Kalman Filter Based State-of-Charge Estimation for

Lithium-ion Batteries in Hybrid Electric Vehicles

Using Pulse Charging Mori W. Yatsui, IEEE Member and Hua Bai, IEEE Member

Department of Electrical and Computer Engineering, Kettering University, Flint, Michigan USA

hbai, yats6501@kettering.edu

Abstract— The battery is one of the most important energy

storage components in EV/HEV. Failing to estimate the state

of the charge accurately will bring the risk of overcharge or

over discharge. The traditional Coulomb counting method will

bring accumulated error over time, therefore high deviation

occurs between the estimated and real state of charge.

Different estimation strategies are compared in this paper, i.e.,

Coulomb counting method, open-circuit-voltage method and

Kalman filter based state of charge estimation. Experimental

results validate the effectiveness of Kalman filter during the

on-line application.

Keywords-component: Lithium-ion Battery, State of Charge,

Plug-in Hybrid Electric Vehicle, Coulomb Counting Method,

Open-circuit-voltage Method, Kalman Filter.

I. INTRODUCTION

Practical mobile systems, such as Plug-in Hybrid Electric Vehicles (PHEVs), are designed to be of most use in harsh environments. The conventional systems used in present vehicles substantially rely on energy storage systems (ESSs), which typically contain a battery pack such as a Lithium-ion (Li-ion) galvanic system. It is of the utmost importance that the battery be stable within its rigorous environment for practical and safe use. A great factor determining the stability of Li-ion battery packs lies within the state-of-charge (SOC). The SOC is dependent on the open-circuit-voltage (OCV), chemistry make-up, number of cells, temperature, age, cycle history, damage, humidity, etc. Failing to predict battery SOC will cause overcharge or over discharge during cycles, which potentially will bring irreversible permanent damage to the battery cell. Damage to the cell reduces battery cell performance and cycle life, therefore posing necessary cost to the customer.

The automotive market has shifted towards electronically-biased vehicles within the last two decades, thus thrusting research into the electrical domain and enforcing the knowledge economy behind battery performance. These advancements will lead to great steps towards software modeling and simulation for batteries and further understanding of battery interfacing and lifetime properties. The most accurate method to measure the SOC is the direct relation of OCV, however this can only be accurately measured offline when the battery has no current transfer. Traditionally, the most commonly applied online SOC estimation is the coulomb counting (CC) method [6], which measures the battery current, integrates it with represent to time, and determines an estimated battery SOC. When the measured battery current has some error, the long-run integration will bring unpredictable deviation of SOC, which will lead to the overcharge or undercharge of the battery. Also, the CC method does not compensate for the

capacity loss over cycling and the temperature. The OCV method and Kalman Filter (KF) are proposed to enhance the battery SOC estimation [8]. Our implementation for the KF will observe the measurement inputs, terminal voltage and current, along with the output measurements of the CC method to better compensate for the non-ideal factors that play a role in the batteries characteristics over time. The OCV can only act as the off-line detection after reasonable chemical settling time, which will depend on battery manufacturing. Thus, the KF must be relied on to reduce the error of CC online, however previous literature still focuses on the constant current or intermittent constant current charging. Pulse charging as shown in this experiment can use the OCV during the offline mode to verify the KF’s validation of operation. This OCV will be measured after the internal impedance is discharged. The effectiveness of the KF in the pulse charging control, which is believed healthy for the battery, still needs be investigated personally [8].

This paper will first present the battery model used for SOC estimation of the selected Li-ion galvanic battery. Secondly, the construction of the OCV, CC, and KF computational algorithm will be presented. Section IV will present the comparison of different methods we used to identify the SOC, along with posing the experimental validation of KF in the pulse charging process. Section V is the conclusion.

II. MODEL OF THE LITHIUM-ION BATTERY

Battery modeling is a crucial development stage for PHEV electrical simulations. Models have been created for studies in different applications with Pb-A, Li-ion, and Ni-MH batteries [2,3]. The present concern with these models in Li-ion is to create a dynamic model for the battery’s SOC, which is directly related to the battery capacity and also dependent upon the cell’s chemical material. The capacity is the quantity of charge delivered by under a set of conditions, defined as Ah or Wh, which are regarded as a time integration of the current and defined as Peukert’s capacity.

Li-ion batteries differ from Nickel-Metal Hydride and Lead-Acid batteries in that these are unable to deliver considerable charge within the entire voltage span. Each Li-ion cell potential is recommended to remain between 2.7 V and 4.15 V with the nominal voltage varying between 3.2 V and 3.8 V depending on the type of Li-ion cell. The failure to follow these recommendations will result in an unserviceable battery that has a large voltage drop when loaded caused by high internal resistance and low self-discharge resistance. This is due to crystalline solids, which form within the cells.

A simulation software program called ADVISOR

developed by NREL Center for Transportation Technologies

and Systems in 1994 uses Matlab-Simulink to run

simulations of battery models. The ADVISOR battery model

shows an equivalent circuit similar to Thevenin’s circuit for a

battery model. It uses an internal resistance to account for the

potential drop from its off-line to on-line application. The

resistance is dependent of SOC, temperature, and current

flow direction. This method neglects the self-impedance,

responses, and proper battery equivalency to run simulations.

Typically, this information is not readily available from

battery datasheets [6]. The simplicity of the simple battery

model compiles the internal resistances into one value and

allows the OCV to be measured immediately upon

disconnecting all sources or loads to the battery. The

impracticality is that the simple battery model does not focus

on self-discharge and actual reactions seen in battery tests. In order to properly treat the given scenario, the idea of a

battery must merge with reality. A capacitor is a source of energy that does not allow its voltage change instantaneously. The merging of ideals to reality for the battery uses the capacity to approximate an equivalent capacitance. Each cell within a battery packaging will have connection resistance, and the chemical polarization of ions cause a transient reaction given a charging unit-step as seen in Fig. 1. Based on stated information, the selected battery model to be used within this study is shown in Fig. 2. with a consistent environment of 25°C with current passing into the positive terminal.

The lithium-ion battery cell is modeled as a large

capacitor Cb to enable the simulation of the state-variables.

Rs represents the summation of contact resistances within the

battery pack, while Rp

and Cp represent chemical

polarization resistance and the chemical polarization

capacitance, respectively. The discharge resistance which

degrades over time is represented as Rd. The equations

stated in this paper are for charging, thus discharging must

have a negative current.

III. COULOMB COUNTING, OCV, AND KF

The Li-ion battery pack selected for this study was the

Valence U1-12XP with an approximate nominal capacity of

40 Ah and a voltage of 14.6 V when SOC reaches 1. The

manufacturer states the battery maintains an internal

resistance less than 15 mΩ. This understanding is based

from a test done to find Rs based on the immediate voltage

increase as a charge is placed. The manufacturer’s internal

connection resistance was tested with pulse charging with

resting periods of 8 seconds, however the battery resistance

becomes significant as SOC approaches 0 as discussed in

the results section.

For the CC method, the time to charge and discharge

follows the general Peukert equation

k

QIt−

−= , (1)

where Q is the Peukert capacity per ampere charge/discharge and k is the ideality factor due to side chemical reactions. The ideality factor of an ideal battery is stated as k=1 [7].

For OCV method, experimentally we could determine

the relationship between OCV and SOC using CC. Varying

the instantaneous charging current will lead to the change of

terminal voltage [8]. Imposing ICHRG1 and ICHRG2 results in

different terminal voltages

222

111

BATOsCHRG

BATOsCHRG

VRIV

VRIV

+×=

+×=

,

(2)

respectively, where VBATO is the real voltage of the battery

and Rs is the internal resistance of the battery. In the case

where VBATO1 and VBATO2 are the same, the internal resistance

of the battery is

21

21

CHRGCHRG

sII

VVR

−

−=

.

(3)

Acquisition of the battery resistance could let the

charging system determine the real battery voltage from the

terminal voltage in the simple battery model. In order to

overcome the voltage loss across the battery resistance, the

targeted apparent terminal voltage (V) should be set as the

actual battery voltage (VBATO), or said to be the OCV, plus

the voltage drop, i.e.,

BATOsCHRG VRIV +×= (4)

Based on the identification of VBATO, the SOC could be

identified with the curve of SOC and OCV, hence the OCV

method.

According to Coulomb counting principles, it does not

contain any reference point and can suffer from long-term

drifting due to error introduced from system noise and

measurement noise. The value of system error can also be

introduced as the differentiating Peukert capacity and

Fig. 2. Schematic of the complex battery (top) and the complex battery

model (bottom).

Fig. 1. Terminal voltage during charging unit-step pulses showing the

desired model characteristics.

Peukert ideality coefficient as the battery’s history

elongates. CC must be reinitialized with an accustomed

Peukert capacity and Peukert coefficient from a newly

acquired log-log chart from battery tests. Currently, there is

no correction for this within the CC simulation and it would

be difficult to find more recent values every few cycles.

Implementation of the KF requires a continuous time

observer to give a prediction to the filter itself for this

simulation, which is called a Kalman-Bucy filter. For

estimation of the true battery OCV, the observer will read

the current, terminal voltage, and the SOC from the CC-

OCV estimation in order to output a value. This is done

using an input of a stochastic state-space model, which

describes an estimation of the battery’s parameters to

estimate the voltage across the battery’s equivalent

capacitance and polarization capacitance. Solving the KCL

and KVL equations,

p

Cp

CpR

vii += (5)

CpCbtS vvviR −−=

,

(6)

respectively, gives system equations that result in the

necessary dynamic equations () = () + () + ()

and () = () + (). The dynamic system is

finalized in the form

)(1

0

01

10

11

tsi

v

C

CR

v

v

CR

CRCR

v

vt

p

bS

Cp

Cb

pp

bSbS

Cp

Cb+

+

−

−−

=

&

&

(7)

)(

0

1)( tm

v

vty

Cp

Cb+

=

,

(8)

where )(ts and )(tm represent system noise and

measurement noise varying in respect to time, respectively.

The observer estimates the rate of change with respect to

time and solves for the state variables, Cov and

Cpv . Then,

the observer calculates the gain of the system in order to

achieve the desired output.

The Kalman filter is used to filter out noise,

measurements, and other inaccuracies resulting in values

closer to real values in dynamic linear systems. Upon

reception of the input values, the Kalman filter estimates the

uncertainty of the values and computes a weighted average

between the measured value and the predicted value. It

compiles all noise influences, even those unconsidered

within this paper, such as thermal differences. The

uncertainty is qualified by using the best linear depiction of

the input value, which allows for cancellation of noise by

narrowing the sensor and noise values to a minimum. The

weighted average is calculated placing heavier weights on

those values that are more likely, which is determined by the

covariance on the uncertainty. The filter then feeds back

and recursively uses prior prediction to determine the new

best guess at each time step. This functionality gives it the

title of an adaptive filter for digital signal processing.

Then, the KF uses a linear-quadratic regular in

combination with a Gaussian controller to form a linear-

quadratic-Gaussian controller, or LQG, to estimate the state-

space variables within our linear dynamic system. The cost

function is used to share the control input history and is

stated as

[ ]

++= ∫T

dttutRtutxtQtxTFxTxEJ0

)()()(')()()(')()('

,

(9)

where F, Q, and R are functions of time and are greater than

0 given ttty <≤ '0),'( and T is the final simulation time.

The LQG controller calculates

[ ]

[ ]

)(ˆ)(

)0()0(ˆ

)(ˆ)()()()(ˆ)(ˆ

txLtu

xEx

txtCtyKtButxAtx

−=

=

−++=&

,

(10)

where K is the Kalman gain, L is the feedback gain, and E is

the average value, or the expectation value.

The Kalman gain is described as

( )( )

)(')()(

)0(')0(),(),(,,)(

1tWCtPtK

xxEwWvVCAftK

−=

=

,

(11)

and to find P(t)

( ))0(')0()0(

)()()(')(')()()( 1

xxEP

tVtCPtWCtPAtPtAPtP

=

+−+=−&

.

(12)

The feedback gain is described as

( )

)()(')()(

),(),(,,)(

1tStBtRtL

FtRtQBAftL

−=

=

,

(13)

and to find S(t)

FTS

tQtStBtBRtSAtStSAtS

=

+−+=−−

)(

)()()(')()()()(')(1&

.

(14)

The dynamic matrix equations for P(t) and S(t) are defined

as Riccati differential equations. Individually, the first

matrix Riccati differential equation solves the linear-

quadratic estimation, whereas the second matrix Riccati

differential equation solves the linear-quadratic regulator.

Appending these calculations to the stochastic state-space

model allows for the LQG control algorithm to be solved,

resulting in a data reduced SOC output [8].

Referring to the previous section, CC, the battery’s

Peukert capacity and Peukert ideality coefficient would now

be considered part of the system error. Thus, the Kalman

filter adjusts for the constantly changing Peukert values and

noise given from the sensors and system.

IV. RESULTS AND DISCUSSION

The pulse charging method made the internal Ohmic

impedance available to determine. It is concluded that Li-

ion battery internal Ohmic resistances are not constant

through all potential ranges, but are relatively within

tolerance range as can be seen in Fig. 3 and Fig. 4. A failed

battery is easily noticeable from Ohmic resistance testing

shown in Fig. 5. However, it is very likely that every

manufacturer holds different resistive properties depending

on battery architecture and materials, and these differences

must be taken into account.

The pulse charging method also determines the OCV vs.

SOC and compares it to the manufacturer’s specifications,

shown in Fig. 3. The test differentiation is caused by

deterioration of the Peukert capacity.

The results determine a difference between the

manufacturer’s and the two trial runs mostly likely due to

the Peukert capacity and ideality coefficient altering over

the battery’s short history or differences in the

manufacturing process. The internal Ohmic resistance

remains below 15Ω until the SOC reaches the overcharge

regions, where the impedance increases drastically to

infinite. It is concluded that the Li-ion SOC performed

properly with a linear depiction with respect to time.

The CC simulation was implemented to accumulate the

SOC measured based on Peukert’s ideality equations. The

automated simulation was developed in National

Instruments’ Labview 2009 SP using a National Instruments

myDAQ, a portable data acquisition unit.

The simulation, shown in accurately models long-term SOC

with introduced error, but is compensated for in the KF

algorithm. The KF was implemented to correct the CC SOC’s error

and result in a more “true” value with consideration to noise in the system. This is done using the algorithm stated previously. Several tests were done to observe the influence of induced scenarios or errors on the OCV-CC-KF system.

The first test was done through Labview to implement a random signal generation in the readings to simulate large noise measurements in an automotive application. The test was performed in both charging and discharging scenarios. The result was a precise but less accurate system by a magnitude of approximately ±0.14% weighted mean.

The second test was done through automated pulse charging and data acquisition to determine the difference of SOC values. The KF SOC was measured during the charging moments and the OCV was measured during the idle moments given an 8 second rest prior to reading. This test was performed in both charging and discharging scenarios. The results of this test were in agreement with the research done by Smith, Rahn, and Wang on pulse charging of lithium-ion batteries [8]. According to the data found, the KF SOC estimation had an error / tolerance of ±1.76% in comparison to the OCV method estimation, which was calculated to be statistically insignificant.

The third test was done through Labview to implement a constant DC offset on the KF inputs to determine the correction abilities for large CC estimation error. The DC offset implemented to each of the KF inputs results in a percent error as shown in Table 1.

Fig. 6. Labview schematics for CC SOC estimation for cycling.

Fig. 5. Internal connection resistance testing at various currents of a bad battery (experimental data using OCV).

Fig. 4. Internal connection resistance testing at various currents of a good

battery (experimental data using OCV).

Fig. 3. SOC CC-OCV estimation method verification of Peukert’s ideality

principles.

Table 1 shows the dependency of the KF on specific inputs

and these values are well understood to be results of the

weighting average principles behind the functions.

The Labview 2009 code was proven to be useful during

this process. Fig. 7 and Fig. 8 show estimation of the SOC

from OCV, OCV-CC, and OCV-CC-KF accumulated

methods at a low SOC and a high SOC using the original

prototype algorithm. Fig. 9 is the newest prototype using

Labview for pulse charging and discharging. This

representation allows us to easily view the battery SOC

during online testing and verification with OCV SOC

detection.

V. CONCLUSION

This paper shows the implementation and results of

combining open circuit voltage, Coulomb counting, and

Kalman filter methods in order to more accurately estimate

the SOC of Li-ion battery cells by various factors into

consideration. The combination of the methods accurately

estimates the SOC with an error of ±1.76% in comparison to

the OCV method estimation. This error may vary

depending on hardware used for the data acquisition. This

set up was done with a National Instruments myDAQ. This

error tolerance was calculated to be statistically insignificant

and therefore usable. The KF observes the terminal voltage

rate of change as well as discharging or charging currents to

calculate electrode surface dynamics, electro-chemical

reactions, and electrode particle transfer, along with other

side reactions. In addition to the rate of change, it also

receives the output of SOC from the CC method to “edit”

the needed values for the filtered SOC output. The tests in

this paper used the single polarization impedance model

shown at the beginning of this paper in Fig. 2, called the

complex battery model assembled within this study.

VI. RECOMMENDATIONS

It is recommended to enhance the KF processing

capabilities by incorporating thermal differentiation and

capacity differentiation. This is an essential part of our goal

to produce an overall battery management system for

PHEVs. This technology can be additionally supportive

within small electronic devices, which use an ESS.

REFERENCES

[1] Li, J. Murphy, E. Winnick, J. Kohl, P.A. J. Power Source 102 (2001) 302-309.

[2] H. L. Martin and R. E. Goodson, “New concept in battery performance simulation and monitoring,” in EV EXPO 80, Cervantes Convention Center, St. Louis, Missouri, May 1980.

[3] B. Thomas, W. B. Gu, J. Anstrom, C. Y. Wang, and D. A. Streit, “In- vehicle testing and computer modeling of electric vehicle batteries,” in Proc. 17th Int. Electric Vehicle Symp., Montreal, Oct. 2000, [Online]. Available: http://mtrl1.me.psu.edu.

[4] V. S. Bagotzky and A. M. Skundin, Chemical Power Sources: Academic Press, 1980.

[5] NREL, “ADVISOR HEV Simulation Model,” 1996.

[6] X. He and J.W. Hodgson, “Modeling and Simulation for Hybrid Electric Vehicles-Part I: Modeling,” IEEE Trans. Intelligent TransportationSystems, 3.4 December 2002.

[7] C. J. Hoff, “Hybrid Vehicle Propulsion,” Kettering University 2010.

[8] K. A. Smith, C. D. Rahn, and C. Y. Wang, Model-based electrochemical estimation and constraint management for pulse

operation of lithium-ion batteries: IEEE Trans. Control Systems Technoloy. 18.3, May 2010.

Fig. 9. Second prototype code using pulse charging and discharging.

Fig. 8. Pulse charging at stage 2 constant voltage for SOC accuracy

estimation. Note that charging current is negative.

Fig. 7. Pulse charging at stage 1 constant current for SOC accuracy

estimation. Note that charging current is negative.

TABLE 1

SOC ERROR FROM FORCED DC OFFSETS IN INPUTS OF CODE.