IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008
2027
Nonlinear State of Charge Estimator for
Hybrid Electric Vehicle Battery
Il-Song Kim, Member, IEEE
Abstract—A new method for battery state of charge estimation
using a sliding mode observer has been developed. A nonlinear
battery dynamic modeling technique is established and design
methodology with the sliding mode observer is presented. Contrary to the conventional methods using complicated battery
modeling, a simple resistor-capacitor battery model was used
in this work. The modeling errors caused by the simple model
are compensated by the sliding mode observer. The structure of
the sliding mode observer is simple, but it shows robust control
property against modeling errors and uncertainties. The convergence of the proposed observer has been proved by the equivalent
control method. The performance of the system has been verified
by the sequence of urban dynamometer driving schedule test. The
test results of the proposed observer system shows robust tracking
performance under real driving environments.
Index Terms—Battery modeling, hybrid electric vehicle, slidingmode observer, state of charge (SOC), state of charge estimation.
NOMENCLATURE
Open-circuit voltage as a function of SOC Z.
Polarization capacitance
Propagation resistor.
Diffusion resistor.
Ohmic resistor.
Cell terminal voltage.
Z
Polarization voltage.
State of charge.
Nominal capacitance of the cell.
Estimate of the cell terminal voltage.
Estimate of the state of charge.
Estimate of the polarization voltage.
Feedback gain of sliding-mode observer.
I. INTRODUCTION
UE to the high fuel efficiency and low-emission requirements, a hybrid electric vehicle (HEV) has great potential
as a new alternative means of transportation. A HEV is mainly
composed of an internal combustion engine, electric motor and
rechargeable battery. The electric motor provides boost energy
to assist the combustion engine and acts as a generator when regenerating brake energy, or when the engine has excess power
D
Manuscript received April 12, 2007; revised November 15, 2007. Published
June 20, 2008. Recommended for publication by Associate Editor S. Choi.
The author is with Chung-Ju National University, Chung-Ju, Republic of
Korea (e-mail: iskim@cjnu.ac.kr).
Digital Object Identifier 10.1109/TPEL.2008.924629
Fig. 1. Configuration of the HEV operation.
to charge the battery. As can be seen in Fig. 1, the HCU (Hybrid
control unit) controls motor power and engine power according
to the vehicle conditions for the best fuel performance. The
amount of motor power is limited by the maximum available
battery charge/discharge power. For a required motor power, the
battery should provide available charge/discharge power to meet
the power requirement. During the cold cranking or regenerative
braking, the discharged or charged power is rated up to 30 [kw]
for 10 s, for example. The battery available power is directly
obtained from the state of charge (SOC) information and therefore it is very important to accurately obtain its value for the
best performance of the HEV. The SOC is calculated from the
BMS by the cell voltage and temperature and other information
such as polarization effect caused by the electrolyte concentration gradient during high rate charge and discharge period. It is
sent to the HCU by CAN communication line. Using this SOC
information, the HCU controls motor power for the best performance and safe operation of the battery.
Since the SOC is an internal chemical state of battery and
thus cannot be directly measured with electric signals, it should
be estimated with the aids of physical measurements such as
voltage and current of the battery terminal. There have been
numerous attempts to estimate the SOC of battery. The most
common methods are the coulomb counting and Kalman filter
approach. The coulomb counting or current integration method
measures the amount of charge taken out or put into a battery
in terms of ampere-hours [1]. If a sufficiently accurate current
sensor is used, this method is reasonably accurate and inexpensive to implement. However, the coulomb counting is an open
loop SOC estimator and thus the errors in the current detector
are accumulated by the estimator. The Kalman filter method is
a well known technology for dynamic system state estimation
such as target tracking, navigation, and battery [2], [3]. It provides a recursive solution to optimal linear filtering for state
observation and prediction problems as well. The unique advantage of the Kalman filter is that it optimally estimates states
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008
Fig. 3. Battery dynamic model structure.
Fig. 2. Open-circuit voltage versus SOC over temperature variation of the
lithium-polymer battery.
affected by broadband noise contained within the system bandwidth. The main disadvantage of the Kalman filter is that it requires high complex mathematical calculation. The gain is obtained by the five steps of the Kalman filter algorithm. There
can be some possibility of divergence due to imperfect modeling and complex calculation loads. If the processors are not
mighty enough, the calculation time could excess the sampling
time and thus it can not track the correct state values. Also the
Kalman filter has some limitations for a real implementation
such as perfect modeling of the plant and Gaussian distribution
of the external noises. If these constraints are not satisfied, the
performance of the Kalman filter would be degraded and thus
cannot be used in the real applications [4].
Other reported methods for estimating the SOC have been
based on artificial neural networks and fuzzy logic principles
[5], [6]. The neural network method incurs large computation
overload on the BMS, it can be a problem for online implementation. Since the fuzzy logic method relies on the training data
battery is operated in unusual way. The empirical method based
on the battery chemical characteristics is also reported in recently [7].
In this paper, a new design method with a sliding-mode
observer has been presented for the battery SOC estimation.
The sliding-mode observer can overcome aforementioned
drawbacks by using sliding-mode techniques. The main characteristics of the sliding-mode observer are simple control
structure and robust tracking performance under uncertain
environments [8], [9].
II. BATTERY MODELING
Dynamic state model of the battery is necessary to develop
a simulation model for the emulation of battery behavior
[10]–[15]. The model is developed from experimental cell
data, where open circuit voltage (OCV) tests are performed
on successive discharge of the battery, by the application of
periodic current discharge. As for the temperature variation
to
, the OCV of a lithium-polymer battery
from
(LI-PB) varies nonlinearly over the battery SOC as can be seen
in Fig. 2. Therefore, the nonlinear RC models are developed to
model nonlinear OCV characteristics of the Li-PB [16], [17].
The proposed model consists of: 1) nonlinear voltage source
as a function of SOC Z to represent nonlinear characto model polarization
teristics of the OCV; 2) a capacitance
to model propagation resiseffect; 3) a propagation resistor
as a function of current I; and
tance; 4) a diffusion resistor
and terminal voltage
as shown in
5) an ohmic resistance
Fig. 3. The self-discharge resistor does not considered in the
model because the self-discharge characteristic of the lithium
battery is extremely low compared to other batteries such as
nickel cadium, lead-acid, nickel metal hydride type.
is denoted as . The
The voltage across the capacitance
terminal voltage equation is given as
(1)
(2)
where is the instantaneous current (positive for charging, negative for discharging).
The SOC is defined as a ratio of the remaining capacity to the
nominal capacity of the cell, where the remaining capacity is the
number of ampere-hours that can be drawn from the cell at room
temperature with the C/30 rate before it is fully discharged [3].
Based on this definition, the mathematical relation on the SOC
is developed as
(3)
is SOC and
is the nominal capacitance of the
where
cell which is defined as the number of ampere-hours that can
be drawn from the cell at room temperature at the C/30 rate,
starting with the cell fully charged [3].
The time derivative for SOC Z can be expressed as follows:
(4)
Equating the two voltage equations (1) and (2) with some algebraic manipulation yields
(5)
From Kirchhoff’s law, it results in
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(6)
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2029
Then, from (5) and (6), obtain
where
(7)
Equating (7) to (4) results in
(8)
Also, by similar way, substituting (7) into (6), we have
(9)
The output voltage equation is given from (2) and (7) by
This model is not accurate compared with the real cell data.
Therefore, the nonlinear unknown disturbances terms are added
to the model in order to compensate for the modeling errors
(10)
The output voltage is expressed as a nonlinear equation of Z
and therefore can be considered as the third state variable. The
rate of change of input current can be negligible due to the
fast sampling interval when implemented into digital system.
Taking the time derivative of the output voltage and assuming
gives
(14)
not only represent nonlinearities caused
where
by linearization error and modeling error, but also time-varying
terms and internal/external disturbances.
III. THEORETICAL BACKGROUND OF THE
SLIDING-MODE OBSERVER
(11)
The above equation is based on the assumption that the OCV
can be considered as a piecewise linear function of the Z and
therefore the following relationship is obtained for the piecewise
region:
(12)
From (2) and (7), solving for
and then substituting into (10),
then the complete state equation including the derivative of the
output voltage is obtained as
Consider the observer problem for a continuous-time single
and measurement
input system
, where
,
is the scalar
model
represents bounded modeling errors and
feedback control,
disturbances, and
[18]. When an
observer for the system is defined as
(15)
where
with
, is an estimate
of ,
is the signum function, and represent vectors
has dynamics
of switching gains. The observer error
, where
,
.
and the switching function is defined as
A local sliding regime exists on the surface
whenever
(16)
(13)
where is the first column of switching gain .
, a local sliding regime exists on
Therefore if
for
.
The set
is sometimes referred to as the sliding patch. The ideal sliding
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008
dynamics are determined by Filippov’s solution concept: for
,
Z,
As OCV is monotonically increasing with respect to the SOC
can be considered as piecewise linear to the
. Therefore
(23)
(17)
represents a projection onto the null space of C along
where
span {L}.
The sliding-mode observer shows robust tracking property by
the sliding regime. The robustness of sliding-mode observer is
to maintain sliding regime in the presence of dynamical disturbances, and to retain robustness properties of the sliding mode.
IV. SLIDING-MODE OBSERVER DESIGN
Since the observability matrix of (13) has always full rank,
the internal state of the battery can be estimated by the observer.
The sliding-mode observer design starts from the output equation [19]–[24]. The corresponding sliding-mode observer for the
is given by
output
(18)
are the estimates for
, and
is a constant
where
positive feedback gain. When defining the error with
, the following error equation is obtained:
The error system for
is given as
(24)
where is a piecewise linear gain and the maximum value is
determined from the experimental result.
. Select
Choose Lyapunov candidate function
, the sign of
and
is opposite. Therefore
, as in the previous case,
and
for all
subsequent time.
Then the following relationship is obtained as
(25)
Finally, the observer for
is built as
(26)
The error system is given as
(27)
(19)
where
.
The convergence of error equation can be proved by the Lyapunov candidate function by choosing
.
, then the sign of
is negative if
is
Select
positive regardless of
is. The sign of
is
positive if
is negative. Therefore
, and after
and
for all subsequent time.
some finite time,
According to the equivalent control method, the error system
is replaced by its equivin sliding mode behaves as if
, which can be calculated from (19)
alent value
and
. Once the sliding surface is
assuming
and
reached, then from the equivalent control concept,
reduced to zero and the uncertainties
vanish, and then the
resulting equation from (19) can be written as follows:
As in the previous case, select
, then the error
goes to zero if
is larger than the uncertainties.
The resultant observer equations are given as
(28)
The range of feedback gain should be
(20)
The next observer equation for
(29)
is obtained as
(21)
are the estimates for
, and
is a constant
where
positive feedback gain. Define error as
and
, then the following error equation is obtained as
(22)
The boundaries of the uncertainties can be determined by
comparing the cell test data with modeling parameters.
The sliding-mode observer switching gain
can be
arbitrarily assigned to attain robustness against disturbances.
However, the restriction on the assignment of the switching gain
comes from the condition that the observer is stable. The practical system is implemented with a digital system which has a
finite sampling time and gives rise to the chattering phenomena.
The magnitude of chattering is highly dependent on the observer
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KIM: NONLINEAR STATE OF CHARGE ESTIMATOR FOR HEV BATTERY
2031
Fig. 5. Block diagram of the proposed observer operation.
as voltage recovery time and resultant value is 1.2 s. Then by
circuit analysis,
Fig. 4. Picture of the test Li-PB.
gain. If the gain is remarkably big, a large amount of ripples may
result, causing estimation errors. Therefore, a tradeoff should be
made between the robustness and stability of the observer [18].
The design methodology for sliding-mode observer can
be summarized as follows. First, build the battery model by
state equation including output state. Second, decompose state
variable equations into corresponding observer equations.
Third, design the feedback switching gain so as to guarantee
the sliding-mode regime by equivalent control method.
V. EXPERIMENTAL RESULT
The cell comprises of a
cathode, an artificial
graphite anode and is designed for high power application. The
nominal capacity and voltage is 5.0 Ah and 3.8 V, respectively.
The picture of the test LI-PB is shown in Fig. 4. The dimension
in mm and the weight of the cell
of the cell is
is 120 g.
The RC model parameters are obtained from the circuit
analysis methods. The nominal capacitance is determined by
analyzing the amount of stored energy using the following
expression:
(30)
by the
The internal resistance of the cell is measured to be 4
DC-IR test which performs 1-C discharge for 10 s and calculates
. It is generally
the ratio of voltage and current change
corresponds to the 25% of the total internal
assumed that
resistance [2]
(31)
and
. The
is set to be
Therefore
the same as
. The polarization capacitance is based on high
frequency excitation test which involves 10-C pulsed discharge
for 500-ms intervals to determine the time constant given by the
and its associated resistance. The time constant is defined
(32)
The modeling parameters are calculated using circuit values
The test was performed using the cell model parameters. The
configuration of experimental setup is shown in Fig. 5. The
thermal chamber and the Nittetsu cycler were used as chargedischarge equipments for temperature regulations. Nittetsu cycler has 0 5 V and 0 120 A of voltage and current measurement range, respectively. The cycler’s voltage measurement accuracy is 5 mV and its current measurement accuracy is 200
mA. It also has precision ampere-hour counter for direct SOC
calculation. True SOC was directly obtained from this amperehour counter. The test was performed with fully charged condition to set the SOC to one. As the test proceeds, the true
SOC was calculated by the ampere-hour counter. The controller
has been built with Infineon 16-bit microprocessor XC167-40
[Mhz]. It has been used for automotive parts such as engine control unit, transmission control unit, air bag, and so on. It contains
internal A/D converter for 10-bit and other peripherals such as
PWM and communications. The total control loop takes less
than 10 ms including measurement and calculation time of the
sliding-mode observer.
Two types of cell tests were performed for the proposed observer. The first type of test comprised a sequence of constant
current discharges for 180 s and rests for 3600 s. The cell was
fully charged up to 4.2 V before the test begins. The discharge
current is 5 A and it corresponds to the 1-C rate of the nominal
capacity. This amounts to 5% decrease of SOC for each period.
The sampled data is collected every second. The purpose of this
test is to set the OCV over the entire SOC range and the test result was shown in Fig. 2. The thermal chamber was set to 25 .
This data was used to identify parameters of the cell models.
The cell model parameters are obtained by the cell test results
and the sliding-mode observer equations are established by current cycle of the LI-PB as shown in Fig. 5. The charge-discharge
current is simultaneously applied to the LI-PB and sliding-mode
observer. The terminal voltage of the LI-PB is measured as the
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Fig. 6. Current, voltages for true and model cell, and error.
Fig. 8. Estimated voltage and SOC of the sliding-mode observer.
Fig. 7. Polarization resistance R .
output and fed into the sliding-mode observer to compensate for
the errors, and the output of the observer is the estimated SOC.
The true SOC is obtained directly from the cycler by precision
ampere-hour counter.
The result comparing the electrical modeling with the cell’s
test data is shown in Fig. 6. It shows the discharge current, true
cell voltage, modeling cell voltage and the modeling voltage
error. The shapes of the true cell response and the model output
are similar in general, although many details of the true cell response are different. This is mainly due to the nonlinear charof the true cell and
acteristics of the nominal capacitance
also to the fact that the values of resistances are changed by the
is a nonlinear resistance which varies on the current.
SOC.
over current is shown in Fig. 7.
The plot of
The observer gains are selected to satisfy the condition in
(29). The selected observer gains are
,
,
. The results of SOC estimation using sliding-mode
observer are shown in Fig. 8. The estimated model output is
controlled with respect to cell terminal voltage with switching
ripple, and the estimated SOC follows the true SOC although it
has deviation at the start/end of the rest period. This is caused by
the discontinuous current and is affected by the abrupt change
of . In the discontinuous period, the sliding trajectory is away
from the sliding surface by the discontinuous function, but the
trajectory tracks into the sliding surface in a short time. The
one cycle of Fig. 8 is rendered in Fig. 9. The estimated output
voltage tracks cell voltage with chattering ripples. The estimated
SOC also tracks true SOC with chattering ripples. The average
value of the estimated SOC is close to the true SOC. This result
shows that the proposed sliding-mode observer can track SOC
accurately even in the presence of errors in the cell modeling.
To verify the performance of the proposed observer at the real
driving situation, the second test was performed as a sequence
of 20 urban dynamometer driving schedule (UDDS) cycles. It is
operated by series of charge-discharge pulses and 5-min rests,
and spread over the 100%-0% SOC range. It can be seen that the
SOC decreases by about 5% during each UDDS cycle. Fig. 10
shows the result of overall UDDS cycle current, true cell and
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KIM: NONLINEAR STATE OF CHARGE ESTIMATOR FOR HEV BATTERY
Fig. 9. One-cycle plot of Fig. 7.
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Fig. 11. One UDDS test cycle result for model voltage.
Fig. 12. Estimated SOC and error for UDDS cycles.
Fig. 10. UDDS test result of the model voltage and error.
model cell voltage and their voltage error. The modeling error
is less than 20 mV for 20 80% of SOC range.
For a clear view, one cycle of UDDS for model voltage is
shown in Fig. 11. The proposed sliding-mode observer was applied to the overall UDDS cycle. The resultant SOC for whole
UDDS cycle is shown in Fig. 12. The estimated SOC and error
for the whole UDDS cycles are shown in the figure. The SOC
error is bounded to 3% in all cases. The trajectory of the estimated SOC and error for the one UDDS cycle are shown in
Fig. 13 in order to show clear view of the sliding-mode observer behavior. The trajectories are always confined to the true
SOC with the chattering value. The magnitude of chattering is
dependent on the sampling time. If the high performance microprocessor is used for the controller, the chattering would be
Fig. 13. One-cycle result of the estimated SOC and error.
smaller. This chattering can be smoothed by a saturation function instead of a sign function. In another way, the average value
of the estimated SOC can be close to the true SOC. In this way,
the suggested sliding-mode observer can be directly applied to
the HEV environment with superior performance.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008
Fig. 14. Tracking performance for an incorrect initial condition.
In order to show the tracking performance of the proposed
system for an incorrect initial condition, the test was performed
to verify the robustness of proposed system. The true SOC is
0.4 and the true cell voltage is 3.75 V at the starting time. The
initial value of sliding-mode observer is set to far away from
the true value. The initial value of estimated SOC is 0.8 and the
estimated cell voltage is 4.17 V. As can be seen in the Fig. 14,
the proposed system converges to the true value about 1 min.
The convergence time would be shortened for the higher value
of charge-discharge current. However, the current integration
method fails to converge to the true value.
VI. CONCLUSION
A new method for battery modeling has been presented
to compensate for the nonlinear Li-PB characteristics. The
modeling parameters are extracted by a series of tests. The
sliding-mode observer equations are obtained from the battery model and the cell output voltage. The design method of
sliding-mode observer has been shown step by step and the convergence of observer has been proved by an equivalent control
method. The proposed method shows robust tracking performance under modeling uncertainties and noisy environments
compared to the conventional methods. The performances of
the proposed system are confirmed by the UDDS cycle test.
SOC error is confined to the acceptable level, less than 3% in
most cases which is applicable to the real environments.
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IL-Song Kim (M’04) was born in Korea in 1968. He
received the B.S. degree in electronics engineering
from Yonsei University, Seoul, Korea, in 1991, and
the M.S. and Ph.D. degrees in electrical engineering
from the Korea Advanced Institute of Science and
Technology (KAIST), Daejon, in 1994 and 2005,
respectively.
From 1994 to 1999, he was with the Satellite
Business Division of Hyundai Electronics. From
1999 to 2003, he was a Power Team Leader at the
KITSAT-4 satellite project in the Satellite Research
Center (SATREC), where he designed a solar battery charger and manufactured
battery pack. From 2005 to 2007, he joined Battery R&D, LG Chemical,
where he was involved in the development of the Battery Management System
of Hybrid Electric Vehicles with Hyundai Motors. Since 2007, he has been
Assistant Professor at Electrical Engineering at Chung-Ju National University.
His research interests include photovoltaic systems, satellite power and system
engineering, aerospace electronic equipment, and control systems.
Dr. Kim is registered in the Marquis Who’s Who in Science and Engineering
and IBC in 2000 outstanding scientists.
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