# Patent

## Title
Battery SOC estimation with automatic correction

## Abstract

An embodiment contemplates a method of determining a state-of-charge of a battery for a vehicle. (a) An OCV is measured for a current vehicle ignition startup after ignition off for at least eight hours. (b) An SOCOCV is determined for the current vehicle ignition startup. (c) An SOCOCV—est is determined for a current vehicle ignition startup. (d) A determination is made whether the difference in the SOCOCV for the current startup and the SOCOCVest for the current startup is less than a predefined error bound using. Steps (a)-(d) is performed in response to the difference being greater than the predefined error; otherwise, determining an ignition-off current for the current vehicle ignition startup as a function of the SOCOCV of the current vehicle ignition startup and previous vehicle ignition startup, and a SOC based on current integration over time. Determining an SOCest of the current vehicle ignition startup using the processor.

## Background
BACKGROUND OF INVENTION
An embodiment relates generally to external device integration within a vehicle.
Determining a state-of-charge (SOC) for a battery can be performed utilizing various techniques utilizing coulomb counting or parameter estimations techniques. Coulomb counting involves the use of one measurement (i.e., battery current) to estimate the battery state-of-charge. The accuracy of the battery current is critical to determining a state-of-charge. If there is measurement error, such as the current sensor not accurate integration error accumulates quickly. Furthermore, the coulomb counting is not carried out during the vehicle ignition off in order to save battery energy, which may bring additional SOC estimation error. Most vehicles utilize low end current and voltage sensors which do not provide accurate results. Therefore, many systems utilize high cost current sensors to monitor SOC all the time to overcome this deficiency.
SUMMARY OF INVENTION
An advantage of an embodiment is a determination of the state-of-charge (SOC) of a battery utilizing an estimation technique without the use of expensive and high accuracy sensors. The estimation technique utilizes a previous SOC estimation, a present SOC estimation, and a current integration estimation for determining an estimated SOC. The current integration utilizes an ignition-on current integration and an ignition-off current integration determination. The ignition-off integration is determined as a function of a previous open circuit voltage SOC estimation, a present open circuit voltage SOC estimation, and a current integration estimation where the previous and present open circuit voltages are based on open circuit voltage measurements after at least an 8 hour ignition-off period. A comparison is made between the present open circuit voltage SOC measurement and the previous open circuit voltage SOC measurement to determine whether the data from either SOC is skewed by the battery not being at equilibrium. If so, then a next open circuit voltage SOC will be obtained at a next ignition off for generating a next open circuit voltage SOC which may be used to determine the ignition-off current.
An embodiment contemplates a method of determining a state-of-charge of a battery for a vehicle. The vehicle is in a charging state when the engine is operating and a non-charging state when the engine is not operating, the method comprising the steps of: (a) measuring an OCV for a current vehicle ignition startup using a voltmeter, wherein the current vehicle ignition start-up is performed after the vehicle is in the non-charging state for at least eight hours; (b) determining an SOCOCV for the current vehicle ignition startup using the processor; (c) determining an SOCOCV—est for a current vehicle ignition startup using the processor; (d) determining whether the difference in the SOCOCV for the current vehicle ignition startup and the SOCOCVest for the current vehicle ignition startup is less than a predefined error bound using the processor; (e) performing steps (a)-(d) in response to the difference being greater than the predefined error; otherwise, proceeding to step (f); (f) determining an ignition-off current for the current vehicle ignition startup as a function of the SOCOCV of the current vehicle ignition startup, an SOCOCV of a previous vehicle ignition startup, and a state-of-charge based on current integration over time using the processor technique; and (g) determining an SOCest of the current vehicle ignition startup using the processor.



BRIEF DESCRIPTION OF DRAWINGS
FIG. 1 is a block diagram of a state-of-charge estimation system.
FIG. 2 is a flowchart for estimating the state-of-charge (SOC) over time.
FIG. 3 is a flowchart for identifying a first state-of-charge and a second state-of-charge.
FIG. 4 is a flowchart for identifying an ignition-off current.



DETAILED DESCRIPTION
FIG. 1 illustrates a block diagram of an embodiment of a vehicle 10 incorporating a state-of-charge (SOC) estimation system. The vehicle 10 includes a battery 12 for starting the vehicle. The battery 12 is a lead-acid battery. The battery 12 is typically made up of cells that contain electrodes (cathode and anode) of lead (Pb) and lead oxide (PbO2) in an electrolyte of sulfuric acid. A chemical reaction takes place to store energy within the battery. The concept is to convert lead sulphate that forms on the plates of a discharged battery into lead dioxide which forms the plates of a charged battery.
The vehicle battery 12 is electrically coupled to a plurality of devices 14 which utilize the battery as a power source. The vehicle 10 may further include a voltage meter 16, a current sensor 18, a temperature sensor 19, and a control module 20.
The plurality of devices 14 include, but are not limited to, power outlets adapted to an external device, accessories, components, subsystems, and systems of a vehicle. The current sensor 16 is used to monitor the current leaving the vehicle battery 12. The voltmeter 18 measures a voltage so that an open circuit voltage (OCV) may be determined. The temperature sensor 19 senses the temperature of the battery and can be used as a factor in determining the state-of-charge of the battery. A control module 20, or similar module, obtains, derives, monitors, and/or processes a set of parameters associated with the vehicle battery 12. These parameters may include, without limitation, current, voltage, state-of-charge (SOC), battery capacity, battery internal resistances, battery internal reactance, battery temperature, and power output of the vehicle battery. The control module 20 includes a processor for executing for executing a vehicle state-of-charge (SOC) estimation technique.
The control module 20 utilizes the OCV of the battery for determining the SOC. The SOC may be derived by determining the OCV and then applying OCV mapping or current integration may be applied. To accurately determine the SOC, the OCV may be accurately measured only after the OCV equilibrium is obtained, which occurs a predetermined time after battery charging has been discontinued (i.e., either by an ignition off operation or other charging device). Typically the predetermined time to obtain OCV equilibrium includes 24 hours after charging the battery is discontinued. That is, an open-circuit voltage measurement is accurate only when the battery voltage is under the equilibrium conditions.
Electrical charges on the surface of the battery's plates cause false voltmeter readings. When a battery is charged, the surface of the plates may have a higher charge than the inner portions of the plates. After a period of time after charging has been discontinued, the surface charge on the surface of the plates will become slightly discharged as a result of the charged energy penetrating deeper into the plates. Therefore, the surface charge, if not dissipated to the inner portion of the plates, may make a weak battery appear good. As a result, to obtain an accurate OCV measurement that can be used to determine the SOC, the vehicle typically must be at rest for a long duration of time >8 hours.
Furthermore, for lead acid batteries, the battery transforms the chemical energy into electrical energy as the result of a chemical reaction between the electrolyte solution and the lead of the plates. During the energy conversion and discharge of electrical energy from the battery, the acid reacts with the lead of the plates to build up a sulfate composition. As a load is connected across the terminals, a current flow of electrons is produced to equalize the difference in the charges on the plates. Excess electrons flow from the negative plate to the positive plate. During current flow, the plates can be measured by the poles of the battery to determine the voltage. Stratification of the battery plates occurs if the electrolyte solution is stratified. Since acid is denser than water, the acid build up and layering is greater on bottom of the battery solution than in comparison to the bottom of the battery. The high acid concentration in the lower portion of the battery artificially raises an open circuit voltage and the battery voltage appears to be fully charged and operable, but this is not the case. The amount of current available that the battery can deliver for a defined duration of time while maintaining a terminal-to-terminal voltage when significant stratification is present is very low as opposed a newly produced battery. As a result a false SOC reading may be detected while stratification is present within the battery.
Typical routines assume that open circuit voltage is measured when the battery is in an equilibrium state (i.e., no surface charge and no acid stratification). These typical routines will use the following formula to determine the running state-of-charge which can be represented follows:




  
    S
    OC
  
  =
  
    
      f
      ⁡
      
        (
        
          
            
              V
              OC
            
            ⁡
            
              (
              0
              )
            
          
          ,
          T
        
        )
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        ∫
        
          ρ
          ·
          I
          ·
          
            d
            t
          
        
      
    
  




where f(VOC(0),T) is the present startup





  SOC
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    1
    
      C
      norm
    
  
  ⁢
  
    ∫
    
      ρ
      ·
      I
      ·
      
        d
        t
      
    
  




is the state of charge that is determined by coulomb counting while the charging is occurring. These routines measure the open circuit voltage (OCV) after a long ignition key off such as 8 or 16 hours; however, depending on the charging history, a battery may not reach the equilibrium stage at the 8th or 16th hour. In addition, if the current sensor is not accurate, then integration error accumulates over time with respect to the coulomb counting. Moreover, current measurements during the ignition off are sparse and inaccurate. The following procedure overcomes deficiencies of low cost current sensors, surface charge and acid stratification.

FIG. 2 illustrates a flowchart of a general overview for estimating the state-of-charge (SOC) over time. In step 30, data for determining the SOC is obtained. In step 31, algorithm 1, shown in FIG. 3, is executed for identifying SOC0 and SOC1. SOC0 is a state-of-charge at a first instance of time for a respective ignition cycle, and SOC1 is a state-of-charge at a later instance of time for later ignition cycle. The initial objective is find an SOC0 and SOC1 that are within a predetermined error of one another. If a respective set of SOC0 and SOC1 are not within a predetermined error, then the likelihood that SOC1 is not an accurate determination based on equilibrium of the battery and a next ignitions cycle is analyzed for identifying a next SOC1.
In step 32, a determination is made as to whether SOC0 and SOC1 are in agreement with one another. That is, a determination is made whether the respective SOC values are offset by a predetermined amount, and if so, would indicate that a respective set of values are invalid and that a calculation for an estimated state-of-charge would also be incorrect. If the determination is made that SOC0 and SOC1 are not in agreement with one another, then a return is made to step 30 for determining a state-of-charge at a next ignition cycle. If the determination is made in step 32 that the SOC0 and SOC1 are in agreement with one another, then the routine proceeds to step 33.
In step 33, a determination is made as to whether the ignition off time >8 hours and whether the open circuit voltage SOCOCV is within the error bound. The SOCOCV is the state-of-charge value calculated as a function of the open circuit voltage (OCV) and the battery estimated temperature. The OCV is the battery voltage which is measured before the current ignition cycle (k) but after at least eight hours since the last charging state. After at least eight hours, the battery current is very low (<20 ma), so the battery voltage is the OCV. Therefore, the SOCOCV may be determined from the determined OCV. If the determination is that made that either one of the conditions are not satisfied, then the routine proceeds to step 35; otherwise the routine proceeds to step 34.
In step 34, algorithm 2, as described in detail later, is utilized for updating the ignition time off current Iign—off. After Iign—off is updated, the routine proceeds to step 35.
In step 35, the state-of-charge estimation SOCest is updated utilizing the following equation:




  
    
      SOC
      
        (
        est
        )
      
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      
        SOC
        est
      
      ⁡
      
        (
        
          k
          -
          1
        
        )
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        ∫
        
          ρ
          ·
          
            I
            on
          
          ·
          
            d
            t
          
        
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        
          I
          off
        
        ·
        Δ
      
      ⁢
      
        
      
      ⁢
      
        
          t
          off
        
        ⁡
        
          (
          
            k
            -
            1
          
          )
        
      
    
  




where (k) is the number of ignition cycles with at least an eight hour ignition off time before a next cycle is initiated, SOCest(k−1) is the state-of-charge at the k−1 ignition start, Cnorm is the battery normal capacity, ρ is the charge efficiency, Ion is the ignition on-current, Ioff is the ignition off-current, and Δtoff is the ignition off time between (k−1) ignition-on cycle and (k) ignition-on cycle.

In step 36, a determination is made as to whether SOCest confidence is high (e.g., the length of time since the last SOC0 and SOC1 have been used). If the confidence is high, then SOC0 may be utilized again for updating Iign—off. The routine then returns to step 33. If the confidence is low, then the routine proceeds to step 30 for determining a new SOC0 and SOC1.
In step 37, the state-of-charge may be output on a display device of the vehicle for identifying the state-of-charge to the operator. Alternatively, the state-of-charge may be provided to other vehicle systems for use in other vehicle operations where the battery state-of-charge is required for its operation.
FIG. 3 is a flowchart of Algorithm 1, as described earlier, for identifying whether SOC0 and SOC1 is found. In step 40, the routine is initiated and the flag is set to 0 (e.g., FlagSOC0—found=0). This flag identifies whether SOC0 and SOC1 are valid and therefore the flag is set to 1, or if invalid, the flag is set to 0.
In step 41, the ignition cycles are sequentially numbered for determining an estimated open circuit voltage SOCOCVest. For i=1 to N, the SOCOCVest is determined by the following formula:

SOCOCV—est(k)=SOCOCV(k−i)+ΔSOC,

where

SOCOCV(k−i) is the OCV based SOC at ignition k−i,
ΔSOC is the integration of ignition−on current from ignition k−i to ignition k.
It should be understood that between the (k−1) and (k) ignition cycle, the engine may crank/start several times but if the ignition off time between two neighbor cranks is less than eight hours, then the OCV is unavailable.
In step 42, a determination is made as to whether difference between the SOCOCV at the kth ignition and the SOCOCVest is less than a predefined error bound. The formula for the above determination is represented as follows:

|SOCOCV(k)−SOCOCV—est(k)|<ε,

where ε is the predefined error bound. The following parameters utilized for determining the above inequality is as follows:

    
    
        FlagSOC0—found=1
        SOC0=SOCOCV(k−i)
        SOC1=SOCOCV(k)

where SOCOCV(k) is the SOC at the kth ignition start, and SOCOCV(k−i) is the SOC at a prior ignition start.

    
    


If the determination in step 42 is that the difference is less than the predetermined error bound ε, then the routine proceed to step 43, otherwise the routine returns to step 41 for re-estimating an open circuit voltage state-of-charge.
In step 43, the subroutine exits and the SOC values for obtained for SOCOCV and SOCOCVest are used for determining for determining the ignition-off current Ioff.
The following embodiments describe various embodiments for determining the ignition-off current Ioff. If error or noise is not present in any of the measurement data, then a straightforward model may be utilized. The following formula may be used if the error and bias is not present for determining the ignition off current:




  
    
      I
      off
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      
        [
        
          
            SOC
            ⁡
            
              (
              k
              )
            
          
          -
          
            SOC
            ⁡
            
              (
              
                k
                -
                1
              
              )
            
          
          -
          
            
              1
              
                C
                norm
              
            
            ⁢
            
              ∫
              
                ρ
                ·
                
                  
                    I
                    on
                  
                  ⁡
                  
                    (
                    
                      
                        k
                        -
                        1
                      
                      ,
                      t
                    
                    )
                  
                
                ·
                
                  d
                  t
                
              
            
          
        
        ]
      
      ·
      
        
          C
          norm
        
        /
        Δ
      
    
    ⁢
    
      
    
    ⁢
    
      
        t
        off
      
      ⁡
      
        (
        
          k
          -
          1
        
        )
      
    
  




where SOC(k) is the state-of-charge at the kth ignition start, SOC(k−1) is the state-of-charge at the k−1 ignition start, Cnorm is the battery, ρ is the charge efficiency, and Δtoff is the time.

Alternatively, if any error is present in the measurement data, then the following embodiments may be used to for determining Ioff. The following model represents a particle filter that may be used if the noise/error is not Gaussian (i.e., normal distribution). The model follows a state space model and the equations that represent the state space model are follows:




  
    {
    
      
        
          
            
              
                SOC
                
                  (
                  est
                  )
                
              
              ⁡
              
                (
                k
                )
              
            
            =
            
              
                
                  SOC
                  est
                
                ⁡
                
                  (
                  
                    k
                    -
                    1
                  
                  )
                
              
              +
              
                
                  1
                  
                    C
                    norm
                  
                
                ⁢
                
                  ∫
                  
                    ρ
                    ·
                    
                      I
                      on
                    
                    ·
                    
                      d
                      t
                    
                  
                
              
              +
              
                
                  1
                  
                    C
                    norm
                  
                
                ⁢
                
                  
                    I
                    off
                  
                  ·
                  Δ
                
                ⁢
                
                  
                
                ⁢
                
                  t
                  off
                
              
              +
              
                ɛ
                SOC
              
            
          
        
      
      
        
          
            
              
                I
                off
              
              ⁡
              
                (
                k
                )
              
            
            =
            
              
                
                  I
                  off
                
                ⁡
                
                  (
                  
                    k
                    -
                    1
                  
                  )
                
              
              +
              
                ɛ
                1
              
            
          
        
      
    
    }
  
   




the measurement model is represented as follows:

SOCest(k)=SOCOCV(k)+εOCV 

where SOCest(k) is the ignition-off current for the current vehicle startup, and SOCOCV(k−1) is the ignition-off current for the previous vehicle startup, and εOCV is the current sensor error of the current sensor

Once the state model and measurement model formulas are defined, the model is applied to determine the Ioff(k) using the following routine as shown in FIG. 5.
In step 50, the following particle set is initialized:

{SOCesti,i=1,2, . . . ,N}{Ioffi,i=1,2, . . . ,N}

In step 51, the particles are updated based on the state space model in the equation set forth above. Utilizing the state space model, SOCesti(k) and Ioffi(k) are determined from the particle set. Ion is an ignition-on measurement, SOCesti(k) is calculated utilizing the model.
In step 52, the weights used to compensate for the error/bias at each ignition start. The weights are calculated based on the difference between (k) and SOC(k). The larger the difference, the smaller the weight will be. is represented by the following formula:




  
    
      w
      ′
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      1
      
        σ
        ⁢
        
          
            2
            ⁢
            
              
            
            ⁢
            π
          
        
      
    
    ⁢
    
      e
      
        
          
            -
            
              
                [
                
                  
                    
                      
                        SOC
                        l
                      
                      ⋀
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  -
                  
                    SOC
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                
                ]
              
              2
            
          
          /
          2
        
        ⁢
        
          σ
          2
        
      
    
  




where σ is the standard deviation of the {SOCesti}.

In step 53, the particle set is resampled based on the weights. As the particles are resampled for a next iteration, there likelihood will be increased to obtain the particles closer to the true value of SOC(k). The estimation of SOC(k),k+1,k+2 will converge to the true value.
In step 54, an estimated weighted average ignition off current is determined using the following formula:

Ioff(k)=Σiwi(k)Ioffi(k)

SOCest(k)=Σiwi(k)SOCoffi(k)

where wi (k) is the weights for each particle at each respective ignition start, and Ioffi(k) is the measured ignition off current for each particle at each respective ignition start.

In the event that the noise/error is Gaussian which follows a normal distribution, then a Kalman filter may be used. The Kalman filter is utilizes a series of measurements that are observed over time. The measurements contain noise and other inaccuracies. The Kalman filter that operates recursively utilizing streams of noise input data to produce an estimate of the system. The Kalman filter produces estimates of unknown variables and are often more precise than estimates based on a single measurements.
While certain embodiments of the present invention have been described in detail, those familiar with the art to which this invention relates will recognize various alternative designs and embodiments for practicing the invention as defined by the following claims.

## Claims

1. A method of determining a state-of-charge of a battery for a vehicle, the vehicle being in a charging state when the engine is operating and a non-charging state when the engine is not operating, the method comprising the steps of:
(a) measuring an open circuit voltage (OCV) for a current vehicle ignition startup using a voltmeter, wherein the current vehicle ignition start-up is performed after the vehicle is in the non-charging state for at least eight hours;
(b) determining an open circuit voltage-based state of charge (SOCOCV) for the current vehicle ignition startup using the processor;
(c) determining an estimated open circuit-based state of charge (SOCOCV—est) for a current vehicle ignition startup using the processor;
(d) determining whether the difference in the SOCOCV for the current vehicle ignition startup and the SOCOCVest for the current vehicle ignition startup is less than a predefined error bound using the processor;
(e) performing steps (a)-(d) in response to the difference being greater than the predefined error; otherwise, proceeding to step (f);
(f) determining an ignition-off current for the current vehicle ignition startup as a function of the SOCOCV of the current vehicle ignition startup, an SOCOCV of a previous vehicle ignition startup, and a state-of-charge based on current integration over time using the processor technique;
(g) determining an estimated state of charge (SOCest) of the current vehicle ignition startup using the processor; and
(h) providing the SOCest to a vehicle subsystem for use in a vehicle operation where the battery state-of-charge is utilized.

2. The method of claim 1 wherein estimating a SOCest of the current vehicle ignition startup is determined as a function of the SOCest of a previous vehicle ignition startup, a state-of-charge based on an ignition-on current integration over time, and a state-of-charge based on the ignition-off current integration over time.

3. The method of claim 2 wherein estimating a SOCest of the current vehicle ignition startup is represented by the following formula:




  
    
      SOC
      
        (
        est
        )
      
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      
        SOC
        est
      
      ⁡
      
        (
        
          k
          -
          1
        
        )
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        ∫
        
          
            ρ
            ·
            
              I
              on
            
          
          ⁢
          
            
              (
              
                k
                -
                1
              
              )
            
            ·
            
              d
              t
            
          
        
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        
          
            I
            off
          
          ⁡
          
            (
            
              k
              -
              1
            
            )
          
        
        ·
        Δ
      
      ⁢
      
        
      
      ⁢
      
        
          t
          off
        
        ⁡
        
          (
          
            k
            -
            1
          
          )
        
      
    
  




where SOCest(k−1) is the estimated state-of-charge at the k−1 ignition start, Cnorm is the battery capacity, ρ is the charge efficiency, Ion is the ignition on-current of the previous vehicle ignition start, Ioff is the ignition on-current of the previous vehicle ignition start, and Δt is the time.

4. The method of claim 1 wherein the SOCOCV—est for a current vehicle ignition startup is determined by the following formula:


SOCOCV—est(k)=SOCOCV(k−i)+ΔSOC


where SOCOCV(k−i) is and open circuit voltage based SOC at ignition k−i, and ΔSOC is the integration of ignition-on current from ignition k−i to ignition k, and i is a count of the vehicle ignition startup.

5. The method of claim 1 wherein if no error is present in the measurement data, then the formula for determining the vehicle ignition-off current is determined as follows:




  
    
      I
      off
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      
        [
        
          
            SOC
            ⁡
            
              (
              k
              )
            
          
          -
          
            SOC
            ⁡
            
              (
              
                k
                -
                1
              
              )
            
          
          -
          
            
              1
              
                C
                norm
              
            
            ⁢
            
              ∫
              
                ρ
                ·
                
                  
                    I
                    on
                  
                  ⁡
                  
                    (
                    
                      
                        k
                        -
                        1
                      
                      ,
                      t
                    
                    )
                  
                
                ·
                
                  d
                  t
                
              
            
          
        
        ]
      
      ·
      
        
          C
          norm
        
        /
        Δ
      
    
    ⁢
    
      
    
    ⁢
    
      
        t
        off
      
      ⁡
      
        (
        
          k
          -
          1
        
        )
      
    
  




where SOC(k) is the state-of-charge at the kth ignition start, SOC(k−1) is the state-of-charge at the k−1 ignition start, Ion is the ignition-on current, Ioff is the ignition-off current, ρ is the charge efficiency, and t is the time.

6. The method of claim 1 wherein if measurement data error is present and an error distribution is non-uniform, then a particle filter is used for determining the ignition-off current.

7. The method of claim 6 wherein a state space model is defined and a measurement model is defined for determining the ignition-off current.

8. The method of claim 7 wherein the state space model of the SOC for current vehicle ignition startup is defined by the following formula:




  
    SOC
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      SOC
      ⁡
      
        (
        
          k
          -
          1
        
        )
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        ∫
        
          ρ
          ·
          
            I
            on
          
          ·
          
            d
            t
          
        
      
    
    +
    
      
        1
        
          C
          norm
        
      
      ⁢
      
        
          I
          off
        
        ·
        Δ
      
      ⁢
      
        
      
      ⁢
      
        t
        off
      
    
    +
    
      ɛ
      SOC
    
  




where SOC(k) is the state-of-charge at the kth ignition start, SOC(k−1) is the state-of-charge at the k−1 ignition start, Cnorm is the battery, Ion is the ignition-on current, Ioff is the ignition-off current, ρ is the charge efficiency, Δt is the time, and εSOC is an error the measurement data of a current sensor in the SOC determination.

9. The method of claim 8 wherein the state space model of the ignition-off current for the vehicle ignition startup is defined by the following formula:


Ioff(k)=Ioff(k−1)+ε1 


where Ioff(k) is the ignition-off current for the current vehicle startup, and Ioff (k−1) is the ignition-off current for the previous vehicle startup, and ε1 is the measurement error of a current sensor.

10. The method of claim 7 wherein the measurement model is defined by the following formula:


SOCoff(k)=SOCOCV(k)−εOCV 


where SOCoff(k) is the ignition-off current for the current vehicle startup, and SOCOCV(k−1) is the ignition-off current for the previous vehicle startup, and εOCV is the measurement error of the current sensor for the OCV.

11. The method of claim 7 wherein determining the ignition-off current using the particle filter comprises the following steps of:
initializing a particle set;
updating a respective particle in the particle set based on the state space model for determining a estimate of the SOCest at a kth ignition start using the state space model;
calculate a weighting factor that is a function of the SOC at a kth ignition start and the estimate SOCest at a kth ignition start;
resampling a particle set as a function of the weighting factor; and
estimating the weighted average of the ignition-off current at current ignition startup.

12. The method of claim 11 wherein the particle set of Ioffi(0) for i=1, 2, . . . , N.

13. The method of claim 12 wherein the weighting factor is represented by the following formula:




  
    
      w
      i
    
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      1
      
        σ
        ⁢
        
          
            2
            ⁢
            
              
            
            ⁢
            π
          
        
      
    
    ⁢
    
      e
      
        
          
            -
            
              
                [
                
                  
                    SO
                    ⁢
                    
                      
                        
                          C
                          ^
                        
                        l
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  -
                  
                    SOC
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                
                ]
              
              2
            
          
          /
          2
        
        ⁢
        
          σ
          2
        
      
    
  




where σ is the standard deviation of the SOCest.

14. The method of claim 13 wherein the estimated weighted average of the ignition-off current is represented by the following formula:


Ioff(k)=Σiwi(k)Ioffi(k)


where wi(k) is a respective weight for a particle at each respective ignition start, and Ioffi(k) is the measured ignition off current for each particle at each respective ignition start.

15. The method of claim 1 wherein if current sensor error is determined and an error distribution is uniform, then a Kalman filter is used for determining the ignition-off current.

16. The method of claim 1 wherein the step of providing the SOCest to a vehicle subsystem for use in a vehicle operation includes outputting the SOCest to a display device for identifying the state-of-charge to the operator of the vehicle.

17. The method of claim 1 further comprising the step of determining whether the estimated SOC is utilized for a duration of time greater than a predetermined period of time, and wherein a return is made to step (a) if the estimated SOC being utilized for a duration of time that is greater than the predetermined period of time.

18. The method of claim 17 wherein if the estimated SOC is utilized for a duration of time that is less than a predetermined period of time, the method comprises the following steps:
determining whether the current ignition-off time is greater than eight hours and the SOCOCV is within an error bound; and
proceeding to step (f) in response to determining that the current ignition-off time is greater than eight hours and the SOCOCV is within the error bound, otherwise proceeding to step (a).