Patent Publication Number: US-9834112-B2

Title: Battery state of charge estimation based on reduced order electrochemical models

Description:
TECHNICAL FIELD 
     This application is generally related to battery state of charge estimation using reduced order battery models. 
     BACKGROUND 
     Hybrid-electric and pure electric vehicles rely on a traction battery to provide power for propulsion. The traction battery typically includes a number of battery cells connected in various configurations. To ensure optimal operation of the vehicle, various properties of the traction battery may be monitored. One useful property is the battery state of charge (SOC), which indicates the amount of charge stored in the battery. The SOC may be calculated for the traction battery as a whole and for each of the cells. The SOC of the traction battery provides an indication of the charge remaining. The SOC for each individual cell provides information that is useful for balancing the SOC between the cells. In addition to the SOC, battery allowable charging and discharging power limits can be used to determine the range of battery operation and to prevent battery excessive operation. 
     SUMMARY 
     A vehicle may include a battery having positive and negative electrodes. A controller may be programmed to charge and discharge the battery according to a state of charge (SOC). The SOC may be calculated from an effective lithium-ion concentration profile of one of the electrodes that is estimated by a closed-loop estimator based on an electrochemical battery model. The closed-loop estimator may be designed to estimate effective surface lithium-ion concentration of the positive electrode or negative electrode. An effective surface lithium-ion concentration of one of the electrodes may be calculated from an effective lithium-ion concentration profile of the other of the electrodes via a non-linear relationship mapping the center-to-surface lithium-ion concentration profile of the one of the electrodes to the effective surface lithium-ion concentration of the other of the electrodes. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of a hybrid vehicle illustrating typical drivetrain and energy storage components. 
         FIG. 2  is a diagram of a possible battery pack arrangement comprised of multiple cells, and monitored and controlled by a Battery Energy Control Module. 
         FIG. 3  is a diagram of a closed-loop state-based control framework in a Kalman filter system. 
         FIG. 4  is an illustration of a cross section of a metal-ion battery with porous electrodes. 
         FIG. 4A  is an illustration of Li-ion concentration profiles inside representative particles in the negative electrode resulting from the Li-ion diffusion process during discharging. 
         FIG. 4B  is an illustration of Li-ion concentration profiles inside representative particles in the positive electrode resulting from the Li-ion diffusion process during discharging. 
         FIG. 4C  is an illustration of an active material solid particle and Li-ion transfer and diffusion processes. 
         FIG. 5  is a graph of the over-potential in relation to the cell thickness in response to a 10 second current impulse input. 
         FIG. 6  is a graph of the voltage drop in the electrolyte in relation to the cell thickness in response to a 10 second current impulse input. 
         FIG. 7  is a graph illustrating an open circuit potential curve at the positive electrode and negative electrode in relation to the normalized ion concentration for the anode and cathode of an electro-chemical battery. 
         FIG. 8  is a graph illustrating battery state of charge (SOC) and estimated Li-ion concentration profiles at representative electrode particles at the positive electrode and the negative electrode in relation to time. 
         FIG. 9  is an illustration and graph of the ion concentration of an even discretization and an uneven discretization along the radius of an active material particle. 
         FIG. 10  is graph showing a terminal voltage profile of the battery and a terminal voltage profile of the battery with added noise used as the input to the closed-loop state estimator. 
         FIG. 11  is a graph showing the terminal voltage estimation error and the comparison between the terminal voltage measurement and the terminal voltage estimation of the Extended Kalman Filter. 
         FIG. 12  is a graph showing the battery SOC error and the comparison between the SOC based on current integration and the SOC estimated by the Extended Kalman Filter. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present disclosure are described herein. It is to be understood, however, that the disclosed embodiments are merely examples and other embodiments can take various and alternative forms. The figures are not necessarily to scale; some features could be exaggerated or minimized to show details of particular components. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for teaching one skilled in the art to variously employ the present invention. As those of ordinary skill in the art will understand, various features illustrated and described with reference to any one of the figures can be combined with features illustrated in one or more other figures to produce embodiments that are not explicitly illustrated or described. The combinations of features illustrated provide representative embodiments for typical applications. Various combinations and modifications of the features consistent with the teachings of this disclosure, however, could be desired for particular applications or implementations. 
       FIG. 1  depicts a typical plug-in hybrid-electric vehicle (HEV). A typical plug-in hybrid-electric vehicle  112  may comprise one or more electric machines  114  coupled to a hybrid transmission  116 . The electric machines  114  may be capable of operating as a motor or a generator. In addition, the hybrid transmission  116  is coupled to an engine  118 . The hybrid transmission  116  is also coupled to a drive shaft  120  that is coupled to the wheels  122 . The electric machines  114  can provide propulsion and deceleration capability when the engine  118  is turned on or off. The electric machines  114  also act as generators and can provide fuel economy benefits by recovering energy that would normally be lost as heat in the friction braking system. The electric machines  114  may also reduce vehicle emissions by allowing the engine  118  to operate at more efficient conditions (engine speeds and loads) and allowing the hybrid-electric vehicle  112  to be operated in electric mode with the engine  118  off under certain conditions. 
     A traction battery or battery pack  124  stores energy that can be used by the electric machines  114 . A vehicle battery pack  124  typically provides a high voltage DC output. The traction battery  124  is electrically connected to one or more power electronics modules. One or more contactors  142  may isolate the traction battery  124  from other components when opened and connect the traction battery  124  to other components when closed. The power electronics module  126  is also electrically connected to the electric machines  114  and provides the ability to bi-directionally transfer energy between the traction battery  124  and the electric machines  114 . For example, a typical traction battery  124  may provide a DC voltage while the electric machines  114  may use a three-phase AC current to function. The power electronics module  126  may convert the DC voltage to a three-phase AC current used by the electric machines  114 . In a regenerative mode, the power electronics module  126  may convert the three-phase AC current from the electric machines  114  acting as generators to the DC voltage used by the traction battery  124 . The description herein is equally applicable to a pure electric vehicle. For a pure electric vehicle, the hybrid transmission  116  may be a gear box connected to an electric machine  114  and the engine  118  may not be present. 
     In addition to providing energy for propulsion, the traction battery  124  may provide energy for other vehicle electrical systems. A vehicle may include a DC/DC converter module  128  that converts the high voltage DC output of the traction battery  124  to a low voltage DC supply that is compatible with other vehicle loads. Other high-voltage electrical loads  146 , such as compressors and electric heaters, may be connected directly to the high-voltage without the use of a DC/DC converter module  128 . The electrical loads  146  may have an associated controller that operates the electrical load  146  when appropriate. The low-voltage systems may be electrically connected to an auxiliary battery  130  (e.g., 12V battery). 
     The vehicle  112  may be an electric vehicle or a plug-in hybrid vehicle in which the traction battery  124  may be recharged by an external power source  136 . The external power source  136  may be a connection to an electrical outlet. The external power source  136  may be electrically connected to electric vehicle supply equipment (EVSE)  138 . The EVSE  138  may provide circuitry and controls to regulate and manage the transfer of energy between the power source  136  and the vehicle  112 . The external power source  136  may provide DC or AC electric power to the EVSE  138 . The EVSE  138  may have a charge connector  140  for plugging into a charge port  134  of the vehicle  112 . The charge port  134  may be any type of port configured to transfer power from the EVSE  138  to the vehicle  112 . The charge port  134  may be electrically connected to a charger or on-board power conversion module  132 . The power conversion module  132  may condition the power supplied from the EVSE  138  to provide the proper voltage and current levels to the traction battery  124 . The power conversion module  132  may interface with the EVSE  138  to coordinate the delivery of power to the vehicle  112 . The EVSE connector  140  may have pins that mate with corresponding recesses of the charge port  134 . Alternatively, various components described as being electrically connected may transfer power using a wireless inductive coupling. 
     One or more wheel brakes  144  may be provided for decelerating the vehicle  112  and preventing motion of the vehicle  112 . The wheel brakes  144  may be hydraulically actuated, electrically actuated, or some combination thereof. The wheel brakes  144  may be a part of a brake system  150 . The brake system  150  may include other components that work cooperatively to operate the wheel brakes  144 . For simplicity, the figure depicts one connection between the brake system  150  and one of the wheel brakes  144 . A connection between the brake system  150  and the other wheel brakes  144  is implied. The brake system  150  may include a controller to monitor and coordinate the brake system  150 . The brake system  150  may monitor the brake components and control the wheel brakes  144  to decelerate or control the vehicle. The brake system  150  may respond to driver commands and may also operate autonomously to implement features such as stability control. The controller of the brake system  150  may implement a method of applying a requested brake force when requested by another controller or sub-function. 
     The various components discussed may have one or more associated controllers to control and monitor the operation of the components. The controllers may communicate via a serial bus (e.g., Controller Area Network (CAN)) or via discrete conductors. In addition, a system controller  148  may be present to coordinate the operation of the various components. A traction battery  124  may be constructed from a variety of chemical formulations. Typical battery pack chemistries may be lead acid, nickel-metal hydride (NIMH) or Lithium-Ion. 
       FIG. 2  shows a typical traction battery pack  200  in a simple series configuration of N battery cells  202 . Battery packs  200 , may be composed of any number of individual battery cells connected in series or parallel or some combination thereof. A typical system may have a one or more controllers, such as a Battery Energy Control Module (BECM)  204  that monitors and controls the performance of the traction battery  200 . The BECM  204  may monitor several battery pack level characteristics such as pack current  206  that may be monitored by a pack current measurement module  208 , pack voltage  210  that may be monitored by a pack voltage measurement module  212  and pack temperature that may be monitored by a pack temperature measurement module  214 . The BECM  204  may have non-volatile memory such that data may be retained when the BECM  204  is in an off condition. Retained data may be available upon the next ignition cycle. A battery management system may be comprised of the components other than the battery cells and may include the BECM  204 , measurement sensors and modules ( 208 ,  212 ,  214 ), and sensor modules  216 . The function of the battery management system may be to operate the traction battery in a safe and efficient manner. 
     In addition to the pack level characteristics, there may be battery cell  220  level characteristics that are measured and monitored. For example, the voltage, current, and temperature of each cell  220  may be measured. A system may use a sensor module  216  to measure the characteristics of individual battery cells  220 . Depending on the capabilities, the sensor module  216  may measure the characteristics of one or multiple of the battery cells  220 . The battery pack  200  may utilize up to N c  sensor modules  216  to measure the characteristics of each of the battery cells  220 . Each sensor module  216  may transfer the measurements to the BECM  204  for further processing and coordination. The sensor module  216  may transfer signals in analog or digital form to the BECM  204 . In some embodiments, the functionality of the sensor module  216  may be incorporated internally to the BECM  204 . That is, the sensor module  216  hardware may be integrated as part of the circuitry in the BECM  204  wherein the BECM  204  may handle the processing of raw signals. 
     The battery cell  200  and pack voltages  210  may be measured using a circuit in the pack voltage measurement module  212 . The voltage sensor circuit within the sensor module  216  and pack voltage measurement circuitry  212  may contain various electrical components to scale and sample the voltage signal. The measurement signals may be routed to inputs of an analog-to-digital (A/D) converter within the sensor module  216 , the sensor module  216  and BECM  204  for conversion to a digital value. These components may become shorted or opened causing the voltage to be measured improperly. Additionally, these problems may occur intermittently over time and appear in the measured voltage data. The sensor module  216 , pack voltage sensor  212  and BECM  204  may contain circuitry to ascertain the status of the voltage measurement components. In addition, a controller within the sensor module  216  or the BECM  204  may perform signal boundary checks based on expected signal operating levels. 
       FIG. 3  is an illustration of a (Extended) Kalman Filter  300  used with a reduced order electrochemical model. Feedback loops are used to estimate system state while minimizing process and measurement noises existent in a real system. Predict state  302  feeds update state  304  with predicted values {circumflex over (x)} k   −   308 , which is calculated based on the control input and estimated states at the previous time step and measured voltage output, y k    310 , which is the measurement signal at the present time step, and feedback values P k-1|k-1    314 , which is the covariance of the estimated error of states at the previous time step. The predict state  302  is fed with control signal or current input u k-1    306  at the previous time step, feedback predicted value {circumflex over (x)} k-1   −   322  at the previous time step, and feedback value P k-1    318 . Update state  304  outputs predicted value {circumflex over (x)} k   + , and P k . 
     An example electrochemical method is disclosed.  FIG. 4  is an illustration of the cross section of the laminated structure of a Metal-ion battery cell  400  or cell. This Metal-ion battery cell  400  may be a Li-ion battery cell. The laminated structure may be configured as a prismatic cell, a cylindrical cell or other cell structure with respect to various packaging methods. The cell geometry or physical structure may be different (e.g. cylindrical, rectangular, etc.), but the basic structure of the cell is the same. Generally, the Metal-ion cell  400 , for example a Li-ion battery, includes a positive current collector  402  which is typically aluminum, but may be another suitable material or alloy, a negative current collector  404  which is typically copper, but may be another suitable material or alloy, a negative electrode  406  which is typically carbon, graphite or graphene, but may be another suitable material, a separator  408 , and a positive electrode  410  which is typically a metal oxide (e.g. lithium cobalt oxide (LiCoO 2 ), Lithium iron phosphate (LiFePO 4 ), lithium manganese oxide (LMnO 2 ), Nickel Manganese Cobalt Oxide (NMC), Nickel Manganese Cobalt Oxide (NMC)), but may be another suitable material. Each electrode ( 406 ,  410 ) may have a porous structure increasing the surface area of each electrode, in which Metal-ions (e.g. Li-ions) travel across the electrode though the electrolyte and diffuse into/out of electrode solid particles ( 412 ,  414 ). 
     There are multiple ranges of time scales existent in electrochemical dynamic responses of a Metal-ion battery  400 . For example with a Li-ion battery, factors which impact the dynamics include but are not limited to the electrochemical reaction in active solid particles  412  in the electrodes and the mass transport of Lithium-ion across the electrodes  416 . When considering these aspects, the basic reaction in the electrodes may be expressed as
 
Θ+Li++ e −⇄=Θ−Li  (1)
 
     In which Θ is the available site for intercalation, Li +  is the Li-ion, e −  is the electron, and Θ−Li is the intercalated Lithium in the solid solution. 
     This fundamental reaction expressed by equation (1) is governed by multiple time scale processes. This is shown in  FIG. 4C , in which the categories of the processes include charge transfer  416 , diffusion  418 , and polarization  420 . These terms differ from the definitions used by the electrochemical society to facilitate a reduced-order electrochemical battery model derivation. Here, the charge transfer process  416  represents the Metal-ion exchange behavior across the solid-electrolyte interface (SEI)  422  at each active solid particle ( 412 ,  414 ). The charge transfer process is fast (e.g. less than, but not limited to, 100 milliseconds) under most cases and directly affected by the reaction rate at each electrode ( 406  &amp;  410 ). There are multiple frequency components for the charge transfer, the charge transfer consists of both fast and slow dynamics, or in other words the charge transfer has frequency components less and greater than a predetermined frequency. The diffusion process  418  represents the Metal-ion transfer from the surface to the center of the solid particle or vice versa. The diffusion process is slow (e.g. greater than, but not limited to, 1 second) and is determined by the size and material of active solid particle ( 412 ,  414 ), and the Metal-ion intercalation level. There are multiple frequency components for the diffusion process, the diffusion process consists of both fast and slow dynamics, or in other words the diffusion process has frequency components less and greater than a predetermined frequency. The polarization  420  process includes all other conditions having inhomogeneous Metal-ion concentrations in the electrolyte or electrode in space. The polarization  420  caused by the charge transfer  416  and the diffusion  418  is not included in this categorization. There are multiple frequency components for the polarization, the polarization consists of both fast and slow dynamics, or in other words the polarization has frequency components less and greater than a predetermined frequency. 
     The anode  406  and cathode  410  may be modeled as a spherical material (i.e. spherical electrode material model) as illustrated by the anode spherical material  430  and the cathode spherical material  432 . However other model structures may be used. The anode spherical material  430  has a metal-ion concentration  434  which is shown in relation to the radius of the sphere  436 . The concentration of the Metal-ion  438  changes as a function of the radius  436  with a metal-ion concentration at the surface to electrolyte interface of  440 . Similarly, the cathode spherical material  432  has a metal-ion concentration  442  which is shown in relation to the radius of the sphere  444 . The concentration of the Metal-ion  446  changes as a function of the radius  444  with a metal-ion concentration at the surface to electrolyte interface of  448 . 
     The full-order electrochemical model of a Metal-ion battery  400  is the basis of a reduced-order electrochemical model. The full-order electrochemical model resolves Metal-ion concentration through the electrode thickness ( 406  &amp;  410 ) and assumes the Metal-ion concentration is homogeneous throughout the other coordinates. This model accurately captures the key electrochemical dynamics. The model describes the electric potential changes and the ionic mass transfer in the electrode and the electrolyte by four partial differential equations non-linearly coupled through the Butler-Volmer current density equation. 
     The model equations include Ohm&#39;s law for the electronically conducting solid phase which is expressed by equation (2), 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     in which φ is the electric potential, c is the Metal-ion concentration, subscript s and e represent the active electrode solid particle and the electrolyte respectively, σ eff  is the effective electrical conductivity of the electrode, κ eff  is the effective electrical conductivity of the electrolyte, κ D   eff  is the liquid junction potential term, D s  is the diffusion coefficient of Metal-ion in the electrode, D e   eff  is the effective diffusion coefficient of Metal-ion in the electrolyte, t 0  is the transference number, F is the Faraday constant, α a  is the transfer coefficient for anodic reaction, α s  is the transfer coefficient for cathodic reaction, R is the gas constant, T is the temperature, η=φ s −φ e −U(c se ) is the over potential at the solid-electrolyte interface at an active solid particle, and j 0 =k(c e ) α     a   (c s,max −c se ) α     a    (c se ) α     c   . 
     Fast and slow dynamic responses were evaluated and validated by comparing the dynamic responses to test data under the same test conditions, for example, a dynamic response under a ten second discharging pulse are computed using a full-order battery model to investigate the battery dynamic responses. 
     The analysis of the dynamic responses includes the diffusion overpotential difference and the electric potential difference of the electrolyte.  FIG. 5  is a graphical representation of the change in overpotential with respect to distance on an axis, in this example, the radius of the spherical battery model. Here, the overpotential difference between the current collectors  500  is expressed as η p | x=L −η n | x=0 . The x-axis represents the electrode thickness  502 , and the y-axis represents the overpotential  504 . At the positive current collector when a 10 sec current pulse is applied, the instantaneous voltage drop is observed. At zero seconds  506 , the voltage is influenced by the Ohmic term  508 . As time increases, as shown at 5 seconds  510 , the voltage is additional influenced by the polarization term  512  wherein the voltage is influenced by both the Ohmic and the polarization term, until the voltage influence reaches steady state as shown at time 100 seconds  514 . The voltage drop at the positive current collector is slightly changing while input current is applied. Two dominant time scales, instantaneous and medium-to-slow, are observed in the over potential difference responses. 
       FIG. 6  is a graphical representation of the change in electrolyte electrical potential (electrical potential) with respect to distance on an axis, in this example, the radius of the spherical battery model. The electrolyte electrical potential difference of the electrolyte between the current collectors  600 , expressed as φ e | x=L −φ e | x=0 , is shown in  FIG. 6 . The x-axis represents the electrode thickness  602 , and the y-axis represents the electrical potential  604 . There is an instantaneous voltage drop at zero second  606 . The instantaneous voltage drop is mainly governed by the electrical conductivity of the electrolyte  608 . The voltage change after the initial drop, as shown at 5 seconds  610 , is governed by Metal-ion transport across the electrodes  612 . The steady state potential is shown at 100 seconds  614 . The electrochemical dynamics, such as local open circuit potential, over potential and electrolyte potential, include both instantaneous-to-fast dynamics and slow-to-medium dynamics. 
     The use of the full-order dynamics in a real-time control system is computationally difficult and expensive using modern microprocessors and microcontrollers. To reduce complexity and maintain accuracy, a reduced-order electrochemical battery model should maintain data relevant to physical information throughout the model reduction procedure. A reduced-order model for battery controls in electrified vehicles should be valid under a wide range of battery operation to maintain operational accuracy. The model structure may be manipulated to a state-space form for control design implementation. Although significant research has been conducted to develop reduced-order electrochemical battery models, an accurate model has previously not been available for use in a vehicle control system. For example, single particle models typically are only valid under low current operating conditions due to the assumption of uniform Metal-ion concentration along the electrode thickness. Other approaches (relying on model coordinate transform to predict terminal voltage responses) lose physically relevant information of the electrochemical process. 
     A new approach is disclosed to overcome aforementioned limitations of previous approaches. This newly disclosed model reduction procedure is designed: (1) to capture broad time scale responses of the electrochemical process; (2) to maintain physically relevant state variables; and (3) to be formulated in a state-space form. 
     The reduction procedure starts from the categorization of electrochemical dynamic responses in a cell. The electrochemical dynamics are divided into “Ohmic” or instantaneous dynamics  506  and  606 , and “Polarization” or slow-to-medium dynamics  510  and  610 . The battery terminal voltage may be expressed by equation (7),
 
 V=φ   s | x=L φ s | x=0 ,  (7)
 
the over potential at each electrode may be expressed by equation (8),
 
η i =φ s,i −φ e,i   −U   i (θ i ),  (8)
 
in which U i (θ i ) is the open-circuit potential of electrode as a function of a normalized metal-ion concentration. From eqns. (7) and (8), the terminal voltage may be expressed by equation (9),
 
     
       
         
           
             
               
                 
                   
                     
                       
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                           ⁢ 
                           
                             - 
                             
                               η 
                               n 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                           ⁢ 
                           
                             + 
                             
                               ϕ 
                               e 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               L 
                             
                           
                           ⁢ 
                           
                             - 
                             
                               ϕ 
                               e 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The battery terminal voltage in eqn. (9) includes the open-circuit potential difference between the current collectors which may be expressed as (U p (θ p )| x=L −U n (θ n )| x=0 ), the over potential difference between the current collectors which may be expressed as (η p | x=L −η n | x=0 ), and the electrolyte electrical potential difference between the current collectors which may be expressed as (φ e | x=L −φ e | x=0 ) 
     The terminal voltage may be reduced to equation (10), 
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         = 
                           
                         ⁢ 
                         
                           
                             
                               U 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 p 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               L 
                             
                           
                           ⁢ 
                           
                             - 
                             
                               
                                 U 
                                 n 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   θ 
                                   n 
                                 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                           ⁢ 
                           
                             + 
                             
                               η 
                               p 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               L 
                             
                           
                           ⁢ 
                           
                             - 
                             
                               η 
                               n 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                           ⁢ 
                           
                             + 
                             
                               ϕ 
                               e 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               L 
                             
                           
                           ⁢ 
                           
                             - 
                             
                               ϕ 
                               e 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               U 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 p 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               L 
                             
                           
                           ⁢ 
                           
                             - 
                             
                               
                                 U 
                                 n 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   θ 
                                   n 
                                 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             | 
                             
                               x 
                               = 
                               0 
                             
                           
                           ⁢ 
                           
                             
                               + 
                               Δη 
                             
                             + 
                             
                               
                                 Δϕ 
                                 e 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7  illustrates a graphical representation of the surface potentials of the active solid particles at the current collectors  700 . The x-axis represents the normalized metal-ion concentration  702 , and the y-axis represents the electrical potential  704 . The surface potential of the anode  706  may be expressed by U n (θ n )| x=0  and the surface potential of the cathode  708  may be expressed by U p (θ p )| x=L . The x-axis represents the normalized Metal-ion concentration  706 , and the y-axis represents the surface potential in volts  708 . The difference of surface potential  710  may be expressed by U p (θ p )| x=L −U n (θ n )| x=0  in which the normalized Metal-ion concentration in each electrode is expressed as θ s,p =c s,p   eff /c s,p,max  and θ s,n   eff =c s,n   eff /c s,n,max  respectively. The normalized metal-ion concentration of the anode when the battery state of charge is at 100% is shown at point  712  and the normalized metal-ion concentration of the anode when the battery state of charge is at 0% is shown at point  714 , with an operating point at a moment in time being shown as  716 , as an example. Similarly, the normalized metal-ion concentration of the cathode when the battery state of charge is at 100% is shown at point  720  and the normalized metal-ion concentration of the cathode when the battery state of charge is at 0% is shown at point  718 , with an operating point at the moment in time being shown as  722 , as an example. Viewing a change of concentration along the anode  706  and cathode  708 , as the SOC increases, the anode operating point at a moment in time  716  moves from left to right, and the cathode operating point at the moment in time  722  moves from right to left. Due to many factors including chemistry and composition, the current operating point of the cathode  722  can be expressed as a function of the current operating point of the normalized anode concentration  716  and battery SOC. Similarly, the current operating point of the anode  716  can be expressed as a function of the current operating point of the normalized cathode concentration  722  and battery SOC. 
     The normalized Metal-ion concentration θ is mainly governed by the diffusion dynamics and slow dynamics across the electrodes. Resolving Δη and Δφ from equation (10) into “Ohmic” and “Polarization” terms is expressed by equations (11) and (12),
 
Δη=Δη Ohm +Δη polar ,  (11)
 
Δφ e =Δφ e   Ohm +Δφ e   polar .  (12)
 
     The “Ohmic” terms include instantaneous and fast dynamics, the “Polarization” terms include medium to slow dynamics. The terminal voltage of equation (10) may then be expressed as equation (13),
 
 V=U   p (θ p )| x=L   −U   n (θ n )| x=0 +Δη polar +Δφ e   polar +Δη Ohm +Δφ e   Ohm   (13)
 
     Equation (13) represents the battery terminal voltage response without loss of any frequency response component. The first four components of equation (13) are related to the slow-to-medium dynamics, including diffusion and polarization. The slow-to-medium dynamics are represented as “augmented diffusion term”. The last two components of equation (13) represent the instantaneous and fast dynamics. The instantaneous and fast dynamics are represented as “Ohmic term”. 
     The augmented diffusion term may be modeled using a diffusion equation to maintain physically relevant state variables. 
                         ∂     c   s   eff         ∂   t       =         ∇   ρ     r     ⁢     (       D   s   eff     ⁢         ∇   ρ     r     ⁢     c   s   eff         )         ,           (   14   )               
in which c s   eff  is the effective Metal-ion concentration accounting for all slow-to-medium dynamics terms, and D s   eff  is the effective diffusion coefficient accounting for all slow-to-medium dynamics terms. The boundary conditions for equation (14) are determined as
 
                           ∂     c   s   eff         ∂   r       ⁢     |     r   =   0         =   0     ,           (     15   ⁢   a     )                       ∂     c   s   eff         ∂   r       ⁢     |     r   =     R   s           =     -     I     δ   ⁢           ⁢     AFa   s     ⁢     D   s   eff             ,           (     15   ⁢   b     )               
in which A is the electrode surface area, δ is the electrode thickness, R s  is the active solid particle radius, and
 
                 a   s     =       3   ⁢     ɛ   s         R   s         ,         
in which ε s  is the porosity of the electrode. The Ohmic term is modeled as
 
− R   0   eff   I,   (16)
 
in which R 0   eff  is the effective Ohmic resistance accounting for all instantaneous and fast dynamics terms, and I is the battery current. R 0   eff  is obtained by deriving the partial differential equation (13) with respect to the battery current I and expressed as
 
                     R   0   eff     =     -       (         ∂     Δη   Ohm         ∂   I       +       ∂     Δϕ   e   Ohm         ∂   I         )     .               (   17   )               
The effective Ohmic resistance can be modeled based on equation (17), or can be determined from test data.
 
     The terminal voltage may then be expressed as
 
 V=U   p (θ se,p )− U   n (θ se,n )− R   0   eff   I,   (18)
 
in which the normalized Metal-ion concentration at the solid/electrolyte interface of the cathode is θ se,p =c se,p   eff /c s,p,max , the normalized Metal-ion concentration at the solid/electrolyte interface of the anode is θ se,n =c se,n   eff /c s,n,max , c s,p,max  is the maximum Metal-ion concentration at the positive electrode, c s,n,max  is the maximum Metal-ion concentration at the negative electrode, and c se   eff  is the effective Metal-ion concentration at the solid-electrolyte interface.
 
     Equation (18) may be expressed as three model parameters, the anode effective diffusion coefficients (D s,n   eff ), the cathode effective diffusion coefficients (D s,p   eff ), effective internal resistance of both the anode and cathode (R 0   eff ), and one state vector, the effective Metal-ion concentration (c s   eff ). The state vector effective Metal-ion concentration (c s   eff ) includes the anode state vector effective Metal-ion concentration (c s,n   eff ), which may be governed by the anode effective diffusion coefficients (D s,n   eff ), and cathode state vector effective Metal-ion concentration (c s,p   eff ), which may be governed by the cathode effective diffusion coefficients (D s,p   eff ) based on the application of equation (14). The parameters may be expressed as functions of, but not limited to, temperature, SOC, battery life, battery health and number of charge cycles applied. The parameters (D s,n   eff , D s,p   eff , R 0   eff ) may be determined by modeling, experimentation, calibration or other means. 
     Referring back to  FIG. 7 , the normalized Metal-ion concentration at the solid/electrolyte interface of the anode θ se,n  may be expressed as a function of the normalized Metal-ion concentration at the solid/electrolyte interface of the cathode θ se,p  and the battery state of charge SOC ave . An example of the augmented diffusion dynamics, as the Metal-ion concentration of the cathode at the current collector increases along the normalized Metal-ion concentration line  708  (e.g. from 0.7 to 0.8), the Metal-ion concentration of the anode at the current collector will correspondingly decreases along the normalized Metal-ion concentration line  706 . The corresponding decrease of the anode will be a function of the increase of the cathode, but may not be equal to the amount increased in the cathode. This functional relationship allows the status or operation of one electrode (i.e. a representative electrode) to provide information to determine the status or operation of the other electrode. A change of the open circuit voltage of the anode (ΔU n )  726  corresponds to a change in the normalized metal-ion concentration at the surface to electrolyte interface (Δθ se,n )  724 . 
     If the metal-ion concentration of the anode is expressed by θ se,n =f(c se,p ,SOC ave ) to relate the metal-ion dynamics at the cathode to the metal-ion dynamics at the anode, the dynamic responses of the anode may be calculated from the dynamic response of the cathode. The terminal voltage may then be expressed from eqn. (18) as
 
 V=U   p (θ se,p )− U   n ( f ( c   se,p ,SOC ave ))− R   0   eff   I,   (19)
 
where
 
θ se =θ 0% +SOC se (θ 100% −θ 0% )  (20)
 
     f(c se,p ,SOC ave ) in the second term in eqn. (19) may be calculated by
 
θ se,n   =f ( c   p,se   eff ,SOC ave )= w   1   c   p,se   eff   +w   2 SOC ave   (21)
 
c se,p  and SOC ave  is defined by
 
 c   se,p   =G   1 ( c   s   eff )  (22)
 
SOC ave   =G   2 ( c   s   eff )  (23)
 
where G 1  is a function to map c s   eff  to c se,p  and G2 is a function to map c s   eff  to SOC ave , and, the weight w 1 =(SOC ave ) m  in which m may be an exponent to tune the response and the weight w 2 =1−w 1 .
 
     By combining eqns. (22) and (23), eqn. (21) becomes
 
 f ( c   p,se   eff ,SOC ave )= w   1   G   1 ( c   s   eff )+ w   2   G   2 ( c   s   eff )  (24)
 
Then, equation (19) is expressed as
 
 V=U   p (θ se,p )− U   n ( w   1   G   1 ( c   s   eff )+ w   2   G   2 ( c   s   eff ))− R   0   eff   I   (25)
 
Equation (25) is now a function of c s   eff  with the definition of θ se,p =c se,p   eff /c s,p,max  
 
       FIG. 8  is a graphical illustration of the battery state of charge (SOC)  804  in relation to time  802 . This graphical illustration shows the average battery state of charge  806 , the battery state of charge at the solid to electrolyte interface of the cathode  808  and the battery state of charge at the solid to electrolyte interface of the anode  810 . A computed electrochemical dynamic from the model at one electrode  814 , for example the cathode, allows predicted electrochemical dynamics of the other electrode  812 , based on equations (19), (20), and (21). 
     Using equations (19), (20), and (21), different electrochemical dynamics between electrodes are captured, and the difference results in ΔSOC se,n  along the line A-A′  816 . In other words, the dynamics difference between the electrodes and resulting the difference in battery state of charge (ΔSOC se,n )  818  are captured by the proposed methodology. The difference of the normalized Li-ion concentration at the negative electrode can be computed from ΔSOC se,n    818 , and the difference results in ΔU n  in  726 . Thus, the terminal voltage in equation (19) is computed. 
     Further model reduction may be possible by reducing the number of discretization using uneven discretization. The objectives of uneven discretization are to achieve a compact model structure while maintaining the model accuracy. Thus, the uneven discretization may produce a more compact battery model form and lower the required processor bandwidth. Other model reduction approaches could capture similar battery dynamics. However, the uneven discretization can maintain physically meaningful states to represent Metal-ion diffusion dynamics. 
     Equation (14) is expressed as a set of ordinary differential equations (ODE) by using the finite difference method for the spatial variable r in order to be used as the battery control-oriented model. The derived state-space equations using uneven discretization are
 
 c &amp; s   eff   =Ac   s   eff   +Bu   (26)
 
 Y=h ( c   s   eff )= U   p (θ se,p )− U   n (θ se,n )  (27)
 
where c s   eff  is the effective Li-ion concentration n-by-1 vector accounting for the slow-to-medium dynamics terms, A is the n-by-n system matrix that characterize the slow-to-medium dynamics of the battery, B is the n-by-1 input matrix that directly relates the input to the rate of state variables, and u is the input to the system, i.e., the battery current. A is also the function of the parameters related to battery capacity and dynamics. The number of states, n, in the developed model is optimized with the consideration of the balance between the computational efficiency and prediction fidelity.
 
     A state estimator may be designed from eqns. (26) and (27). By linearizing eqn. (27), a closed-loop estimator may be designed. The linearized expression of eqn. (27) is
 
δ Y=Hδx   (28)
 
where H matrix is computed by linearizing Y around x 0  to design Extended Kalman Filter (EKF).
 
     The output matrix, H, may be derived from: 
                   H   =       ⁢         ∂   Y       ∂   x       =       ⁢         ∂       U   p     ⁡     (     θ     se   ,   p       )           ∂   x       -       ∂       U   n     ⁡     (     θ     se   ,   n       )           ∂   x                   (   29   )               
where θ se,p =c se,p   eff /c s,p,max  and θ se,n =f(c p,se   eff ,SOC ave )=w 1 c p,se   eff +w 2 SOC ave .
 
     Equation (29) may be further converted to 
                   H   =       ⁢         ∂   Y       ∂   x       =       ⁢         ∂       U   p     ⁡     (     θ     se   ,   p       )           ∂   x       -         ∂       U   n     ⁡     (     θ     se   ,   n       )           ∂   f       ⁢       ∂     f   ⁡     (     c   s   eff     )           ∂   x                     (   30   )               
Then, expressed in a form of
 
                   H   =           ∂     U   p         ∂     θ     se   ,   p           ⁢     1     c     p   ,   max         ⁢     G   1       -         ∂     U   n         ∂     θ     se   ,   n           ⁢     1     c     p   ,   max         ⁢     (         w   1     ⁢     G   1       +       w   2     ⁢     G   2         )                 (   31   )               
by combining eqns. (21), (22), and (23).
 
where
 
               ∂     f   ⁡     (     c   s   eff     )           ∂   x           
is the nonlinear function to map c s   eff  to θ se,n . The non-linear function may be determined or identified to capture the dynamics of the output as accurately as possible. Implementing this equation allows modeling the dynamics of both electrodes in the battery with the dynamics of one electrode.
 
     The resulting expression in eqn. (31) is a function of c s   eff  of only one electrode. From the estimated of ĉ s   eff , the battery SOC may be estimated from a non-linear function of eqn. (23). 
     Now referring to  FIG. 10 , a graph  1000  includes terminal voltage on the y-axis  1002  with respect to time  1004  on the x-axis. A battery terminal voltage  1008  is shown. The battery terminal voltage  1008  fluctuates over time. The terminal voltage  1008  may include noise  1006 , which may impede proper operation of the battery control module. The EKF, as disclosed above, may improve the battery terminal voltage estimation by minimizing the noise therein may recover the terminal voltage signal  1008  without noises. Noise may be caused by sensor characteristics, model mismatch, or environmental impacts. 
     One example of a resulting battery terminal voltage is depicted in  FIG. 11 .  FIG. 11  includes a graph  1100  having terminal voltage on the y-axis  1102  with respect to time  1104  on the x-axis. For clarity, a hypothetical test signal  1106  depicts the terminal voltage of the battery over time in charge depleting or charge sustaining states, as signified by line  1110 . An estimate of the terminal voltage  1108 , with removed noise from the sensor. The estimate matches the test voltage (as shown without noise). A graph  1120  shows the error  1126  of the estimation of battery terminal voltage  1108  when compared with test signal  1106 . The error  1126  is shown on the y-axis  1122  over time on the x-axis  1124 . The estimation error  1126  is mostly constant whether to the right or left of the charge depleting or charge sustaining line  1130 . 
     Now referring to  FIG. 12 , a graph  1200  depicts the battery SOC estimation. As derived from the EKF above the model SOC  1208  is shown as a percentage of SOC on the y-axis  1202  over time of the x-axis  1204 . The model SOC  1208  is shown in view of the current integration method SOC  1206 . Both operating modes, charge depleting or charge sustaining, are shown on either side of line  1210  with the estimation results of the model SOC  1208 . Graph  1200  is shown comparatively next to graph  1220 . Graph  1220  depicts the battery SOC estimation error  1226  by virtue of the model SOC  1208  estimate. The battery SOC estimation error  1226  is shown on the y-axis  1222  as a percentage of error with respect to time  1224  on the x-axis. Although minimal errors are present for charge sustaining operation, estimation errors can reach 0.5% in the charge depleting range. These operating modes are shown on either side of line  1230 . 
     The processes, methods, or algorithms disclosed herein can be deliverable to/implemented by a processing device, controller, or computer, which can include any existing programmable electronic control unit or dedicated electronic control unit. Similarly, the processes, methods, or algorithms can be stored as data and instructions executable by a controller or computer in many forms including, but not limited to, information permanently stored on non-writable storage media such as Read Only Memory (ROM) devices and information alterably stored on writeable storage media such as floppy disks, magnetic tapes, Compact Discs (CDs), Random Access Memory (RAM) devices, and other magnetic and optical media. The processes, methods, or algorithms can also be implemented in a software executable object. Alternatively, the processes, methods, or algorithms can be embodied in whole or in part using suitable hardware components, such as Application Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs), state machines, controllers or other hardware components or devices, or a combination of hardware, software and firmware components. 
     While exemplary embodiments are described above, it is not intended that these embodiments describe all possible forms encompassed by the claims. The words used in the specification are words of description rather than limitation, and it is understood that various changes can be made without departing from the spirit and scope of the disclosure. As previously described, the features of various embodiments can be combined to form further embodiments of the invention that may not be explicitly described or illustrated. While various embodiments could have been described as providing advantages or being preferred over other embodiments or prior art implementations with respect to one or more desired characteristics, those of ordinary skill in the art recognize that one or more features or characteristics can be compromised to achieve desired overall system attributes, which depend on the specific application and implementation. These attributes may include, but are not limited to cost, strength, durability, life cycle cost, marketability, appearance, packaging, size, serviceability, weight, manufacturability, ease of assembly, etc. As such, embodiments described as less desirable than other embodiments or prior art implementations with respect to one or more characteristics are not outside the scope of the disclosure and can be desirable for particular applications.