Patent Publication Number: US-11660979-B2

Title: System and method for estimating state of charge of a lithium battery cell of a vehicle

Description:
INTRODUCTION 
     The present disclosure relates lithium batteries and, more particularly, systems and methods of estimating state of charge of lithium batteries for vehicles. 
     In the automotive industry, it has been challenging to estimate state of charge of a lithium battery cell of a vehicle. For example, Some state of charge estimations are not as reliable as desired. Moreover, relatively significant calibration effort is required for existing systems and methods. 
     SUMMARY 
     Thus, while current state of charge estimations achieve their intended purpose, there is a need for a new and improved system and method for estimating state of charge of lithium batteries for vehicles. 
     One aspect of the present disclosure includes a method for estimating state of charge of a lithium battery cell of a vehicle. The method comprises providing a non-linear function of the lithium battery cell, a normalized lithium surface density and an actual voltage at a current of the lithium battery cell having an internal resistance. The non-linear function may be represented as 
               V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡     (       c   s   +       c     s   ,   max     +       )             
where V is the actual voltage,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density, I is the current, R ohmic  is the internal resistance of the battery cell. In this example, the method further comprises determining an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function represented as
 
     
       
         
           
             
               
                 c 
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                 c 
                 
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     In this aspect, the method further comprises determining a varying parameter based on the actual lithium surface density relative to a first diffusion model. The first diffusion model may be represented as 
                     c   s   +     (   k   )       c   s     +   max         -         c   s   +     (   0   )       c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k       I   ⁡   (   n   )           =       -     I   ⁡   (   k   )       ⁢     b   0             
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     In this aspect of the present disclosure, the method further comprises determining a predicted lithium surface density and an estimated state of charge based on the varying parameter relative to a second diffusion model. The second diffusion model may be represented as 
                 x   ⁡   (     k   +   1     )     =       x   ⁡   (   k   )     -       I   ⁡   (   k   )     ⁢         and         ⁢   
         [                 c   s   +     (   k   )       c   s     +   max                   S   ⁢   o   ⁢       C   +     (   k   )             ]     =         [           h   ⁢   Δ   ⁢   t               h   ⁢   Δ   ⁢   t           ]     ⁢     x   ⁡   (   k   )       +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢       I   ⁡   (   k   )     .       where     ⁢             c   s   +       c     s   ,       max     +                   
is the normalized lithium surface density and SoC +  is the estimated state of charge.
 
     In this aspect, the method further comprises determining an estimated state of charge of the lithium battery cell when a difference between the predicted and actual lithium surface densities is less than a first threshold. 
     In another example of this aspect, the step of determining the estimated state of charge of the lithium battery cell comprises comparing the predicted lithium surface density with the actual lithium surface density in real time. Moreover, the step of determining the estimated state of charge of the lithium battery cell further comprises correcting the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is greater than a second threshold. 
     In yet another example, the step of determining the actual lithium surface density comprises applying the actual voltage to the inverse of the non-linear function. 
     In still another example, the step of determining the varying parameter comprises applying the actual lithium surface density to the first diffusion model of the lithium battery cell. 
     In yet another example of this aspect, the step of determining the predicted lithium surface density and the estimated state of charge comprises applying the varying parameter to the second diffusion model of the lithium battery cell. 
     In another example of this aspect, the method further comprises transmitting a signal of the estimated state of charge to a receiver having a display in the vehicle and displaying the estimated state of charge based on the signal. 
     Another aspect of the present disclosure, a method for estimating state of charge of a lithium battery cell of a vehicle is provided. The method comprises providing a non-linear function of the lithium battery cell, a normalized lithium surface density and an actual voltage at a current of the lithium battery cell having an internal resistance. The non-linear function may be represented as 
                 V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡   (       c   s   +       c     s   ,       max     +       )       ,         
where V is the actual voltage,
 
               c   S   +       c     s   ,       max     +           
is the normalized lithium surface density, I is the current, and R ohmic  is the internal resistance of the battery cell. In this aspect, the method further comprises determining an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function which may be represented as
 
     
       
         
           
             
               
                 c 
                 s 
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                 c 
                 
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     Moreover, the method further comprises determining a varying parameter based on the actual lithium surface density relative to a first diffusion model which may be represented as 
                     c   s   +     (   k   )       c   s     +   max         -         c   s   +     (   0   )       c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k       1   ⁢     (   n   )             =       -     I   ⁡   (   k   )       ⁢     b   0             
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     In this aspect, the method further comprises determining a predicted lithium surface density and estimated state of charge based on the varying parameter relative to a second diffusion model which may be represented as 
                 x   ⁡   (     k   +   1     )     =       x   ⁡   (   k   )     -       I   ⁡   (   k   )     ⁢         and         ⁢   
         [                 c   s   +     (   k   )       c   s     +   max                   S   ⁢   o   ⁢       C   +     (   k   )             ]     =         [           h   ⁢   Δ   ⁢   t               h   ⁢   Δ   ⁢   t           ]     ⁢     x   ⁡   (   k   )       +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢       I   ⁡   (   k   )     .       where     ⁢             c   s   +       c     s   ,       max     +                   
is the normalized lithium surface density and SoC +  is the s, max estimated state of charge.
 
     Moreover, the method further comprises determining an estimated state of charge of the lithium battery cell when a difference between the predicted and actual lithium surface densities is less than a first threshold. Furthermore, the method comprises displaying the estimated state of charge of the lithium battery cell in the vehicle to monitor battery cell state of charge. 
     In one example of this aspect, the step of determining the estimated state of charge of the lithium battery cell comprises comparing the predicted lithium surface density with the actual lithium surface density in real time. 
     In another example, the step of determining the estimated state of charge of the lithium battery cell comprises correcting the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is greater than a second threshold. 
     In another example, the step of determining the actual lithium surface density comprises applying the actual voltage to the inverse of the non-linear function. 
     In yet another example of this aspect, the step of determining the varying parameter comprises applying the actual lithium surface density to the first diffusion model of the lithium battery cell. 
     In still another example, the step of determining the predicted lithium surface density and the estimated state of charge comprises applying the varying parameter to the second diffusion model of the lithium battery cell. 
     In another example, the step of displaying comprises transmitting a signal of the estimated state of charge to a receiver having a display in the vehicle and displaying the estimated state of charge based on the signal. 
     In another aspect of the present disclosure, a system for estimating state of charge of a lithium battery cell of a vehicle is provided. The system comprises a display disposed in the vehicle for indicating the estimated state of charge of the lithium battery cell. The system further comprises a control unit disposed in the vehicle. In this aspect, the control unit is connected to the battery cell and the display. The electronic control unit is programmed to include or store a non-linear function of the lithium battery cell, a normalized lithium surface density and an actual voltage at a current of the lithium battery cell having an internal resistance. The non-linear function may be represented as 
                 V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡   (       c   s   +       c     s   ,       max     +       )       ,         
where V is the actual voltage,
 
               c   S   +       c     s   ,       max     +           
is the normalized lithium surface density, I is the current, R ohmic  is the internal resistance of the battery cell. The control unit is programmed to determine an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function represented as
 
     
       
         
           
             
               
                 c 
                 s 
                 + 
               
               
                 c 
                 
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                   , 
                     
                   max 
                 
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             = 
             
               
                 
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                   V 
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                         m 
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     In this example, the control unit is programmed to determine a varying parameter based on the actual lithium surface density relative to a first diffusion model represented as 
                     c   s   +     (   k   )       c   s     +   max         -         c   s   +     (   0   )       c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k       1   ⁢     (   n   )             =       -     I   ⁡   (   k   )       ⁢     b   0             
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     Also in this example, the control unit is programmed to determine a predicted lithium surface density and an estimated state of charge based on the varying parameter relative to a second diffusion model. The second diffusion model may be represented as 
                 x   ⁡   (     k   +   1     )     =       x   ⁡   (   k   )     -       I   ⁡   (   k   )     ⁢         and         ⁢   
         [                 c   s   +     (   k   )       c   s     +   max                   S   ⁢   o   ⁢       C   +     (   k   )             ]     =         [           h   ⁢   Δ   ⁢   t               h   ⁢   Δ   ⁢   t           ]     ⁢     x   ⁡   (   k   )       +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢     I   ⁡   (   k   )           ⁢   
     where   ⁢             c   s   +       c     s   ,       max     +               
is the normalized lithium surface density and SoC +  is the s, max estimated state of charge. In this example of this aspect, the control unit is programmed to determine an estimated state of charge of the lithium battery cell when a difference between the predicted and actual lithium surface densities is less than a first threshold.
 
     Moreover, the control unit is programmed to transmit a signal of the estimated state of charge of the lithium battery cell to the display in the vehicle. Additionally, the display is configured to indicate the estimated state of charge based on the signal. 
     In one example of this aspect, the step of determining the estimated state of charge of the lithium battery cell comprises comparing the predicted lithium surface density with the actual lithium surface density in real time. 
     In another example, the step of determining the estimated state of charge of the lithium battery cell comprises correcting the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is greater than a second threshold. 
     In yet another example, the step of determining the actual lithium surface density comprises applying the actual voltage to the inverse of the non-linear function. 
     In still another example of this aspect, the step of determining the varying parameter comprises applying the actual lithium surface density to the first diffusion model of the lithium battery cell. 
     In another example, the step of determining the predicted lithium surface density and the estimated state of charge comprises applying the varying parameter to the second diffusion model of the lithium battery cell. 
     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIG.  1    is a schematic diagram of a system for estimating state of charge of a lithium battery cell of a vehicle in accordance with one example of the present disclosure. 
         FIG.  2 A  is a graph of a voltage-normalized lithium surface density relationship in accordance with a non-linear function of the lithium battery cell in  FIG.  1   . 
         FIG.  2 B  is another graph of a actual lithium surface density-voltage relationship in accordance with an inverse of the non-linear function of the lithium battery cell in  FIG.  1   . 
         FIG.  3    is a flowchart for a method of estimating state of charge for the system in  FIG.  1   . 
         FIG.  4    is a flowchart for another method of estimating state of charge for the system in  FIG.  1   . 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     In accordance with one example of the present disclosure,  FIG.  1    depicts a system  10  for estimating state of charge of a lithium battery cell  12  of a vehicle  14 . As shown, the system  10  comprises a display  16  disposed in the vehicle  14  for indicating the estimated state of charge of the lithium battery cell  12 . The system  10  further comprises a control unit  20  disposed in the vehicle  14 . As depicted, the control unit  20  is connected to the battery cell  12  and the display  16 . 
     Generally, the control unit  20  may include an observer  22  that may be programmed to estimate the state of charge of a lithium battery cell of a vehicle. Such estimation is based on relationships between battery cell current, state of charge of the battery cell, and lithium surface density. The relationships may be described in linear models and non-linear functions discussed in greater detail below. 
     Moreover, the control unit  20  or the observer  22  of the control unit  20  estimates the state of charge by comparing actual lithium surface density and predicted lithium surface density by way of given parameters, linear models and non-linear functions discussed below. The control unit  20  or the observer  22  corrects error between the actual and predicted lithium surface densities in real time to a first predetermined percent error or first threshold which may be greater than or equal to zero. The estimated state of charge is estimated when the actual and predicted lithium surface densities are within a second threshold or second predetermined percent error. It is to be understood that either the control unit  20  or the observer  22  may be programmed as described herein without departing from the spirit or scope of the present disclosure. 
     More specifically, the control unit  20  is programmed to include or store a non-linear function of the lithium battery cell  12 , a normalized lithium surface density, and an actual voltage at a current of the lithium battery cell  12  having an internal resistance. In this example, the non-linear function may be represented as 
                 V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡   (       c   s   +       c     s   ,       max     +       )       ,         
where V is the actual voltage,
 
               c   s   +       c     s   ,       max     +           
is the normalized lithium surface density, I is the current, and R ohmic  is the internal resistance of the battery cell  12 .  FIG.  2 A  is a graph depicting a voltage-normalized lithium surface density relationship in accordance with the non-linear function.
 
     In this example, the control unit  20  is programmed to determine an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function represented as 
     
       
         
           
             
               
                 c 
                 s 
                 + 
               
               
                 c 
                 
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                   , 
                     
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                   f 
                   
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                 ( 
                 
                   V 
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                       R 
                       
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                         ⁢ 
                         h 
                         ⁢ 
                         m 
                         ⁢ 
                         i 
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                         c 
                       
                     
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     The control determines the actual lithium surface density by applying the actual voltage to the inverse of the non-linear function.  FIG.  2 B  is another graph depicting an actual lithium surface density-voltage relationship in accordance with the inverse of the non-linear function. 
     Moreover, the control unit  20  is programmed to determine a varying parameter based on the actual lithium surface density relative to a first diffusion model. The first diffusion model may be represented as 
                           c   s   +     ⁡     (   k   )         c   s     +   max         -         c   s   +     ⁡     (   0   )         c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k     ⁢     I   ⁡     (   n   )             =       -     I   ⁡     (   k   )         ⁢     b   0                               
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     In one example, the control determines the varying parameter b 0  by applying the actual lithium surface density to the first diffusion model of the lithium battery cell  12 . In this example, recursive least squares (RLS) approximation is employed using the first diffusion model to track the varying parameter b 0 . Additionally, the varying parameter b 0  is updated in real time before the next execution to improve estimation accuracy. 
     Moreover, the control unit  20  is programmed to determine a predicted lithium surface density based on the varying parameter relative to a second diffusion model of the lithium battery cell  12 . The second diffusion model is a diffusion dynamic model that includes relationships between current, state of charge and lithium surface density of the battery cell  12 . The second diffusion model may be represented as 
               x   ⁡     (     k   +   1     )       =         x   ⁡     (   k   )       -       I   ⁡     (   k   )       ⁢           ⁢     and   ⁢     
     [               c   s   +     ⁡     (   k   )         c   s     +   max                   S   ⁢   o   ⁢       C   +     ⁡     (   k   )               ]         =         [           h   ⁢           ⁢   Δ   ⁢           ⁢   t               h   ⁢           ⁢   Δ   ⁢           ⁢   t           ]     ⁢     x   ⁡     (   k   )         +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢     I   ⁡     (   k   )                   
where k is the discrete-time index, h is the known constant, Δt is the time difference between discrete-time indices, I(k) is the measured current at k time index,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density and SoC +  is the estimated state of charge, and x is a state of system. For example, the control unit  20  may determine the predicted lithium surface density by applying the varying parameter b 0  to the second diffusion model of the lithium battery cell  12 .
 
     In this example, the control unit  20  is programmed to determine an estimated state of charge of the lithium battery cell  12  when a difference between the predicted and actual lithium surface densities is less than a first threshold which may be greater than or equal to zero. In one example, the control unit  20  determines the estimated state of charge of the lithium battery cell  12  by comparing the predicted lithium surface density with the actual lithium surface density in real time. Then, the control unit  20  corrects the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is equal to or greater than a second threshold. 
     Moreover, the control unit  20  is programmed to transmit a signal of the estimated state of charge of the lithium battery cell  12  to the display  16  in the vehicle  14 . In this example, the display  16  may include a receiver (not shown) to receive the signal from the control unit  20 . Additionally, the display  16  is configured to indicate the estimated state of charge based on the signal. It is to be understood that the display  16  may indicate the estimated state of charge by any suitable manner such as a LED (light emitting diode) display without departing from the spirit or scope of the present disclosure. 
     With one example of the present disclosure,  FIG.  3    depicts a method  110  for estimating state of charge of a lithium battery cell  12  of a vehicle  14  in accordance with the system  10  of  FIG.  1   . As shown, the method  110  comprises in box  111  providing a non-linear function of the lithium battery cell  12 , a normalized lithium surface density and an actual voltage at a current of the lithium battery cell  12  having an internal resistance. The non-linear function may be represented as 
               V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡     (       c   s   +       c     s   ,   max     +       )             
where V is the actual voltage,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density, I is the current, R ohmic  is the internal resistance of the battery cell. As discussed above,  FIG.  2 A  depicts a voltage-normalized lithium surface density relationship in accordance with the non-linear function.
 
     In this example, the method  110  further comprises in box  112  determining an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function represented as 
     
       
         
           
             
               
                 c 
                 s 
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                 c 
                 
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                   max 
                 
                 + 
               
             
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                   f 
                   
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     For example, the step of determining the actual lithium surface density comprises applying the actual voltage to the inverse of the non-linear function.  FIG.  2 B  depicts an actual lithium surface density-voltage relationship in accordance with the inverse of the non-linear function. 
     In this aspect, the method  110  further comprises in box  114  determining a varying parameter based on the actual lithium surface density relative to a first diffusion model. The first diffusion model may be represented as 
                     c   s   +     ⁡     (   k   )         c   s     +   max         -         c   s   +     ⁡     (   0   )         c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k     ⁢     I   ⁡     (   n   )             =       -     I   ⁡     (   k   )         ⁢     b   0             
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     For example, the step of determining the varying parameter may comprise applying the actual lithium surface density to the first diffusion model of the lithium battery cell  12 . In this example, recursive least squares (RLS) approximation is employed using the first diffusion model to track the varying parameter b 0 . Additionally, the varying parameter b 0  is updated in real time before the next execution to improve estimation accuracy. 
     In this aspect of the present disclosure, the method  110  further comprises in box  116  determining a predicted lithium surface density based on the varying parameter relative to a second diffusion model. The second diffusion model is a diffusion dynamic model that includes relationships between current, state of charge and lithium surface density of the battery cell  12 . The second diffusion model may be represented as 
               x   ⁡     (     k   +   1     )       =         x   ⁡     (   k   )       -       I   ⁡     (   k   )       ⁢           ⁢     and   ⁢     
     [               c   s   +     ⁡     (   k   )         c   s     +   max                   S   ⁢   o   ⁢       C   +     ⁡     (   k   )               ]         =         [           h   ⁢           ⁢   Δ   ⁢           ⁢   t               h   ⁢           ⁢   Δ   ⁢           ⁢   t           ]     ⁢     x   ⁡     (   k   )         +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢       I   ⁡     (   k   )       .                 
where k is the discrete-time index, h is the known constant, Δt is the time difference between discrete-time indices, I(k) is the measured current at k time index,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density and SoC +  is the estimated state of charge, and x is a state of system. In this example, the step of determining the predicted lithium surface density comprises applying the varying parameter to the second diffusion model of the lithium battery cell  12 .
 
     In this aspect, the method  110  further comprises in box  120  determining an estimated state of charge of the lithium battery cell  12  when a difference between the predicted and actual lithium surface densities is less than a first threshold which may be greater than or equal to zero. In one example, the step of determining the estimated state of charge of the lithium battery cell  12  comprises comparing the predicted lithium surface density with the actual lithium surface density in real time. Moreover, the step of determining the estimated state of charge of the lithium battery cell  12  further comprises correcting the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is equal to or greater than a second threshold. 
     In this example, the method  110  may further comprise transmitting a signal of the estimated state of charge of the lithium battery cell  12  to the display  16 . In this example, the display  16  may include a receiver (not shown) to receive the signal from the control unit  20 . The method  110  further comprises displaying the estimated state of charge based on the signal. It is to be understood that the display may indicate the estimated state of charge by any suitable manner such as a LED (light emitting diode) display without departing from the spirit or scope of the present disclosure. 
     With another example of the present disclosure,  FIG.  4    depicts a method  210  for estimating state of charge of a lithium battery cell  12  of a vehicle  14  in accordance with the system  10  of  FIG.  1   . As shown, the method  210  comprises in box  211  providing a non-linear function of the lithium battery cell  12 , a normalized lithium surface density and an actual voltage at a current of the lithium battery cell  12  having an internal resistance. The non-linear function may be represented as 
               V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       =     f   ⁡     (       c   s   +       c     s   ,   max     +       )             
where V is the actual voltage,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density, I is the current, R ohmic  is the internal resistance of the battery cell. As discussed above,  FIG.  2 A  depicts a voltage-normalized lithium surface density relationship in accordance with the non-linear function.
 
     In this example, the method  210  further comprises in box  212  determining an actual lithium surface density based on the actual voltage relative to an inverse of the non-linear function represented as 
                 c   s   +       c     s   ,   max     +       =         f     -   1       ⁡     (     V   -       R     o   ⁢   h   ⁢   m   ⁢   i   ⁢   c       ⁢   I       )       .           
For example, the step of determining the actual lithium surface density comprises applying the actual voltage to the inverse of the non-linear function.  FIG.  2 B  depicts an actual lithium surface density-voltage relationship in accordance with the inverse of the non-linear function.
 
     In this aspect, the method  210  further comprises in box  214  determining a varying parameter based on the actual lithium surface density relative to a first diffusion model. The first diffusion model may be represented as 
                     c   s   +     ⁡     (   k   )         c   s     +   max         -         c   s   +     ⁡     (   0   )         c   s     +   max         +     h   ⁢   Δ   ⁢   t   ⁢       ∑     n   =   1     k     ⁢     I   ⁡     (   n   )             =       -     I   ⁡     (   k   )         ⁢     b   0             
where b 0  is the varying parameter dependent on temperature and aging, k is a discrete-time index, h is a known constant, Δt is a time difference between discrete-time indices, I(k) is a measured current at k time index, and n is a summation variable.
 
     For example, the step of determining the varying parameter comprises applying the actual lithium surface density to the first diffusion model of the lithium battery cell  12 . In this example, recursive least squares (RLS) approximation is employed using the first diffusion model to track the varying parameter b 0 . Additionally, the varying parameter b 0  is updated in real time before the next execution to improve estimation accuracy. 
     In this aspect of the present disclosure, the method  210  further comprises in box  216  determining a predicted lithium surface density based on the varying parameter relative to a second diffusion model. The second diffusion model is a diffusion dynamic model that includes relationships between current, state of charge and lithium surface density of the battery cell  12 . The second diffusion model may be represented as 
               x   ⁡     (     k   +   1     )       =         x   ⁡     (   k   )       -       I   ⁡     (   k   )       ⁢           ⁢     and   ⁢     
     [               c   s   +     ⁡     (   k   )         c   s     +   max                   S   ⁢   o   ⁢       C   +     ⁡     (   k   )               ]         =         [           h   ⁢           ⁢   Δ   ⁢           ⁢   t               h   ⁢           ⁢   Δ   ⁢           ⁢   t           ]     ⁢     x   ⁡     (   k   )         +       [             -     b   0       -     h   ⁢   Δ   ⁢   t               0         ]     ⁢       I   ⁡     (   k   )       .                 
where k is the discrete-time index, h is the known constant, Δt is the time difference between discrete-time indices, I(k) is the measured current at k time index,
 
               c   s   +       c     s   ,   max     +           
is the normalized lithium surface density and SoC +  is the estimated state of charge, and x is a state of system. In this example, the step of determining the predicted lithium surface density comprises applying the varying parameter to the second diffusion model of the lithium battery cell  12 .
 
     In this aspect, the method  210  further comprises in box  220  determining an estimated state of charge of the lithium battery cell  12  when a difference between the predicted and actual lithium surface densities is less than a first threshold, e.g., 0.5% In one example, the step of determining the estimated state of charge of the lithium battery cell  12  comprises comparing the predicted lithium surface density with the actual lithium surface density in real time. Moreover, the step of determining the estimated state of charge of the lithium battery cell  12  further comprises correcting the predicted lithium surface density in real time if a difference between the predicted and actual lithium surface densities is equal to or greater than a second threshold. 
     In this example, the method  210  may further comprise transmitting a signal of the estimated state of charge of the lithium battery cell  12  to the display  16 . In this example, the display  16  may include a receiver (not shown) to receive the signal from the control unit  20 . Moreover, the method  210  further comprises in box  222  displaying the estimated state of charge based on the signal. It is to be understood that the display  16  may indicate the estimated state of charge by any suitable manner such as a LED (light emitting diode) display without departing from the spirit or scope of the present disclosure. 
     The description of the present disclosure is merely exemplary in nature and variations that do not depart from the gist of the present disclosure are intended to be within the scope of the present disclosure. Such variations are not to be regarded as a departure from the spirit and scope of the present disclosure.