 1B  is a block diagram of another battery system with the sensing circuitry provided external to the battery cell, in accordance with other embodiments. 
           [0014]      FIG. 2A  is a functional block diagrams of a battery system that applies a combined estimation structure to jointly estimate both physical parameters of the battery cell and battery state information that are then used by a controller to regulate battery operation, in accordance with some embodiments. 
           [0015]      FIG. 2B  is a functional block diagram of a battery system that estimates physical parameters of the battery cell and battery state information separately, in accordance with some embodiments. 
           [0016]      FIG. 3  is a flow chart of a method for operating a battery management system to estimate battery states and parameters, in accordance with some embodiments. 
       
    
    
     DETAILED DESCRIPTION 
       [0017]    One or more specific embodiments will be described below. Various modifications to the described embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the spirit and scope of the described embodiments. Thus, the described embodiments are not limited to the embodiments shown, but are to be accorded the widest scope consistent with the principles and features disclosed herein. 
         [0018]    An embodiment of a battery system  100 A is shown in  FIG. 1A . The battery system  100 A includes an anode tab  110 A, an anode  120 A, a separator  130 A, a cathode  150 A, a cathode tab  160 A, a sensing circuitry  170 A, and a battery management system  180 A. In some examples, the separator  130 A may be an electrically insulating separator. In some embodiments, the electrically insulating separator comprises a porous polymeric film. In various embodiments the thickness dimension of the components of a battery cell  102 A may be for the anode  120 A about 5 to about 110 micrometers, for the separator  130 A less than about 50 micrometers or in certain embodiments less than about 10 micrometers, and for the cathode  150 A about 50 to about 110 micrometers. 
         [0019]    During the discharge of the battery cell  102 A, lithium is oxidized at the anode  120 A to form a lithium ion. The lithium ion migrates through the separator  130 A of the battery cell  102 A to the cathode  150 A. During charging the lithium ions return to the anode  120 A and are reduced to lithium. The lithium may be deposited as lithium metal on the anode  120 A in the case of a lithium anode  120 A or inserted into the host structure in the case of an insertion material anode  120 A, such as graphite, and the process is repeated with subsequent charge and discharge cycles. In the case of a graphitic or other Li-insertion electrode, the lithium cations are combined with electrons and the host material (e.g., graphite), resulting in an increase in the degree of lithiation, or “state of charge” of the host material. For example, x Li + +x e − +C 6 →Li x C 6 . 
         [0020]    The anode  120 A may comprise an oxidizable metal, such as lithium or an insertion material that can insert Li or some other ion (e.g., Na, Mg, or other suitable ion). The cathode  150  may comprise various materials such as sulfur or sulfur-containing materials (e.g., polyacrylonitrile-sulfur composites (PAN-S composites), lithium sulfide (Li 2 S)); vanadium oxides(e.g., vanadium pentoxide (V 2 O 5 )); metal fluorides (e.g., fluorides of titanium, vanadium, iron, cobalt, bismuth, copper and combinations thereof); lithium-insertion materials (e.g., lithium nickel manganese cobalt oxide (NMC), lithium-rich NMC, lithium nickel manganese oxide (LiNi 0.5 Mn 1.5 O 4 )); lithium transition metal oxides (e.g., lithium cobalt oxide (LiCoO 2 ), lithium manganese oxide (LiMn 2 O 4 ), lithium nickel cobalt aluminum oxide (NCA), and combinations thereof); lithium phosphates (e.g., lithium iron phosphate (LiFePO 4 )). 
         [0021]    The particles may further be suspended in a porous, electrically conductive matrix that includes polymeric binder and electronically conductive material such as carbon (carbon black, graphite, carbon fiber, etc.). In some examples, the cathode may comprise an electrically conductive material having a porosity of greater than 80% to allow the formation and deposition/storage of oxidation products such as lithium peroxide (Li 2 O 2 ) or lithium sulfide, (Li 2 S) in the cathode volume. The ability to deposit the oxidation product directly determines the maximum power obtainable from the battery cell. Materials which provide the needed porosity include carbon black, graphite, carbon fibers, carbon nanotubes, and other non-carbon materials. The pores of the cathode  150 A, separator  130 A, and anode  120 A are filled with an ionically conductive electrolyte that contains a salt such as lithium hexafluorophosphate (LiPF 6 ) that provides the electrolyte with an adequate conductivity which reduces the internal electrical resistance of the battery cell. The electrolyte solution enhances ionic transport within the battery cell  120 A. Various types of electrolyte solutions are available including, non-aqueous liquid electrolytes, ionic liquids, solid polymers, glass-ceramic electrolytes, and other suitable electrolyte solutions. 
         [0022]    The separator  130 A may comprise one or more electrically insulating ionic conductive materials. In some examples, the suitable materials for separator  130 A may include porous polymers, ceramics, and two dimensional sheet structures such as graphene, boron nitride, and dichalcogenides. In certain examples the pores of the separator  130  may be filled with an ionically conductive electrolyte that contains a lithium salt such as lithium hexafluorophosphate (LiPF 6 ) that provides the electrolyte with an adequate conductivity which reduces the internal electrical resistance of the battery cell. 
         [0023]    The battery management system  180 A is communicatively connected to the battery cell  102 A. In one example, the battery management system  180 A is electrically connected to the battery cell  102 A via electrical links (e.g., wires). In another example, the battery management system  180 A may be wirelessly connected to the battery cell  102 A via a wireless communication network. The battery management system  180 A may include for example a microcontroller (with memory and input/output components on a single chip or within a single housing) or may include separately configured components, for example, a microprocessor, memory, and input/output components. The battery management system  180 A may also be implemented using other components or combinations of components including, for example, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other circuitry. Depending on the desired configuration, the processor may include one or more levels of caching, such as a level cache memory, one or more processor cores, and registers. The example processor core may include an arithmetic logic unit (ALU), a floating point unit (FPU), or any combination thereof. The battery management system  180 A may also include a user interface, a communication interface, and other computer implemented devices for performing features not defined herein may be incorporated into the system. In some examples, the battery management system  180 A may include other computer implemented devices such as a communication interface, a user interface, a network communication link, and an interface bus for facilitating communication between various interface devices, computing implemented devices, and one or more peripheral interfaces to the microprocessor. 
         [0024]    In the example of  FIG. 1A , a memory of the battery management system  180  stores computer-readable instructions that, when executed by the electronic processor of the battery management system  180 A, cause the battery management system and, more particularly the electronic processor, to perform or control the performance of various functions or methods attributed to battery management system  180 A herein (e.g., calculate a state or parameter of the battery system, regulate the operation of the battery system, detect an internal short from a dendrite formation). The memory may include any transitory, non-transitory, volatile, non-volatile, magnetic, optical, or electrical media, such as a random access memory (RAM), read-only memory (ROM), non-volatile RAM (NVRAM), electrically-erasable programmable ROM (EEPROM), flash memory, or any other digital or analog media. The functions attributed to the battery management system  180 A herein may be embodied as software, firmware, hardware or any combination thereof. In one example, the battery management system  180 A may be embedded in a computing device and the sensing circuity  170 A is configured to communicate with the battery management system  180 A of the computing device external to the battery cell  102 A. In this example, the sensing circuitry  170 A is configured to have wireless and/or wired communication with the battery management system  180 A. For example, the sensing circuitry  170 A and the battery management system  180 A of the external device are configured to communicate with each other via a network. In yet another example, the battery management system  180 A is remotely located on a server and the sensing circuitry  170 A is configured to transmit data of the battery cell  102 A to the battery management system  180 A. In the above examples, the battery management system  180 A is configured to receive the data and send the data to an electronic device for display as human readable format. The computing device may be a cellular phone, a tablet, a personal digital assistant (PDA), a laptop, a computer, a wearable device, or other suitable computing device. The network may be a cloud computing network, a server, a wireless area network (WAN), a local area network (LAN), an in-vehicle network, a cloud computing network, or other suitable network. 
         [0025]    The battery management system  180 A is configured to receive data from the sensing circuitry  170 A including current, voltage, and/or resistance measurements. Battery management system  180 A is also configured to determine a condition of the battery cell  102 A. Based on the determined condition of battery cell  102 A, the battery management system  180 A may alter the operating parameters of the battery cell  102 A to maintain the internal structure of the battery cell  102 A. The battery management system  180 A may also notify a user of the condition of the battery cell  102 A. 
         [0026]    In other embodiments, the physical placement and configuration of various components may be modified. For example,  FIG. 1B  illustrates another example of a battery system  100 B that includes a battery cell  102 B, an anode tab  110 B, an anode  120 B, a separator  130 B, a cathode  150 B, a cathode tab  160 B, a sensing circuitry  170 B, and a battery management system  180 B. However, in the example of  FIG. 1B  the sensing circuitry  170 B is external to the battery cell  102 B and may be incorporated within the same housing as the battery management system  180 B. 
         [0027]    In some embodiments the battery cell  102 A,  102 B may be closed system. In such a system after the battery cell  102 A,  102 B is produced the casing is sealed to prevent external elements, such as air and moisture, from entering the battery cell  102 A,  102 B and potentially causing degradation of components resulting in reduced performance and shorter life. In the discussion below, examples that refer to components in both battery system  100 A and battery system  100 B will use the reference numeral without the A or B designation (e.g., anode  120  instead of anode  120 A and anode  120 B). 
         [0028]    However, a closed battery cell  102  presents various challenges to the battery management system  180 . The closed system does not allow the direct observation of the condition of the components of the battery cell  102 . Instead, conditions as monitored and measured by the sensing circuitry  170  may be processed or evaluated to determine various characteristics of the battery cell  102 , such as voltage, current, resistance, power, temperature and combinations thereof, during operation or while at rest, and pass those measured characteristics to a battery management system  180  which can interpret the measured characteristics in order to determine the condition of the battery cell  102 . 
         [0029]    Various models have been developed to model the electrochemical reactions occurring within the battery cell  102 . One example, was developed by Fuller, Doyle, and Newman, (the Newman Model), ( J. Electrochem. Soc.,  Vol. 141, No. 1, January 1994, pp. 1-10), the contents of which are hereby incorporated by reference in their entirety. The Newman Model provides a mathematical model which can be used to estimate the electrochemical processes occurring within the battery cell  102 B based on the measured characteristics. 
         [0030]    The charge transfer reactions at the anode  120 , and cathode  150 , may be modelled by an electrochemical model, such as the Newman Model, providing the basis to describe various battery cell  102  parameters during both the charging and discharging of the battery cell  102 . For example, the Newman Model may allow the estimation of various parameters including cathode particle radius, which can vary due to the degree of lithiation of the cathode  150 , which also may be called the state-of-charge of the battery cell  102 , anode particle radius, ion diffusion rates in the anode  120 , cathode  150 , and electrolyte, intercalation current and transference number, solution conductivity in the anode  120 , cathode  150 , and electrolyte, cell porosity of the anode  120  and cathode  150 , and equilibrium potential of the anode  120  and cathode  150 . 
         [0031]    Physics based electrochemical models, such as the Newman Model, may include ordinary and partial differential equations (PDEs) to describe the behavior of the various parameters within the battery cell  102 . The Newman Model is an electrochemical-based model of the actual chemical and electrical processes occurring in the Li-ion batteries. However, the full Newman Model is extremely complex and requires a large number of immeasurable physical parameters to be identified. Identification of such large set of parameters involved in the nonlinear PDE and differential algebraic equations (DAEs) with current computational capacity is impractical. This gives rise to various electrochemical models that approximate the dynamics of the Newman Model. 
         [0032]    For example, the Reduced-Order-Model (ROM), Mayhew, C.; Wei He; Kroener, C.; Klein, R.; Chaturvedi, N.; Kojic, A., “Investigation of projection-based model-reduction techniques for solid-phase diffusion in Li-ion batteries,” American Control Conference (ACC), 2014, pp. 123-128, 4-6 Jun. 2014, the contents of which are hereby incorporated by reference in their entirety, allows the model order reduction of the Newman Model of Li-ion cells while retaining the complete model structure of the of the baseline cell. The ROM of the Newman Model is able to accurately predict behavior of a truth model, compared to less realistic approximate models such as Single Particle Model, while reducing computation time and memory requirements. The Newman Model reduction by ROM, introduces a large number of states and parameters involved in highly nonlinear partial differential equations and differential algebraic equations of the ROM dynamical system. This contributes to the complexity of the parameter and state identification process. Herein we describe methods of parameter and state estimation for the highly nonlinear and complex ROM. These methods are based on online reception of measurement data and achieve a high speed of estimation. 
         [0033]    One specific method that can be used for state and parameter is the Extended Kalman Filter (EKF). An EKF describes the process model as a nonlinear time varying model in discrete time, but uses a local linearization at each time step. The set of outputs from the electrochemical model via the EKF can include estimation of both rapidly varying states of the battery cell  102  and estimation of slowly varying parameters of the battery cell  102 . In some embodiments the state of the battery cell  102  in combination with the present input to the mathematical model allows the model to predict the present output of the battery cell  102 . States of a battery cell may for example include the state-of charge (e.g., for a lithium battery the degree of lithiation) or overpotentials. Parameters of the battery cell  102  typically vary more slowly over time than the states of the battery cell  102 . Additionally, a parameter may not be required for the model to predict the present output of the battery cell  102 . Instead knowledge of the parameters of battery cell, which may be called the state-of-health of the battery, relate to the long term functioning of the battery cell  102 . Additionally, some embodiments comprise parameters which are not directly determinable from the measurement of the current battery cell  102  characteristics. Examples of battery cell  102  parameters include the volume fractions of active materials in the anode and cathode, total cyclable lithium in the cell, electrolyte conductivity and radii of particles in the electrodes. 
         [0034]    An embodiment of a battery system  200 A is shown in  FIG. 2A . The battery system  200  includes a battery management system  205 A that includes a closed loop control module  210 A and a state and parameter estimator  220 A. The closed loop control module  210 A comprises a feedforward module  212 A and a feedback module  214 A. The battery system  200 A additionally comprises a battery  290 A which is in operable communication with the battery management system  205 A. In some embodiments the battery  290  may comprise one or more battery cells  102 . The battery system  200 A may be in operable communication with external sources of inputs to the battery system  200 A. A desired output  230 A by an external source may be input to the battery management system  205 A. An open-loop command  240 A may also be applied to the battery system  200 A via the battery management system  205 A and/or directly to the battery  290 A. 
         [0035]    The battery management system  205 A can comprise the components previously described for the battery management system  180  of  FIG. 1A . Additionally, in certain embodiments of battery system  200 A the battery management system  205 A comprises a closed loop control module  210 A which further comprises a feedforward module  212 A and a feedback module  214 A. The feedforward module  212 A is in operable communication with the state and parameter estimator  220 A as well as with external sources which can provide as inputs various commands such as desired outputs  230 A and/or open loop commands  240 A. The feedforward module  212 A may provide at least one control signal to the battery  290 A. The feedforward module  212  may also provide at least one control signal to the state and parameter estimator  220 A. 
         [0036]    The feedback control module  214 A is in operable communication with the state and parameter estimator  220 A. The feedback control module  214 A receives the estimated states and parameters calculated by the state and parameter estimator  220 A and may provide at least one control signal to the battery  290 A. The feedback control module  214 A may also provide at least one control signal to the state and parameter estimator  220 A. 
         [0037]    Another embodiment of a battery system  200 B is shown in  FIG. 2B .  FIG. 2B  is identical to  FIG. 2A  except the state and parameter estimator  220 A of  FIG. 2A  is replaced by a state estimator  222 B and a parameter estimator  224 B. The state estimator  222 B and parameter estimator  224 B subsequently provide state and parameter estimates to the feedback control module  214 B in similar manner to that described for the state and parameter estimator  220 A of  FIG. 2A . 
         [0038]    The closed loop control module  210 B includes both a feedforward module  212 B and feedback module  214 B. The closed loop control module  210 B may also include set points received from open-loop sources, such as external sources. The closed loop control module  210 B supplies at least one control signal based on the feedforward module  212 B and feedback module  214 B. The feedforward component may be derived from a mathematical model, or from pre-determined set points. The feedback component is based on internal state and parameter estimates based on a physical model, such as from an electrochemical model of the battery  290 B. 
         [0039]    In the example of  FIG. 2A  the state and parameter estimator  220 A may be an EKF-based estimator that simultaneously estimates the states and parameters of the battery  290 A at every time step. In order to make the overall implementation less computationally intensive, dual estimation may be employed such as in the example of  FIG. 2B  (i.e., where states and parameters are estimated separately). By separating the estimation into state estimation  222  and parameter estimation  224  the computational intensity may be reduced. The rate of change of the states of the battery systems  200 A and  200 B is typically faster than the rate of change of the parameters of the battery systems  200 A and  200 B. In some embodiments the frequency of updates of the parameters of the battery system  200 B is less than the frequency of updates of the states of the battery system  200 B. 
         [0040]    In some embodiments, the separation of state and parameter estimation into separate estimators  222 B and  224 B may allow one or both of the state estimator  222 B and/or parameter estimator  224 B to be located remotely to the battery  290 B. In certain embodiments the state estimator  222 B and/or parameter estimator  224 B may be in operable communication with the other elements of the battery system  200 B by wireless communication. 
         [0041]    An example of a method of estimating parameters and states using physics-based techniques by performing a time update and a measurement update is shown in  FIG. 3 . 
         [0042]    At the first time step, an initial estimate of the states and parameters are defined as part of the time update (step  310 ). The Jacobian of the system is evaluated (step  320 ) and an initial estimate of the noise covariance matrix is determined (step  330 ). At time t, given the current state and parameter estimates based on the most recent measurement, the time update (step  310 ) involves simulating the physics-based battery model over one time step, (t+1). 
         [0043]    For this example, the system modelling/estimation is performed based on differential algebraic equations (DAEs)—however, the estimation may be applied to other model forms in other implementations. The DAE model can be represented by the equations: 
         [0000]        {dot over (x)}   1   =f ( x   1   , x   2   , I, P )   (1)
 
         [0000]      {dot over (P)}=0   (2)
 
         [0000]      0= g ( x   1   , x   2   , I, P )   (3)
 
         [0000]        y=h ( x   1   , x   2   , I, P )   (4)
 
         [0000]    where x 1  represents the differential states, x 2  represents the algebraic states, P represents the parameter vector and I represents the current inputs that are used to control battery operation. 
         [0044]    As noted above, updating the covariance matrix (step  330 ) requires derivation of the Jacobian of the system with respect to the differential states and parameters (step  320 ). Since the model is defined in terms of a set of DAEs rather than ordinary differential equations (ODEs), a modified Jacobian is used as represented by the equation: 
         [0000]    
       
         
           
             
               
                 
                   
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                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               f 
                               
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                   ( 
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         [0000]    where f x1 , f x2 , f P , g x1 , g x2 , and g P  represent derivatives of f and g with respect to x 1 , x 2  and P. These derivatives can be obtained either analytically based on the functional forms of f and g, or numerically through simulation of the model. 
         [0045]    After the Jacobian of the system is derived (step  320 ), the noise covariance matrix is then updated (step  330 ) as represented by the equation: 
         [0000]        COV   t+1|t   =J   f   COV   t|t   J   f   T   +Q    (6)
 
         [0000]    where COV t+1|t  is the covariance at t+1, J f  is the Jacobian of f, J T   f  is the transpose of the Jacobian of f, COV t|t  is the covariance at t, and Q is a positive definite matrix that can either be fixed, or time-varying. 
         [0046]    The time-update estimate is further updated in the measurement update step based on available measurements of the characteristics of the battery  290  or battery cell  102  (step  370 ). The characteristics may include, but are not limited to, a voltage, current, resistance, temperature, or combinations thereof of the battery  290  or battery cell  102 . As subsequent measurements are obtained, the model prediction can be used to obtain a measurement error  350  (i.e., an error between a measured characteristic of the battery  340  and a corresponding value as determined by the DAE model) (step  350 ). This calculated measurement error is then used by the system to update and refine the measurement update (step  370 ) as discussed in further detail below. 
         [0047]    Again, with a DAE based system, a modified Jacobian (J h ) is derived (step  260 ) with respect to the output equation as represented by the equation: 
         [0000]    
       
         
           
             
               
                 
                   
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                   = 
                   
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                               h 
                               
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         [0000]    where h x1 , h x2 , h P , g x1 , g x2 , and g P  represent derivatives of h and g with respect to x 1 , x 2  and P. These derivatives can be obtained either analytically based on the functional forms of h and g, or numerically through simulation of the model. 
         [0048]    During the measurement update, the covariance matrix (obtained in step  330 ) and the Jacobian (J h ) (obtained in step  360 ) are used to derive the Kalman gain (K k ), which is represented by the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     K 
                     k 
                   
                   = 
                   
                     
                       
                         COV 
                         
                           t 
                           + 
                           
                             1 
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                   8 
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         [0000]    where K k  is the Kalman gain, R is a measure of the noise associated with each element. The other variables are as defined above for Equation (6). 
         [0049]    The measurement update (step  370 ) causes the Kalman gain (K k ) to act on the measurement error to then provide up-to-date, error corrected state and parameter estimates as represented by the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0050]    The covariance matrix for t+1 is then updated again based on the measurements at t+1 (step  380 ), as represented by the equation: 
         [0000]        COV   t+1|t+1 =( I−K   k   J   h ) COV   t+1|t    (10)
 
         [0051]    The updated states, parameters, and COV are then used as the basis (t) for the next time update  310  as the next iteration of the estimation is performed. 
         [0052]    While the process of  FIG. 3  describes the joint estimation of states and parameters such as in the example of  FIG. 2A , it can be extended to a dual estimation framework of  FIG. 2B  where the states and parameters are estimated separately in similar fashion with two different EKFs. In the discussion below, examples that refer to components in both battery system  200 A and battery system  200 B will use the reference numeral without the A or B designation. Simulation and experimental data shows that different parameters and states of the electro-chemical model have different noise levels and different influence on the output. The noise and influence levels depend on the battery&#39;s state of operation. Various notions of state/parameter sensitivity can be used to determine which states/parameters are most critical to estimate, as well as to determine the noise covariance matrices that are used to evaluate the filter gains on each time step. For example, different notions of states and parameters&#39; sensitivity that may be employed include a) partial derivatives of output versus states and parameters, and b) variations in the output over one drive cycle due to perturbation in states and parameters to evaluate the filter gain during each time step. 
         [0053]    The accurate estimation of the states and parameters of a battery  290  may allow the battery management system  205  to regulate the operation of the battery  290  such that the life and performance of the battery will be enhanced. For example, the battery management system  205  by minimizing the change in parameters of the battery  290  may allow the battery  290  to undergo an increased number of charge/discharge cycles prior to replacement. In some embodiments the battery management system  205  may regulate the charging of the battery  290  to allow for the efficient intercalation of the oxidizable species rather than deposition on the surface of the electrode. This may minimize the formation of dendrites thus limiting the possibility of the formation of an internal short within the battery  290 . In other embodiments the battery management system  205  may regulate the discharge of the battery  290  in order to obtain for example, the maximum total power output from the battery  290 . 
         [0054]    The embodiments described above have been shown by way of example, and it should be understood that these embodiments may be susceptible to various modifications and alternative forms. It should be further understood that the claims are not intended to be limited to the particular forms disclosed, but rather to cover all modifications, equivalents, and alternatives falling with the spirit and scope of this disclosure. 
         [0055]    It is believed that embodiments described herein and many of their attendant advantages will be understood by the foregoing description, and it will be apparent that various changes may be made in the form, construction and arrangement of the components without departing from the disclosed subject matter or without sacrificing all of its material advantages. The form described is merely explanatory, and it is the intention of the following claims to encompass and include such changes.